# Nabladot Analysis of Hybrid Theories in International Relations

Claudio Cioffi-Revilla

Suffice it to remember what Kant asserted; that progress in every science is measured in terms of its use of mathematics. (Gori 2004, 44)

#### 1. Introduction

International relations investigates a vast universe of political phenomena, most of it constituted by a mix of continuity and discreteness. The duration of diplomatic relations among countries, of peace between states, of international treaties, and of global international regimes in diverse policy domains are continuous variables; as are distance between capitals, speed of great power transitions, and probabilities associated with all international events. By contrast, the formal composition of a country's diplomatic organization, of alliances, governmental and nongovernmental international organizations, as well as the requisites of effective deterrence and other extant policies, are discrete variables. Time, space, territories, and emotions are generally continuous, but with discrete features such as barriers, thresholds, empty spaces, layers, and boundaries, which are discrete. This hybrid texture of continuity and discreteness—i.e., "concreteness," meaning simultaneously *continuous and discrete*—is ubiquitous, consequential, and fundamental in international relations, as reflected by theory and research across the discipline (and throughout social science in general).

Research and analysis of hybrid theories of international relations is conducted through mathematical tools from the infinitesimal calculus of Newton and Leibniz and discrete calculus developed in recent decades. Both are needed to understand real-world international phenomena that are otherwise not knowable through purely historical or narrative discussions (Gillespie 1976; Kline 1985; Cioffi 1998). Until recently, however, infinitesimal and discrete calculi

FUP Best Practice in Scholarly Publishing (DOI 10.36253/fup\_best\_practice)

Claudio Cioffi-Revilla, George Mason University, United States, ccioffi@gmu.edu, 0000-0001-6445-9433 Referee List (DOI 10.36253/fup\_referee\_list)

Claudio Cioffi-Revilla, *Nabladot Analysis of Hybrid Theories in International Relations*, © Author(s), CC BY 4.0, DOI 10.36253/978-88-5518-595-0.04, in Fulvio Attinà, Luciano Bozzo, Marco Cesa, Sonia Lucarelli (edited by), *Eirene e Atena. Studi di politica internazionale in onore di Umberto Gori*, pp. 31-53, 2022, published by Firenze University Press, ISBN 978-88-5518-595-0, DOI 10.36253/978-88-5518-595-0

have remained largely disjoint. Here we demonstrate a new unified calculus of hybrid functions with novel applications to a small, albeit representative and convincing sample of international relations theories. As a scientific system for exploration and discovery, this new analysis uncovers novel, significant, and often surprising features and properties of international phenomena that are otherwise inaccessible and, therefore, remain unknown, through earlier approaches. This investigation demonstrates results through formal mathematical and computational analysis supported by visual analytics, similar to the use of alternative "diagnostic imagery" in medical analyses or different ensembles of observational instruments in scientific research.

The next section provides examples of hybrid phenomena in international relations, followed by a section on the methodology of nabladot calculus for unified hybrid analysis. The fourth section investigates three specific cases that demonstrate hybrid analysis applied to international phenomena. Since our interest is substantive (as in all applied mathematics), we focus mainly on significant features of international political phenomena rather than purely mathematical themes. The last section provides concluding remarks.

### 2. Hybridity and hybrid functions in international relations theories

Continuity and discreteness—ontological hybridity—are present in the following international phenomena and their respective theoretical *explanans*:

Peace and other compound international events. All international events e.g, political integration, alliance formation, conditions for peace, success of international regimes, deterrence requirements, nuclear proliferation containment, and international communication—are *compound events*, in the sense of probability theory, because they are always caused by several (i.e., more than one) conjunctive events (Bittinger and Crown 1982; Bruschi 1990; Goertz and Starr 2003). Consequently, the probability of an international event is a function of some discrete number of causal conditions required for its occurrence and a continuous value of probability associated with each causal event (Wohlstetter 1968; Cioffi 1998, chs. 5–7).

Growth of great powers or empires (Taagepera's law). At the actor-level of analysis, as an empire expands from some initially small size up to its maximum size, its growth is governed by a logistic function, where time is strictly continuous but the rate of polity expansion is discrete—by chunks of territory (provinces, other administrative units, conquered territories bound by natural barriers, fortifications, and other limiting factors (Taagepera 1968; 1978; 1979).

Wright-Snyder crisis theory of war. An inter-state war *never* "comes out of the blue" but, rather, originates from a prior crisis (a crisis being a metastable phase-transition, in complexity-theoretic terminology). The crisis- and bargaining-based theory explains the onset of interstate war as the violent escalation outcome of a process with multiple outcomes, war amongst them. In this theory, the probability of war in a given epoch is a function of some discrete number of crises during the epoch and the probability of escalation to war in each crisis (Wright 1942, 1271–76, fn. 38; Deutsch 1978; Snyder and Diesing 1977, 13–5 *et passim*; Cioffi 1998, 160–63).

Loss-of-power gradient. At the relational or meso-level of analysis, continuity and discreteness over accessible time and space have mixed effects on the exercise of power ("power projection") at distances away from home base. While the rate of decay can assume positive continuous values, distance from home base is discrete (determined by military bases, supply chains, and other discrete systems and networks), so the overall function of these two quantities is hybrid (Boulding 1962; Wohlstetter 1968).

Size of war alliances (Horvath-Foster law). The frequency of war alliances in politico-military history shows a pattern that decreases with the size of the alliance. Informally, there have been many small war alliances, very few large ones, and an intermediate number in between (Horvath & Foster 1963). This is known as a discrete Yule-Simon distribution with continuous parameter and is symptomatic of complex systems and generative processes that are far from equilibrium; otherwise, the size distribution of war alliances would be normal or Gaussian (as in the height or weight of persons). NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS 5. Size of war alliances (Horvath-Foster law). The frequency of war alliances in politico-military history shows a pattern that decreases with the size of the alliance. Informally, there have been many small war alliances, very few large ones, and an intermediate number in between (Horvath & Foster 1963). This is known as a discrete Yule-Simon distribution with continuous NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS 5. Size of war alliances (Horvath-Foster law). The frequency of war alliances in politico-military history shows a pattern that decreases with the size of the alliance. Informally, there have been many small war alliances, very few large ones, and an intermediate number in between (Horvath & Foster 1963). This is known as a discrete Yule-Simon distribution with continuous NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS 5. Size of war alliances (Horvath-Foster law). The frequency of war alliances in politico-military history shows a pattern that decreases with the size of the alliance. Informally, there have been many small war alliances, very few large ones, and an intermediate number in between (Horvath & Foster 1963). This is known as a discrete Yule-Simon distribution with continuous NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS 5. Size of war alliances (Horvath-Foster law). The frequency of war alliances in politico-military history shows a pattern that decreases with the size of the alliance. Informally, there have been many small war alliances, very few large ones, and an intermediate number in between NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS 5. Size of war alliances (Horvath-Foster law). The frequency of war alliances in politico-military history shows a pattern that decreases with the size of the alliance. Informally, there have been many small war alliances, very few large ones, and an intermediate number in between (Horvath & Foster 1963). This is known as a discrete Yule-Simon distribution with continuous NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS 5. Size of war alliances (Horvath-Foster law). The frequency of war alliances in politico-military history shows a pattern that decreases with the size of the alliance. Informally, there have been many small war alliances, very few large ones, and an intermediate number in between NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS

Warfare and international systemic polarity (Midlarsky's law). At the systemic or macro-level of analysis, the annual frequency of warfare in the international system varies in proportion to the number of great powers in the system, known as polarity. However, the frequency of wars increases with marginally decreasing increments in systemic polarity, so this too is a hybrid, nonlinear relationship (Midlarsky 1974). parameter and is symptomatic of complex systems and generative processes that are far from equilibrium; otherwise, the size distribution of war alliances would be normal or Gaussian (as in the height or weight of persons). 6. Warfare and international systemic polarity (Midlarsky's law). At the systemic or macro-level of analysis, the annual frequency of warfare in the international system varies in proportion to the number of great powers in the system, known as polarity. However, the frequency of wars parameter and is symptomatic of complex systems and generative processes that are far from equilibrium; otherwise, the size distribution of war alliances would be normal or Gaussian (as in the height or weight of persons). 6. Warfare and international systemic polarity (Midlarsky's law). At the systemic or macro-level of analysis, the annual frequency of warfare in the international system varies in proportion to the number of great powers in the system, known as polarity. However, the frequency of wars increases with marginally decreasing increments in systemic polarity, so this too is a hybrid, parameter and is symptomatic of complex systems and generative processes that are far from equilibrium; otherwise, the size distribution of war alliances would be normal or Gaussian (as in the height or weight of persons). 6. Warfare and international systemic polarity (Midlarsky's law). At the systemic or macro-level of analysis, the annual frequency of warfare in the international system varies in proportion to the number of great powers in the system, known as polarity. However, the frequency of wars (Horvath & Foster 1963). This is known as a discrete Yule-Simon distribution with continuous parameter and is symptomatic of complex systems and generative processes that are far from equilibrium; otherwise, the size distribution of war alliances would be normal or Gaussian (as in the height or weight of persons). 6. Warfare and international systemic polarity (Midlarsky's law). At the systemic or macro-level of analysis, the annual frequency of warfare in the international system varies in proportion to parameter and is symptomatic of complex systems and generative processes that are far from equilibrium; otherwise, the size distribution of war alliances would be normal or Gaussian (as in the height or weight of persons). 6. Warfare and international systemic polarity (Midlarsky's law). At the systemic or macro-level of analysis, the annual frequency of warfare in the international system varies in proportion to the number of great powers in the system, known as polarity. However, the frequency of wars (Horvath & Foster 1963). This is known as a discrete Yule-Simon distribution with continuous parameter and is symptomatic of complex systems and generative processes that are far from equilibrium; otherwise, the size distribution of war alliances would be normal or Gaussian (as in the height or weight of persons). 6. Warfare and international systemic polarity (Midlarsky's law). At the systemic or macro-level of analysis, the annual frequency of warfare in the international system varies in proportion to 5. Size of war alliances (Horvath-Foster law). The frequency of war alliances in politico-military history shows a pattern that decreases with the size of the alliance. Informally, there have been many small war alliances, very few large ones, and an intermediate number in between (Horvath & Foster 1963). This is known as a discrete Yule-Simon distribution with continuous parameter and is symptomatic of complex systems and generative processes that are far from

Numerous other instances of international phenomena and corresponding theoretical explanations exist in international relations. Here we shall use instances 1, 3, and 6 to demonstrate how and why they are investigated through nabladot analysis, as described in the next section. increases with marginally decreasing increments in systemic polarity, so this too is a hybrid, nonlinear relationship (Midlarsky 1974). Numerous other instances of international phenomena and corresponding theoretical explanations exist in international relations. Here we shall use instances 1, 3, and 6 to demonstrate how and why they are investigated through nabladot analysis, as described in the next section. nonlinear relationship (Midlarsky 1974). Numerous other instances of international phenomena and corresponding theoretical explanations exist in international relations. Here we shall use instances 1, 3, and 6 to demonstrate how and why they are investigated through nabladot analysis, as described in the next section. increases with marginally decreasing increments in systemic polarity, so this too is a hybrid, nonlinear relationship (Midlarsky 1974). Numerous other instances of international phenomena and corresponding theoretical explanations exist in international relations. Here we shall use instances 1, 3, and 6 to demonstrate how and why they are investigated through nabladot analysis, as described in the next section. the number of great powers in the system, known as polarity. However, the frequency of wars increases with marginally decreasing increments in systemic polarity, so this too is a hybrid, nonlinear relationship (Midlarsky 1974). Numerous other instances of international phenomena and corresponding theoretical explanations increases with marginally decreasing increments in systemic polarity, so this too is a hybrid, nonlinear relationship (Midlarsky 1974). Numerous other instances of international phenomena and corresponding theoretical explanations exist in international relations. Here we shall use instances 1, 3, and 6 to demonstrate how and the number of great powers in the system, known as polarity. However, the frequency of wars increases with marginally decreasing increments in systemic polarity, so this too is a hybrid, nonlinear relationship (Midlarsky 1974). Numerous other instances of international phenomena and corresponding theoretical explanations equilibrium; otherwise, the size distribution of war alliances would be normal or Gaussian (as in the height or weight of persons). 6. Warfare and international systemic polarity (Midlarsky's law). At the systemic or macro-level of analysis, the annual frequency of warfare in the international system varies in proportion to the number of great powers in the system, known as polarity. However, the frequency of wars

#### 3. Mathematical methods for hybrid functions in IR 3. Mathematical methods for hybrid functions in IR 3. Mathematical methods for hybrid functions in IR 3. Mathematical methods for hybrid functions in IR why they are investigated through nabladot analysis, as described in the next section. 3. Mathematical methods for hybrid functions in IR why they are investigated through nabladot analysis, as described in the next section. increases with marginally decreasing increments in systemic polarity, so this too is a hybrid, nonlinear relationship (Midlarsky 1974).

exist in international relations. Here we shall use instances 1, 3, and 6 to demonstrate how and

and deeper understanding of each of the hybrid functions.

and differences are simple rates of change.

and differences are simple rates of change.

and differences are simple rates of change.

and differences are simple rates of change.

Second, the point elasticity and the arc elasticity of *'*, denoted by *⌘<sup>x</sup>* and *⌘y*, respectively, are calculated to understand how patterns of variation in percentage change in each independent variable compare *independent of units of measurement* (which is what elasticity operators *⌘x*(*Z*) and *⌘y*(*Z*) are designed to investigate).<sup>2</sup> Additional graphs and visual analytics are used as well to better understand the structure and effects of elasticities—and add to the theoretical landscape of each hybrid function. This second phase results in transformed standardized dimensional space without units of measurements, making all independent variables and their direct effect on the dependent variable directly comparable. These results lead to one or more dominance principles, which are law-like statements that specify which independent variable has greatest causal effect on

<sup>1</sup> We shall restrict attention to *scalar* hybrid functions, although *vector* hybrid functions also arise in nabladot analysis of scalar functions, as we shall see later in section .4., just as they do in classical analysis. Surveys of mathematical methods in political science and international relations include Cioffi (1979), Ashford et al. (1993), Moore and Siegel (2013). <sup>2</sup> Economists call this "comparative statics," a phrase we shall *not* use here because *time* can be an independent

variable of interest (e.g., as in Taagepera's law of empires) which—by definition—is not static.

and differences are simple rates of change.

and standardized effects, respectively. First, the first-order derivative and first-order difference of hybrid function are separately calculated, graphed, and examined, to understand absolute variations with respect to changes in *X* (continuous independent variable) and *Y* (discrete). This phase maintains the original units of measurement corresponding to each variable, since derivatives

constitute theoretical landscapes—complete with singularities, basins, escarpments, canyons, and other topographic features—that provide sometimes surprisingly faceted or nuanced explanations

and deeper understanding of each of the hybrid functions.

Consider a *hybrid function*, Consider a *hybrid function*, *Z* = *'*(*X, Y* ), such that *'* : (*X, Y* ) *! Z 2 R*, where *X* and *Y* are real-valued continuous and discrete *independent variables*, respectively.<sup>1</sup> Nabladot analysis of a hybrid function *'*(*X, Y* ) begins by (1) clarifying the hybrid domain of *'*, specifically its substantive subdomains *x 2 X* and *y 2 Y* along each independent variable—such subdomain always being a bounded subspace of some broader mathematical domain and then, (2) specifying each variable's unit of measurement. This initial phase of analysis normally includes various graphs of *'* for visual analysis, which are typically 2D or 3D surface graphs and contour plots of the hybrid function under investigation. As we shall see, ensembles of these interrelated graphs constitute theoretical landscapes—complete with singularities, basins, escarpments, canyons, and , such that Consider a *hybrid function*, *Z* = *'*(*X, Y* ), such that *'* : (*X, Y* ) *! Z 2 R*, where *X* and *Y* are real-valued continuous and discrete *independent variables*, respectively.<sup>1</sup> Nabladot analysis of a hybrid function *'*(*X, Y* ) begins by (1) clarifying the hybrid domain of *'*, specifically its substantive subdomains *x 2 X*and *y 2 Y* along each independent variable—such subdomain always being a bounded subspace of some broader mathematical domain and then, (2) specifying each variable's unit of measurement. This initial phase of analysis normally includes various graphs of *'* for visual analysis, which are typically 2D or 3D surface graphs and contour plots of the hybrid function under investigation. As we shall see, ensembles of these interrelated graphs constitute theoretical landscapes—complete with singularities, basins, escarpments, canyons, and , where *X* and *Y* are real-valued continuous and discrete *independent variables*, respectively.1 Nabladot analysis of a hybrid function Consider a *hybrid function*, *Z* = *'*(*X, Y* ), such that *'* : (*X, Y* ) *! Z 2 R*, where *X* and *Y* are real-valued continuous and discrete *independent variables*, respectively.<sup>1</sup> Nabladot analysis of a hybrid function *'*(*X, Y* ) begins by (1) clarifying the hybrid domain of *'*, specifically its substantive subdomains *x 2 X* and *y 2 Y* along each independent variable—such subdomain always being a bounded subspace of some broader mathematical domain and then, (2) specifying each variable's unit of measurement. This initial phase of analysis normally includes various graphs of *'* for visual analysis, which are typically 2D or 3D surface graphs and contour plots of the hybrid function under investigation. As we shall see, ensembles of these interrelated graphs constitute theoretical landscapes—complete with singularities, basins, escarpments, canyons, and begins by (1) clarifying the hybrid domain of 3. Mathematical methods for hybrid functions in IR Consider a *hybrid function*, *Z* = *'*(*X, Y* ), such that *'* : (*X, Y* ) *! Z 2 R*, where *X* and *Y* are real-valued continuous and discrete *independent variables*, respectively.<sup>1</sup> Nabladot analysis of a hybrid function *'*(*X, Y* ) begins by (1) clarifying the hybrid domain of *'*, specifically its substantive subdomains *x 2 X* and *y 2 Y* along each independent variable—such subdomain always being a bounded subspace of some broader mathematical domain and then, (2) specifying each variable's unit of measurement. This initial phase of analysis normally includes various graphs of *'* for visual analysis, which are typically 2D or 3D surface graphs and contour plots of the hybrid function under investigation. As we shall see, ensembles of these interrelated graphs , specifically its substantive subdomains Consider a *hybrid function*, *Z* = *'*(*X, Y* ), such that *'* : (*X, Y* ) *! Z 2 R*, where *X* and *Y* are real-valued continuous and discrete *independent variables*, respectively.<sup>1</sup> Nabladot analysis of a hybrid function *'*(*X, Y* ) begins by (1) clarifying the hybrid domain of *'*, specifically its substantive subdomains *x 2 X* and *y 2 Y* along each independent variable—such subdomain always being a bounded subspace of some broader mathematical domain and then, (2) specifying each variable's unit of measurement. This initial phase of analysis normally includes various graphs of *'* for visual analysis, which are typically 2D or 3D surface graphs and contour plots of the hybrid function under investigation. As we shall see, ensembles of these interrelated graphs constitute theoretical landscapes—complete with singularities, basins, escarpments, canyons, and and 3. Mathematical methods for hybrid functions in IR Consider a *hybrid function*, *Z* = *'*(*X, Y* ), such that *'* : (*X, Y* ) *! Z 2 R*, where *X* and *Y* are real-valued continuous and discrete *independent variables*, respectively.<sup>1</sup> Nabladot analysis of a hybrid function *'*(*X, Y* ) begins by (1) clarifying the hybrid domain of *'*, specifically its substantive subdomains *x 2 X* and *y 2 Y* along each independent variable—such subdomain always being a bounded subspace of some broader mathematical domain and then, (2) specifying each variable's unit of measurement. This initial phase of analysis normally includes various graphs of *'* for visual analysis, which are typically 2D or 3D surface graphs and contour plots of the hybrid function under investigation. As we shall see, ensembles of these interrelated graphs along each independent variable—such subdomain always being a bounded subspace of some broader mathematical domain and then, (2) specifying each variable's unit of measurement. This initial phase of analysis normally includes various graphs of Numerous other instances of international phenomena and corresponding theoretical explanations exist in international relations. Here we shall use instances 1, 3, and 6 to demonstrate how and why they are investigated through nabladot analysis, as described in the next section. 3. Mathematical methods for hybrid functions in IR Consider a *hybrid function*, *Z* = *'*(*X, Y* ), such that *'* : (*X, Y* ) *! Z 2 R*, where *X* and *Y* are real-valued continuous and discrete *independent variables*, respectively.<sup>1</sup> Nabladot analysis of a hybrid function *'*(*X, Y* ) begins by (1) clarifying the hybrid domain of *'*, specifically its for visual analysis, which are typically 2D or 3D

and deeper understanding of each of the hybrid functions.

other topographic features—that provide sometimes surprisingly faceted or nuanced explanations

phase maintains the original units of measurement corresponding to each variable, since derivatives

Second, the point elasticity and the arc elasticity of *'*, denoted by *⌘<sup>x</sup>* and *⌘y*, respectively, are calculated to understand how patterns of variation in percentage change in each independent variable compare *independent of units of measurement* (which is what elasticity operators *⌘x*(*Z*) and *⌘y*(*Z*) are designed to investigate).<sup>2</sup> Additional graphs and visual analytics are used as well to better understand the structure and effects of elasticities—and add to the theoretical landscape of each hybrid function. This second phase results in transformed standardized dimensional space without units of measurements, making all independent variables and their direct effect on the dependent variable directly comparable. These results lead to one or more dominance principles, which are law-like statements that specify which independent variable has greatest causal effect on

The next phase—and first properly analytical step in theoretical analysis—is to closely examine

other topographic features—that provide sometimes surprisingly faceted or nuanced explanations

and deeper understanding of each of the hybrid functions.

constitute theoretical landscapes—complete with singularities, basins, escarpments, canyons, and other topographic features—that provide sometimes surprisingly faceted or nuanced explanations

substantive subdomains *x 2 X* and *y 2 Y* along each independent variable—such subdomain always being a bounded subspace of some broader mathematical domain and then, (2) specifying each variable's unit of measurement. This initial phase of analysis normally includes various

exist in international relations. Here we shall use instances 1, 3, and 6 to demonstrate how and

phase maintains the original units of measurement corresponding to each variable, since derivatives

and differences are simple rates of change.

<sup>1</sup> We shall restrict attention to *scalar* hybrid functions, although *vector* hybrid functions also arise in nabladot analysis of scalar functions, as we shall see later in section .4., just as they do in classical analysis. Surveys of mathematical methods in political science and international relations include Cioffi (1979), Ashford et al. (1993), Moore and Siegel (2013). <sup>2</sup> Economists call this "comparative statics," a phrase we shall *not* use here because *time* can be an independent

variable of interest (e.g., as in Taagepera's law of empires) which—by definition—is not static.

variable of interest (e.g., as in Taagepera's law of empires) which—by definition—is not static.

<sup>1</sup> We shall restrict attention to *scalar* hybrid functions, although *vector* hybrid functions also arise in nabladot analysis of scalar functions, as we shall see later in section .4., just as they do in classical analysis. Surveys of mathematical methods in political science and international relations include Cioffi (1979), Ashford et al. (1993), Moore and Siegel (2013). <sup>2</sup> Economists call this "comparative statics," a phrase we shall *not* use here because *time* can be an independent

<sup>1</sup> We shall restrict attention to *scalar* hybrid functions, although *vector* hybrid functions also arise in nabladot analysis of scalar functions, as we shall see later in section .4., just as they do in classical analysis. Surveys of mathematical methods in political science and international relations include Cioffi (1979), Ashford et al. (1993), Moore and Siegel (2013). <sup>2</sup> Economists call this "comparative statics," a phrase we shall *not* use here because *time* can be an independent

variable of interest (e.g., as in Taagepera's law of empires) which—by definition—is not static.

variable of interest (e.g., as in Taagepera's law of empires) which—by definition—is not static.

<sup>1</sup> We shall restrict attention to *scalar* hybrid functions, although *vector* hybrid functions also arise in nabladot analysis of scalar functions, as we shall see later in section .4., just as they do in classical analysis. Surveys of mathematical methods in political science and international relations include Cioffi (1979), Ashford et al. (1993), Moore and Siegel (2013). <sup>2</sup> Economists call this "comparative statics," a phrase we shall *not* use here because *time* can be an independent

Second, the point elasticity and the arc elasticity of *'*, denoted by *⌘<sup>x</sup>* and *⌘y*, respectively, are calculated to understand how patterns of variation in percentage change in each independent variable compare *independent of units of measurement* (which is what elasticity operators *⌘x*(*Z*) and *⌘y*(*Z*) are designed to investigate).<sup>2</sup> Additional graphs and visual analytics are used as well to better understand the structure and effects of elasticities—and add to the theoretical landscape of each hybrid function. This second phase results in transformed standardized dimensional space without units of measurements, making all independent variables and their direct effect on the dependent variable directly comparable. These results lead to one or more dominance principles, which are law-like statements that specify which independent variable has greatest causal effect on

variations with respect to changes in *X* (continuous independent variable) and *Y* (discrete). This phase maintains the original units of measurement corresponding to each variable, since derivatives

the causal effect of each independent variable on the dependent variable of interest, which is how the emergent field (dependent variable *Z*) is generated by the hybrid domain—given that *'* maps the former (causes) onto the latter (effects). This consists of two steps that examine absolute and standardized effects, respectively. First, the first-order derivative and first-order difference of hybrid function are separately calculated, graphed, and examined, to understand absolute variations with respect to changes in *X* (continuous independent variable) and *Y* (discrete). This phase maintains the original units of measurement corresponding to each variable, since derivatives

Second, the point elasticity and the arc elasticity of *'*, denoted by *⌘<sup>x</sup>* and *⌘y*, respectively, are calculated to understand how patterns of variation in percentage change in each independent variable compare *independent of units of measurement* (which is what elasticity operators *⌘x*(*Z*) and *⌘y*(*Z*) are designed to investigate).<sup>2</sup> Additional graphs and visual analytics are used as well to better understand the structure and effects of elasticities—and add to the theoretical landscape of each hybrid function. This second phase results in transformed standardized dimensional space without units of measurements, making all independent variables and their direct effect on the dependent variable directly comparable. These results lead to one or more dominance principles, which are law-like statements that specify which independent variable has greatest causal effect on

Second, the point elasticity and the arc elasticity of *'*, denoted by *⌘<sup>x</sup>* and *⌘y*, respectively, are calculated to understand how patterns of variation in percentage change in each independent variable compare *independent of units of measurement* (which is what elasticity operators *⌘x*(*Z*) and *⌘y*(*Z*) are designed to investigate).<sup>2</sup> Additional graphs and visual analytics are used as well to better understand the structure and effects of elasticities—and add to the theoretical landscape of each hybrid function. This second phase results in transformed standardized dimensional space without units of measurements, making all independent variables and their direct effect on the dependent variable directly comparable. These results lead to one or more dominance principles, which are law-like statements that specify which independent variable has greatest causal effect on

Second, the point elasticity and the arc elasticity of *'*, denoted by *⌘<sup>x</sup>* and *⌘y*, respectively, are calculated to understand how patterns of variation in percentage change in each independent variable compare *independent of units of measurement* (which is what elasticity operators *⌘x*(*Z*) and *⌘y*(*Z*) are designed to investigate).<sup>2</sup> Additional graphs and visual analytics are used as well to better understand the structure and effects of elasticities—and add to the theoretical landscape of each hybrid function. This second phase results in transformed standardized dimensional space without units of measurements, making all independent variables and their direct effect on the dependent variable directly comparable. These results lead to one or more dominance principles, which are law-like statements that specify which independent variable has greatest causal effect on

phase maintains the original units of measurement corresponding to each variable, since derivatives

and differences are simple rates of change.

<sup>1</sup> We shall restrict attention to *scalar* hybrid functions, although *vector* hybrid functions also arise in nabladot analysis of scalar functions, as we shall see later in section .4., just as they do in classical analysis. Surveys of mathematical methods in political science and international relations include Cioffi (1979), Ashford et al. (1993), Moore and Siegel (2013). <sup>2</sup> Economists call this "comparative statics," a phrase we shall *not* use here because *time* can be an independent

variable of interest (e.g., as in Taagepera's law of empires) which—by definition—is not static.

9

other topographic features—that provide sometimes surprisingly faceted or nuanced explanations

and deeper understanding of each of the hybrid functions.

why they are investigated through nabladot analysis, as described in the next section.

other topographic features—that provide sometimes surprisingly faceted or nuanced explanations

phase maintains the original units of measurement corresponding to each variable, since derivatives

Second, the point elasticity and the arc elasticity of *'*, denoted by *⌘<sup>x</sup>* and *⌘y*, respectively, are calculated to understand how patterns of variation in percentage change in each independent variable compare *independent of units of measurement* (which is what elasticity operators *⌘x*(*Z*) and *⌘y*(*Z*) are designed to investigate).<sup>2</sup> Additional graphs and visual analytics are used as well to better understand the structure and effects of elasticities—and add to the theoretical landscape of each hybrid function. This second phase results in transformed standardized dimensional space without units of measurements, making all independent variables and their direct effect on the dependent variable directly comparable. These results lead to one or more dominance principles, which are law-like statements that specify which independent variable has greatest causal effect on

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<sup>1</sup> We shall restrict attention to *scalar* hybrid functions, although *vector* hybrid functions also arise in nabladot analysis of scalar functions, as we shall see later in section .4., just as they do in classical analysis. Surveys of mathematical methods in political science and international relations include Cioffi (1979), Ashford et al. (1993), Moore and Siegel (2013). <sup>2</sup> Economists call this "comparative statics," a phrase we shall *not* use here because *time* can be an independent

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9

variable of interest (e.g., as in Taagepera's law of empires) which—by definition—is not static.

The next phase—and first properly analytical step in theoretical analysis—is to closely examine the causal effect of each independent variable on the dependent variable of interest, which is how the emergent field (dependent variable *Z*) is generated by the hybrid domain—given that *'* maps the former (causes) onto the latter (effects). This consists of two steps that examine absolute and standardized effects, respectively. First, the first-order derivative and first-order difference the causal effect of each independent variable on the dependent variable of interest, which is how the emergent field (dependent variable *Z*) is generated by the hybrid domain—given that *'* maps the former (causes) onto the latter (effects). This consists of two steps that examine absolute and standardized effects, respectively. First, the first-order derivative and first-order difference The next phase—and first properly analytical step in theoretical analysis—is to closely examine the causal effect of each independent variable on the dependent variable of interest, which is how the emergent field (dependent variable *Z*) is generated by the hybrid domain—given that *'* maps the former (causes) onto the latter (effects). This consists of two steps that examine absolute and standardized effects, respectively. First, the first-order derivative and first-order difference and deeper understanding of each of the hybrid functions. The next phase—and first properly analytical step in theoretical analysis—is to closely examine the causal effect of each independent variable on the dependent variable of interest, which is how the emergent field (dependent variable *Z*) is generated by the hybrid domain—given that *'* maps the former (causes) onto the latter (effects). This consists of two steps that examine absolute The next phase—and first properly analytical step in theoretical analysis—is to closely examine the causal effect of each independent variable on the dependent variable of interest, which is how the emergent field (dependent variable *Z*) is generated by the hybrid domain—given that *'* maps the former (causes) onto the latter (effects). This consists of two steps that examine absolute and standardized effects, respectively. First, the first-order derivative and first-order difference and deeper understanding of each of the hybrid functions. The next phase—and first properly analytical step in theoretical analysis—is to closely examine the causal effect of each independent variable on the dependent variable of interest, which is how the emergent field (dependent variable *Z*) is generated by the hybrid domain—given that *'* maps the former (causes) onto the latter (effects). This consists of two steps that examine absolute graphs of *'* for visual analysis, which are typically 2D or 3D surface graphs and contour plots of the hybrid function under investigation. As we shall see, ensembles of these interrelated graphs constitute theoretical landscapes—complete with singularities, basins, escarpments, canyons, and other topographic features—that provide sometimes surprisingly faceted or nuanced explanations <sup>1</sup> We shall restrict attention to *scalar* hybrid functions, although *vector* hybrid functions also arise in nabladot analysis of scalar functions, as we shall see later in section 4, just as they do in classical analysis. Surveys of mathematical methods in political science and international relations include Cioffi (1979), Ashford et al. (1993), Moore and Siegel (2013).

in the height or weight of persons).

in the height or weight of persons).

NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS

5. Size of war alliances (Horvath-Foster law). The frequency of war alliances in politico-military history shows a pattern that decreases with the size of the alliance. Informally, there have been many small war alliances, very few large ones, and an intermediate number in between (Horvath & Foster 1963). This is known as a discrete Yule-Simon distribution with continuous parameter and is symptomatic of complex systems and generative processes that are far from equilibrium; otherwise, the size distribution of war alliances would be normal or Gaussian (as

6. Warfare and international systemic polarity (Midlarsky's law). At the systemic or macro-level of analysis, the annual frequency of warfare in the international system varies in proportion to the number of great powers in the system, known as polarity. However, the frequency of wars increases with marginally decreasing increments in systemic polarity, so this too is a hybrid,

Numerous other instances of international phenomena and corresponding theoretical explanations exist in international relations. Here we shall use instances 1, 3, and 6 to demonstrate how and

why they are investigated through nabladot analysis, as described in the next section.

in the height or weight of persons).

nonlinear relationship (Midlarsky 1974).

3. Mathematical methods for hybrid functions in IR

and differences are simple rates of change.

nonlinear relationship (Midlarsky 1974).

nonlinear relationship (Midlarsky 1974).

3. Mathematical methods for hybrid functions in IR

3. Mathematical methods for hybrid functions in IR

surface graphs and contour plots of the hybrid function under investigation. As we shall see, ensembles of these interrelated graphs constitute theoretical landscapes—complete with singularities, basins, escarpments, canyons, and other topographic features—that provide sometimes surprisingly faceted or nuanced explanations and deeper understanding of each of the hybrid functions. nonlinear relationship (Midlarsky 1974). Numerous other instances of international phenomena and corresponding theoretical explanations exist in international relations. Here we shall use instances 1, 3, and 6 to demonstrate how and why they are investigated through nabladot analysis, as described in the next section. (Horvath & Foster 1963). This is known as a discrete Yule-Simon distribution with continuous parameter and is symptomatic of complex systems and generative processes that are far from equilibrium; otherwise, the size distribution of war alliances would be normal or Gaussian (as in the height or weight of persons). 6. Warfare and international systemic polarity (Midlarsky's law). At the systemic or macro-level are real-valued continuous and discrete *independent variables*, respectively.<sup>1</sup> Nabladot analysis of a hybrid function *'*(*X, Y* ) begins by (1) clarifying the hybrid domain of *'*, specifically its substantive subdomains *x 2 X* and *y 2 Y* along each independent variable—such subdomain always being a bounded subspace of some broader mathematical domain and then, (2) specifying each variable's unit of measurement. This initial phase of analysis normally includes various graphs of *'* for visual analysis, which are typically 2D or 3D surface graphs and contour plots of are real-valued continuous and discrete *independent variables*, respectively.<sup>1</sup> Nabladot analysis of a hybrid function *'*(*X, Y* ) begins by (1) clarifying the hybrid domain of *'*, specifically its substantive subdomains *x 2 X* and *y 2 Y* along each independent variable—such subdomain always being a bounded subspace of some broader mathematical domain and then, (2) specifying each variable's unit of measurement. This initial phase of analysis normally includes various graphs of *'* for visual analysis, which are typically 2D or 3D surface graphs and contour plots of Consider a *hybrid function*, *Z* = *'*(*X, Y* ), such that *'* : (*X, Y* ) *! Z 2 R*, where *X* and *Y* are real-valued continuous and discrete *independent variables*, respectively.<sup>1</sup> Nabladot analysis of a hybrid function *'*(*X, Y* ) begins by (1) clarifying the hybrid domain of *'*, specifically its substantive subdomains *x 2 X* and *y 2 Y* along each independent variable—such subdomain always being a bounded subspace of some broader mathematical domain and then, (2) specifying each variable's unit of measurement. This initial phase of analysis normally includes various are real-valued continuous and discrete *independent variables*, respectively.<sup>1</sup> Nabladot analysis of a hybrid function *'*(*X, Y* ) begins by (1) clarifying the hybrid domain of *'*, specifically its substantive subdomains *x 2 X* and *y 2 Y* along each independent variable—such subdomain always being a bounded subspace of some broader mathematical domain and then, (2) specifying each variable's unit of measurement. This initial phase of analysis normally includes various graphs of *'* for visual analysis, which are typically 2D or 3D surface graphs and contour plots of

why they are investigated through nabladot analysis, as described in the next section.

why they are investigated through nabladot analysis, as described in the next section.

in the height or weight of persons).

5. Size of war alliances (Horvath-Foster law). The frequency of war alliances in politico-military history shows a pattern that decreases with the size of the alliance. Informally, there have been many small war alliances, very few large ones, and an intermediate number in between (Horvath & Foster 1963). This is known as a discrete Yule-Simon distribution with continuous parameter and is symptomatic of complex systems and generative processes that are far from equilibrium; otherwise, the size distribution of war alliances would be normal or Gaussian (as

5. Size of war alliances (Horvath-Foster law). The frequency of war alliances in politico-military history shows a pattern that decreases with the size of the alliance. Informally, there have been many small war alliances, very few large ones, and an intermediate number in between (Horvath & Foster 1963). This is known as a discrete Yule-Simon distribution with continuous parameter and is symptomatic of complex systems and generative processes that are far from equilibrium; otherwise, the size distribution of war alliances would be normal or Gaussian (as

6. Warfare and international systemic polarity (Midlarsky's law). At the systemic or macro-level of analysis, the annual frequency of warfare in the international system varies in proportion to the number of great powers in the system, known as polarity. However, the frequency of wars increases with marginally decreasing increments in systemic polarity, so this too is a hybrid,

6. Warfare and international systemic polarity (Midlarsky's law). At the systemic or macro-level of analysis, the annual frequency of warfare in the international system varies in proportion to the number of great powers in the system, known as polarity. However, the frequency of wars increases with marginally decreasing increments in systemic polarity, so this too is a hybrid,

Numerous other instances of international phenomena and corresponding theoretical explanations exist in international relations. Here we shall use instances 1, 3, and 6 to demonstrate how and

Numerous other instances of international phenomena and corresponding theoretical explanations exist in international relations. Here we shall use instances 1, 3, and 6 to demonstrate how and

NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS

6. Warfare and international systemic polarity (Midlarsky's law). At the systemic or macro-level of analysis, the annual frequency of warfare in the international system varies in proportion to the number of great powers in the system, known as polarity. However, the frequency of wars increases with marginally decreasing increments in systemic polarity, so this too is a hybrid,

Numerous other instances of international phenomena and corresponding theoretical explanations exist in international relations. Here we shall use instances 1, 3, and 6 to demonstrate how and

Consider a *hybrid function*, *Z* = *'*(*X, Y* ), such that *'* : (*X, Y* ) *! Z 2 R*, where *X* and *Y*

5. Size of war alliances (Horvath-Foster law). The frequency of war alliances in politico-military history shows a pattern that decreases with the size of the alliance. Informally, there have been many small war alliances, very few large ones, and an intermediate number in between (Horvath & Foster 1963). This is known as a discrete Yule-Simon distribution with continuous parameter and is symptomatic of complex systems and generative processes that are far from equilibrium; otherwise, the size distribution of war alliances would be normal or Gaussian (as

NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS

NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS

9

*r· ' ⌘ @x'* i + ∆*y'* j*,* (1)

2014; 2017; 2019; 2020; 2020).<sup>3</sup> Note that the resulting nabladot gradient of hybrid function *'* is a striated vector field with a first-order partial derivative component along the *x*-axis (continuous) and a first-order partial difference component along the *y*-axis (discrete), hence the striation of

where, by convention, i and j denote unit vectors along *x*- and *y*-dimensions, respectively, and *@<sup>x</sup>* and ∆*<sup>y</sup>* denote the first-order derivative and first-order difference with respect to *X* and *Y* (Cioffi

The absolute and standardized norms of the hybrid gradient *r· '* are calculated next, along with corresponding graphs for investigating the resulting vector field. Each pair of plots for a vector field and corresponding norm should use identical domains to facilitate understanding through comparative analysis. Cardinal directions (N, E, S, W) are used for simple orientation in graphs. Other hybrid operations of nabladot calculus equivalent to the divergence, curl, Laplacian, Hessian, and Jacobian are subsequently calculated to shed additional (and usually new) light on the original function *Z* = *'*(*X, Y* ) through the medium of nabladot operators, each supported by additional

The main results of nabladot analysis shed new light on fundamental, real-world, substantive properties and features of the original hybrid function under investigation, features that remain hidden or inaccessible through other forms of analysis. Each main formal expression is accompanied by an interpretation in plain English, although this is not always possible without some loss of precision or clarity. Some results can be somewhat complicated nonlinear functions that do not further simplify; we prefer them that way rather than introducing artificial approximations which may be simpler but unrealistic or unnatural objects, unlike real IR phenomena. In most cases an ensemble of images and visual analytics (Thomas and Cook 2005; Wellin 2013) of complicated

Among the most important substantive (and testable) results from nabladot analysis are the dominance principles mentioned above—they explain which independent variable has dominant effect on the dependent variable, a major theoretical (and arguably policy) question impossible to answer *ex ante*—as well as other characteristic phenomena of interest (e.g., discrete striations, inflection or "tipping" points, asymptotes and other singularities, constant or invariant subfields, and others) revealed by geometric and topological information. In addition, interesting scalar and vector fields of *'* become accessible to direct investigation through formal tools of nabladot calculus and analysis. A novel and valuable feature of this approach is that nabladot calculus provides exact results in analytical investigations where the classical infinitesimal calculus of hybrid IR functions would provide approximations with errors over the discrete domain of independent

field and corresponding norm should use identical domains to facilitate understanding through comparative analysis. Cardinal directions (N, E, S, W) are used for simple orientation in graphs. Other hybrid operations of nabladot calculus equivalent to the divergence, curl, Laplacian, Hessian, and Jacobian are subsequently calculated to shed additional (and usually new) light on the original function *Z* = *'*(*X, Y* ) through the medium of nabladot operators, each supported by additional

The main results of nabladot analysis shed new light on fundamental, real-world, substantive properties and features of the original hybrid function under investigation, features that remain hidden or inaccessible through other forms of analysis. Each main formal expression is accompanied by an interpretation in plain English, although this is not always possible without some loss of precision or clarity. Some results can be somewhat complicated nonlinear functions that do not further simplify; we prefer them that way rather than introducing artificial approximations which may be simpler but unrealistic or unnatural objects, unlike real IR phenomena. In most cases an ensemble of images and visual analytics (Thomas and Cook 2005; Wellin 2013) of complicated

Among the most important substantive (and testable) results from nabladot analysis are the dominance principles mentioned above—they explain which independent variable has dominant effect on the dependent variable, a major theoretical (and arguably policy) question impossible to answer *ex ante*—as well as other characteristic phenomena of interest (e.g., discrete striations, inflection or "tipping" points, asymptotes and other singularities, constant or invariant subfields, and others) revealed by geometric and topological information. In addition, interesting scalar and vector fields of *'* become accessible to direct investigation through formal tools of nabladot calculus and analysis. A novel and valuable feature of this approach is that nabladot calculus provides exact results in analytical investigations where the classical infinitesimal calculus of hybrid IR functions would provide approximations with errors over the discrete domain of independent

<sup>3</sup> IR scholars rarely consider the presence of vectors in international relations, other than metaphorically. This analysis demonstrates the rigorous analysis of vectors and vector fields in IR using formal methods from nabladot calculus, as in the next section. To contain notation, we shall use i and j to denote unit vectors along continuous and discrete dimensions, respectively, rather than create new unit vectors for each variable. <sup>4</sup> Use of the partial derivative with respect to *<sup>Y</sup>* (a discrete variable) instead of the partial difference—which is often used in approximations—produces a measurable error that varies in magnitude depending on the structure of *'* and values of *Y* . Measurable discrepancies between the two operators (nabladot and classical

nabla) are demonstrable but beyond the present scope due to space limitations (Cioffi 2021).

<sup>3</sup> IR scholars rarely consider the presence of vectors in international relations, other than metaphorically. This analysis demonstrates the rigorous analysis of vectors and vector fields in IR using formal methods from nabladot calculus, as in the next section. To contain notation, we shall use i and j to denote unit vectors along continuous and discrete dimensions, respectively, rather than create new unit vectors for each variable. <sup>4</sup> Use of the partial derivative with respect to *<sup>Y</sup>* (a discrete variable) instead of the partial difference—which is often used in approximations—produces a measurable error that varies in magnitude depending on the structure of *'* and values of *Y* . Measurable discrepancies between the two operators (nabladot and classical

nabla) are demonstrable but beyond the present scope due to space limitations (Cioffi 2021).

9

9

*r· ' ⌘ @x'* i + ∆*y'* j*,* (1)

9

5. Size of war alliances (Horvath-Foster law). The frequency of war alliances in politico-military history shows a pattern that decreases with the size of the alliance. Informally, there have been many small war alliances, very few large ones, and an intermediate number in between (Horvath & Foster 1963). This is known as a discrete Yule-Simon distribution with continuous parameter and is symptomatic of complex systems and generative processes that are far from equilibrium; otherwise, the size distribution of war alliances would be normal or Gaussian (as

NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS

in the height or weight of persons).

nonlinear relationship (Midlarsky 1974).

3. Mathematical methods for hybrid functions in IR

NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS

6. Warfare and international systemic polarity (Midlarsky's law). At the systemic or macro-level of analysis, the annual frequency of warfare in the international system varies in proportion to the number of great powers in the system, known as polarity. However, the frequency of wars increases with marginally decreasing increments in systemic polarity, so this too is a hybrid,

5. Size of war alliances (Horvath-Foster law). The frequency of war alliances in politico-military history shows a pattern that decreases with the size of the alliance. Informally, there have been many small war alliances, very few large ones, and an intermediate number in between

why they are investigated through nabladot analysis, as described in the next section.

without units of measurements, making all independent variables and their direct effect on the

the hybrid function under investigation. As we shall see, ensembles of these interrelated graphs

symbol), which is defined as follows:

The next phase—and first properly analytical step in theoretical analysis—is to closely examine

of hybrid function are separately calculated, graphed, and examined, to understand absolute variations with respect to changes in *X* (continuous independent variable) and *Y* (discrete). This phase maintains the original units of measurement corresponding to each variable, since derivatives

Second, the point elasticity and the arc elasticity of *'*, denoted by *⌘<sup>x</sup>* and *⌘y*, respectively, are calculated to understand how patterns of variation in percentage change in each independent variable compare *independent of units of measurement* (which is what elasticity operators *⌘x*(*Z*) and *⌘y*(*Z*) are designed to investigate).<sup>2</sup> Additional graphs and visual analytics are used as well to better understand the structure and effects of elasticities—and add to the theoretical landscape of each hybrid function. This second phase results in transformed standardized dimensional space without units of measurements, making all independent variables and their direct effect on the dependent variable directly comparable. These results lead to one or more dominance principles, which are law-like statements that specify which independent variable has greatest causal effect on

graphic analyses.

<sup>1</sup> We shall restrict attention to *scalar* hybrid functions, although *vector* hybrid functions also arise in nabladot analysis of scalar functions, as we shall see later in section .4., just as they do in classical analysis. Surveys of mathematical methods in political science and international relations include Cioffi (1979), Ashford et al. (1993), Moore and Siegel (2013). <sup>2</sup> Economists call this "comparative statics," a phrase we shall *not* use here because *time* can be an independent

functions can add significant clarity.

variable of interest (e.g., as in Taagepera's law of empires) which—by definition—is not static.

graphic analyses.

variable(s).

10

variable(s).

10

9

9

<sup>1</sup> We shall restrict attention to *scalar* hybrid functions, although *vector* hybrid functions also arise in nabladot analysis of scalar functions, as we shall see later in section .4., just as they do in classical analysis. Surveys

functions can add significant clarity.

The next phase—and first properly analytical step in theoretical analysis—is to closely examine the causal effect of each independent variable on the dependent variable of interest, which is how the emergent field (dependent variable *Z*) is generated by the hybrid domain—given that 3. Mathematical methods for hybrid functions in IR Consider a *hybrid function*, *Z* = *'*(*X, Y* ), such that *'* : (*X, Y* ) *! Z 2 R*, where *X* and *Y* are real-valued continuous and discrete *independent variables*, respectively.<sup>1</sup> Nabladot analysis of a hybrid function *'*(*X, Y* ) begins by (1) clarifying the hybrid domain of *'*, specifically its substantive subdomains *x 2 X* and *y 2 Y* along each independent variable—such subdomain always being a bounded subspace of some broader mathematical domain and then, (2) specifying each variable's unit of measurement. This initial phase of analysis normally includes various graphs of *'* for visual analysis, which are typically 2D or 3D surface graphs and contour plots of the hybrid function under investigation. As we shall see, ensembles of these interrelated graphs constitute theoretical landscapes—complete with singularities, basins, escarpments, canyons, and other topographic features—that provide sometimes surprisingly faceted or nuanced explanations and deeper understanding of each of the hybrid functions. maps the former (causes) onto the latter (effects). This consists of two steps that examine absolute and standardized effects, respectively. First, the first-order derivative and first-order difference of hybrid function are separately calculated, graphed, and examined, to understand absolute variations with respect to changes in *X* (continuous independent variable) and *Y* (discrete). This phase maintains the original units of measurement corresponding to each variable, since derivatives and differences are simple rates of change. of analysis, the annual frequency of warfare in the international system varies in proportion to the number of great powers in the system, known as polarity. However, the frequency of wars increases with marginally decreasing increments in systemic polarity, so this too is a hybrid, nonlinear relationship (Midlarsky 1974). Numerous other instances of international phenomena and corresponding theoretical explanations exist in international relations. Here we shall use instances 1, 3, and 6 to demonstrate how and why they are investigated through nabladot analysis, as described in the next section. 3. Mathematical methods for hybrid functions in IR Consider a *hybrid function*, *Z* = *'*(*X, Y* ), such that *'* : (*X, Y* ) *! Z 2 R*, where *X* and *Y* are real-valued continuous and discrete *independent variables*, respectively.<sup>1</sup> Nabladot analysis the hybrid function under investigation. As we shall see, ensembles of these interrelated graphs constitute theoretical landscapes—complete with singularities, basins, escarpments, canyons, and other topographic features—that provide sometimes surprisingly faceted or nuanced explanations and deeper understanding of each of the hybrid functions. The next phase—and first properly analytical step in theoretical analysis—is to closely examine the causal effect of each independent variable on the dependent variable of interest, which is how the emergent field (dependent variable *Z*) is generated by the hybrid domain—given that *'* maps the former (causes) onto the latter (effects). This consists of two steps that examine absolute and standardized effects, respectively. First, the first-order derivative and first-order difference of hybrid function are separately calculated, graphed, and examined, to understand absolute variations with respect to changes in *X* (continuous independent variable) and *Y* (discrete). This phase maintains the original units of measurement corresponding to each variable, since derivatives and differences are simple rates of change. the hybrid function under investigation. As we shall see, ensembles of these interrelated graphs constitute theoretical landscapes—complete with singularities, basins, escarpments, canyons, and other topographic features—that provide sometimes surprisingly faceted or nuanced explanations and deeper understanding of each of the hybrid functions. The next phase—and first properly analytical step in theoretical analysis—is to closely examine the causal effect of each independent variable on the dependent variable of interest, which is how the emergent field (dependent variable *Z*) is generated by the hybrid domain—given that *'* maps the former (causes) onto the latter (effects). This consists of two steps that examine absolute and standardized effects, respectively. First, the first-order derivative and first-order difference of hybrid function are separately calculated, graphed, and examined, to understand absolute variations with respect to changes in *X* (continuous independent variable) and *Y* (discrete). This phase maintains the original units of measurement corresponding to each variable, since derivatives and differences are simple rates of change. graphs of *'* for visual analysis, which are typically 2D or 3D surface graphs and contour plots of the hybrid function under investigation. As we shall see, ensembles of these interrelated graphs constitute theoretical landscapes—complete with singularities, basins, escarpments, canyons, and other topographic features—that provide sometimes surprisingly faceted or nuanced explanations and deeper understanding of each of the hybrid functions. The next phase—and first properly analytical step in theoretical analysis—is to closely examine the causal effect of each independent variable on the dependent variable of interest, which is how the emergent field (dependent variable *Z*) is generated by the hybrid domain—given that *'* maps the former (causes) onto the latter (effects). This consists of two steps that examine absolute and standardized effects, respectively. First, the first-order derivative and first-order difference of hybrid function are separately calculated, graphed, and examined, to understand absolute variations with respect to changes in *X* (continuous independent variable) and *Y* (discrete). This phase maintains the original units of measurement corresponding to each variable, since derivatives the hybrid function under investigation. As we shall see, ensembles of these interrelated graphs constitute theoretical landscapes—complete with singularities, basins, escarpments, canyons, and other topographic features—that provide sometimes surprisingly faceted or nuanced explanations and deeper understanding of each of the hybrid functions. The next phase—and first properly analytical step in theoretical analysis—is to closely examine the causal effect of each independent variable on the dependent variable of interest, which is how the emergent field (dependent variable *Z*) is generated by the hybrid domain—given that *'* maps the former (causes) onto the latter (effects). This consists of two steps that examine absolute and standardized effects, respectively. First, the first-order derivative and first-order difference of hybrid function are separately calculated, graphed, and examined, to understand absolute variations with respect to changes in *X* (continuous independent variable) and *Y* (discrete). This phase maintains the original units of measurement corresponding to each variable, since derivatives and differences are simple rates of change. NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS 5. Size of war alliances (Horvath-Foster law). The frequency of war alliances in politico-military history shows a pattern that decreases with the size of the alliance. Informally, there have

The next phase—and first properly analytical step in theoretical analysis—is to closely examine the causal effect of each independent variable on the dependent variable of interest, which is how the emergent field (dependent variable *Z*) is generated by the hybrid domain—given that *'* maps the former (causes) onto the latter (effects). This consists of two steps that examine absolute and standardized effects, respectively. First, the first-order derivative and first-order difference of hybrid function are separately calculated, graphed, and examined, to understand absolute variations with respect to changes in *X* (continuous independent variable) and *Y* (discrete). This phase maintains the original units of measurement corresponding to each variable, since derivatives and differences are simple rates of change. Second, the point elasticity and the arc elasticity of *'*, denoted by *⌘<sup>x</sup>* and *⌘y*, respectively, are calculated to understand how patterns of variation in percentage change in each independent variable compare *independent of units of measurement* (which is what elasticity operators *⌘x*(*Z*) and *⌘y*(*Z*) are designed to investigate).<sup>2</sup> Additional graphs and visual analytics are used as well to better understand the structure and effects of elasticities—and add to the theoretical landscape of each hybrid function. This second phase results in transformed standardized dimensional space Second, the point elasticity and the arc elasticity of of a hybrid function *'*(*X, Y* ) begins by (1) clarifying the hybrid domain of *'*, specifically its substantive subdomains *x 2 X* and *y 2 Y* along each independent variable—such subdomain always being a bounded subspace of some broader mathematical domain and then, (2) specifying each variable's unit of measurement. This initial phase of analysis normally includes various graphs of *'* for visual analysis, which are typically 2D or 3D surface graphs and contour plots of the hybrid function under investigation. As we shall see, ensembles of these interrelated graphs constitute theoretical landscapes—complete with singularities, basins, escarpments, canyons, and other topographic features—that provide sometimes surprisingly faceted or nuanced explanations and deeper understanding of each of the hybrid functions. The next phase—and first properly analytical step in theoretical analysis—is to closely examine the causal effect of each independent variable on the dependent variable of interest, which is how the emergent field (dependent variable *Z*) is generated by the hybrid domain—given that *'* maps the former (causes) onto the latter (effects). This consists of two steps that examine absolute and standardized effects, respectively. First, the first-order derivative and first-order difference of hybrid function are separately calculated, graphed, and examined, to understand absolute , denoted by Second, the point elasticity and the arc elasticity of *'*, denoted by *⌘<sup>x</sup>* and *⌘y*, respectively, are calculated to understand how patterns of variation in percentage change in each independent variable compare *independent of units of measurement* (which is what elasticity operators *⌘x*(*Z*) and *⌘y*(*Z*) are designed to investigate).<sup>2</sup> Additional graphs and visual analytics are used as well to better understand the structure and effects of elasticities—and add to the theoretical landscape of each hybrid function. This second phase results in transformed standardized dimensional space without units of measurements, making all independent variables and their direct effect on the dependent variable directly comparable. These results lead to one or more dominance principles, which are law-like statements that specify which independent variable has greatest causal effect on <sup>1</sup> We shall restrict attention to *scalar* hybrid functions, although *vector* hybrid functions also arise in nabladot analysis of scalar functions, as we shall see later in section .4., just as they do in classical analysis. Surveys of mathematical methods in political science and international relations include Cioffi (1979), Ashford et al. (1993), Moore and Siegel (2013). <sup>2</sup> Economists call this "comparative statics," a phrase we shall *not* use here because *time* can be an independent and Second, the point elasticity and the arc elasticity of *'*, denoted by *⌘<sup>x</sup>* and *⌘y*, respectively, are calculated to understand how patterns of variation in percentage change in each independent variable compare *independent of units of measurement* (which is what elasticity operators *⌘x*(*Z*) and *⌘y*(*Z*) are designed to investigate).<sup>2</sup> Additional graphs and visual analytics are used as well to better understand the structure and effects of elasticities—and add to the theoretical landscape of each hybrid function. This second phase results in transformed standardized dimensional space without units of measurements, making all independent variables and their direct effect on the dependent variable directly comparable. These results lead to one or more dominance principles, which are law-like statements that specify which independent variable has greatest causal effect on <sup>1</sup> We shall restrict attention to *scalar* hybrid functions, although *vector* hybrid functions also arise in nabladot analysis of scalar functions, as we shall see later in section .4., just as they do in classical analysis. Surveys of mathematical methods in political science and international relations include Cioffi (1979), Ashford et al. (1993), Moore and Siegel (2013). <sup>2</sup> Economists call this "comparative statics," a phrase we shall *not* use here because *time* can be an independent , respectively, are calculated to understand how patterns of variation in percentage change in each independent variable compare *independent of units of measurement* (which is what elasticity operators Second, the point elasticity and the arc elasticity of *'*, denoted by *⌘<sup>x</sup>* and *⌘y*, respectively, are calculated to understand how patterns of variation in percentage change in each independent variable compare *independent of units of measurement* (which is what elasticity operators *⌘x*(*Z*) and *⌘y*(*Z*) are designed to investigate).<sup>2</sup> Additional graphs and visual analytics are used as well to better understand the structure and effects of elasticities—and add to the theoretical landscape of each hybrid function. This second phase results in transformed standardized dimensional space without units of measurements, making all independent variables and their direct effect on the dependent variable directly comparable. These results lead to one or more dominance principles, which are law-like statements that specify which independent variable has greatest causal effect on <sup>1</sup> We shall restrict attention to *scalar* hybrid functions, although *vector* hybrid functions also arise in nabladot analysis of scalar functions, as we shall see later in section .4., just as they do in classical analysis. Surveys of mathematical methods in political science and international relations include Cioffi (1979), Ashford et al. and Second, the point elasticity and the arc elasticity of *'*, denoted by *⌘<sup>x</sup>* and *⌘y*, respectively, are calculated to understand how patterns of variation in percentage change in each independent variable compare *independent of units of measurement* (which is what elasticity operators *⌘x*(*Z*) and *⌘y*(*Z*) are designed to investigate).<sup>2</sup> Additional graphs and visual analytics are used as well to better understand the structure and effects of elasticities—and add to the theoretical landscape of each hybrid function. This second phase results in transformed standardized dimensional space without units of measurements, making all independent variables and their direct effect on the dependent variable directly comparable. These results lead to one or more dominance principles, which are law-like statements that specify which independent variable has greatest causal effect on <sup>1</sup> We shall restrict attention to *scalar* hybrid functions, although *vector* hybrid functions also arise in nabladot analysis of scalar functions, as we shall see later in section .4., just as they do in classical analysis. Surveys of mathematical methods in political science and international relations include Cioffi (1979), Ashford et al. (1993), Moore and Siegel (2013). <sup>2</sup> Economists call this "comparative statics," a phrase we shall *not* use here because *time* can be an independent are designed to investigate).2 Additional graphs and visual analytics are used as well to better understand the structure and effects of elasticities—and add to the theoretical landscape of each hybrid function. This second phase results in transformed standardized dimensional space without units of measurements, making all independent variables and their direct effect on the dependent variable directly comparable. These results lead to one or more dominance principles, which are law-like statements that specify which independent variable has greatest causal effect on the dependent variable of the hybrid function—a fundamental property not always obvious from simple inspection of the hybrid function under investigation. been many small war alliances, very few large ones, and an intermediate number in between (Horvath & Foster 1963). This is known as a discrete Yule-Simon distribution with continuous parameter and is symptomatic of complex systems and generative processes that are far from equilibrium; otherwise, the size distribution of war alliances would be normal or Gaussian (as in the height or weight of persons). 6. Warfare and international systemic polarity (Midlarsky's law). At the systemic or macro-level of analysis, the annual frequency of warfare in the international system varies in proportion to the number of great powers in the system, known as polarity. However, the frequency of wars increases with marginally decreasing increments in systemic polarity, so this too is a hybrid, nonlinear relationship (Midlarsky 1974). Numerous other instances of international phenomena and corresponding theoretical explanations exist in international relations. Here we shall use instances 1, 3, and 6 to demonstrate how and why they are investigated through nabladot analysis, as described in the next section. CLAUDIO CIOFFI-REVILLA the dependent variable of the hybrid function—a fundamental property not always obvious from CLAUDIO CIOFFI-REVILLA CLAUDIO CIOFFI-REVILLA

without units of measurements, making all independent variables and their direct effect on the dependent variable directly comparable. These results lead to one or more dominance principles, which are law-like statements that specify which independent variable has greatest causal effect on <sup>1</sup> We shall restrict attention to *scalar* hybrid functions, although *vector* hybrid functions also arise in nabladot analysis of scalar functions, as we shall see later in section .4., just as they do in classical analysis. Surveys of mathematical methods in political science and international relations include Cioffi (1979), Ashford et al. (1993), Moore and Siegel (2013). <sup>2</sup> Economists call this "comparative statics," a phrase we shall *not* use here because *time* can be an independent variable of interest (e.g., as in Taagepera's law of empires) which—by definition—is not static. variations with respect to changes in *X* (continuous independent variable) and *Y* (discrete). This phase maintains the original units of measurement corresponding to each variable, since derivatives and differences are simple rates of change. Second, the point elasticity and the arc elasticity of *'*, denoted by *⌘<sup>x</sup>* and *⌘y*, respectively, are calculated to understand how patterns of variation in percentage change in each independent variable compare *independent of units of measurement* (which is what elasticity operators *⌘x*(*Z*) and *⌘y*(*Z*) are designed to investigate).<sup>2</sup> Additional graphs and visual analytics are used as well to better understand the structure and effects of elasticities—and add to the theoretical landscape of each hybrid function. This second phase results in transformed standardized dimensional space variable of interest (e.g., as in Taagepera's law of empires) which—by definition—is not static. variable of interest (e.g., as in Taagepera's law of empires) which—by definition—is not static. (1993), Moore and Siegel (2013). <sup>2</sup> Economists call this "comparative statics," a phrase we shall *not* use here because *time* can be an independent variable of interest (e.g., as in Taagepera's law of empires) which—by definition—is not static. 9 variable of interest (e.g., as in Taagepera's law of empires) which—by definition—is not static. The analytical process thus far has focused on scalar properties of the IR hybrid function under investigation. The first nabladot operation is to calculate the hybrid gradient of *Z* to discover the magnitude *and* direction of changes in *Z* as a function of changes in *X* and *Y*. The result of applying the nabladot operator (a vector operator) to scalar hybrid function 3. Mathematical methods for hybrid functions in IR Consider a *hybrid function*, *Z* = *'*(*X, Y* ), such that *'* : (*X, Y* ) *! Z 2 R*, where *X* and *Y* are real-valued continuous and discrete *independent variables*, respectively.<sup>1</sup> Nabladot analysis of a hybrid function *'*(*X, Y* ) begins by (1) clarifying the hybrid domain of *'*, specifically its substantive subdomains *x 2 X* and *y 2 Y* along each independent variable—such subdomain always being a bounded subspace of some broader mathematical domain and then, (2) specifying each variable's unit of measurement. This initial phase of analysis normally includes various graphs of *'* for visual analysis, which are typically 2D or 3D surface graphs and contour plots of is a hybrid vector function simple inspection of the hybrid function under investigation. The analytical process thus far has focused on scalar properties of the IR hybrid function under investigation. The first nabladot operation is to calculate the hybrid gradient of *Z* to discover the magnitude *and* direction of changes in *Z* as a function of changes in *X* and *Y* . The result of applying the nabladot operator (a vector operator) to scalar hybrid function *'* is a hybrid vector function Φ = *r· '* with *x*- and *y*-components. The hybrid gradient in two dimensions is the scalar vector product calculated using the new *nabladot vector operator r·* (note the dot within the nabla symbol), which is defined as follows: with *x*- and *y*-components. The hybrid gradient in two dimensions is the scalar vector product calculated using the new *nabladot vector operator* the dependent variable of the hybrid function—a fundamental property not always obvious from simple inspection of the hybrid function under investigation. The analytical process thus far has focused on scalar properties of the IR hybrid function under investigation. The first nabladot operation is to calculate the hybrid gradient of *Z* to discover the magnitude *and* direction of changes in *Z* as a function of changes in *X* and *Y* . The result of applying the nabladot operator (a vector operator) to scalar hybrid function *'* is a hybrid vector function Φ = *r· '* with *x*- and *y*-components. The hybrid gradient in two dimensions is the scalar vector product calculated using the new *nabladot vector operator r·* (note the dot within the nabla (note the dot within the nabla symbol), which is defined as follows: the dependent variable of the hybrid function—a fundamental property not always obvious from simple inspection of the hybrid function under investigation. The analytical process thus far has focused on scalar properties of the IR hybrid function under investigation. The first nabladot operation is to calculate the hybrid gradient of *Z* to discover the magnitude *and* direction of changes in *Z* as a function of changes in *X* and *Y* . The result of applying the nabladot operator (a vector operator) to scalar hybrid function *'* is a hybrid vector function Φ = *r· '* with *x*- and *y*-components. The hybrid gradient in two dimensions is the scalar vector product calculated using the new *nabladot vector operator r·* (note the dot within the nabla symbol), which is defined as follows:

$$
\nabla \varphi \equiv \partial\_x \varphi \text{ i} + \Delta\_y \varphi \text{ j},\tag{1}
$$

and deeper understanding of each of the hybrid functions.

field and corresponding norm should use identical domains to facilitate understanding through comparative analysis. Cardinal directions (N, E, S, W) are used for simple orientation in graphs. Other hybrid operations of nabladot calculus equivalent to the divergence, curl, Laplacian, Hessian, and Jacobian are subsequently calculated to shed additional (and usually new) light on the original function *Z* = *'*(*X, Y* ) through the medium of nabladot operators, each supported by additional

where, by convention, i and j denote unit vectors along *x*- and *y*-dimensions, respectively, and *@<sup>x</sup>* and ∆*<sup>y</sup>* denote the first-order derivative and first-order difference with respect to *X* and *Y* (Cioffi 2014; 2017; 2019; 2020; 2020).<sup>3</sup> Note that the resulting nabladot gradient of hybrid function *'* is a

and differences are simple rates of change.

The main results of nabladot analysis shed new light on fundamental, real-world, substantive properties and features of the original hybrid function under investigation, features that remain hidden or inaccessible through other forms of analysis. Each main formal expression is accompanied by an interpretation in plain English, although this is not always possible without some loss of precision or clarity. Some results can be somewhat complicated nonlinear functions that do not further simplify; we prefer them that way rather than introducing artificial approximations which may be simpler but unrealistic or unnatural objects, unlike real IR phenomena. In most cases an ensemble of images and visual analytics (Thomas and Cook 2005; Wellin 2013) of complicated

Among the most important substantive (and testable) results from nabladot analysis are the dominance principles mentioned above—they explain which independent variable has dominant effect on the dependent variable, a major theoretical (and arguably policy) question impossible to answer *ex ante*—as well as other characteristic phenomena of interest (e.g., discrete striations, inflection or "tipping" points, asymptotes and other singularities, constant or invariant subfields, and others) revealed by geometric and topological information. In addition, interesting scalar and vector fields of *'* become accessible to direct investigation through formal tools of nabladot calculus and analysis. A novel and valuable feature of this approach is that nabladot calculus provides exact results in analytical investigations where the classical infinitesimal calculus of hybrid IR functions would provide approximations with errors over the discrete domain of independent

<sup>3</sup> IR scholars rarely consider the presence of vectors in international relations, other than metaphorically. This analysis demonstrates the rigorous analysis of vectors and vector fields in IR using formal methods from nabladot calculus, as in the next section. To contain notation, we shall use i and j to denote unit vectors along continuous and discrete dimensions, respectively, rather than create new unit vectors for each variable. <sup>4</sup> Use of the partial derivative with respect to *<sup>Y</sup>* (a discrete variable) instead of the partial difference—which is often used in approximations—produces a measurable error that varies in magnitude depending on the structure of *'* and values of *Y* . Measurable discrepancies between the two operators (nabladot and classical

nabla) are demonstrable but beyond the present scope due to space limitations (Cioffi 2021).

graphic analyses.

variable(s).

10

functions can add significant clarity.

of mathematical methods in political science and international relations include Cioffi (1979), Ashford et al. (1993), Moore and Siegel (2013). <sup>2</sup> Economists call this "comparative statics," a phrase we shall *not* use here because *time* can be an independent variable of interest (e.g., as in Taagepera's law of empires) which—by definition—is not static. the causal effect of each independent variable on the dependent variable of interest, which is how the emergent field (dependent variable *Z*) is generated by the hybrid domain—given that *'* maps the former (causes) onto the latter (effects). This consists of two steps that examine absolute and standardized effects, respectively. First, the first-order derivative and first-order difference the vector field's topology.<sup>4</sup> The absolute and standardized norms of the hybrid gradient *r· '* are calculated next, along with corresponding graphs for investigating the resulting vector field. Each pair of plots for a vector 2014; 2017; 2019; 2020; 2020).<sup>3</sup> Note that the resulting nabladot gradient of hybrid function *'* is a striated vector field with a first-order partial derivative component along the *x*-axis (continuous) and a first-order partial difference component along the *y*-axis (discrete), hence the striation of the vector field's topology.<sup>4</sup> striated vector field with a first-order partial derivative component along the *x*-axis (continuous) and a first-order partial difference component along the *y*-axis (discrete), hence the striation of the vector field's topology.<sup>4</sup> <sup>2</sup> Economists call this "comparative statics," a phrase we shall *not* use here because *time* can be an independent variable of interest (e.g., as in Taagepera's law of empires) which—by definition—is not static.

in the height or weight of persons).

nonlinear relationship (Midlarsky 1974).

simple inspection of the hybrid function under investigation.

where, by convention, **i** and **j** denote unit vectors along *x*- and *y*-dimensions, respectively, and symbol), which is defined as follows: *r· ' ⌘ @x'* i + ∆*y'* j*,* (1) where, by convention, i and j denote unit vectors along *x*- and *y*-dimensions, respectively, and *@<sup>x</sup>* and ∆*<sup>y</sup>* denote the first-order derivative and first-order difference with respect to *X* and *Y* (Cioffi 2014; 2017; 2019; 2020; 2020).<sup>3</sup> Note that the resulting nabladot gradient of hybrid function *'* is a striated vector field with a first-order partial derivative component along the *x*-axis (continuous) and a first-order partial difference component along the *y*-axis (discrete), hence the striation of and symbol), which is defined as follows: *r· ' ⌘ @x'* i + ∆*y'* j*,* (1) where, by convention, i and j denote unit vectors along *x*- and *y*-dimensions, respectively, and *@<sup>x</sup>* and ∆*<sup>y</sup>* denote the first-order derivative and first-order difference with respect to *X* and *Y* (Cioffi 2014; 2017; 2019; 2020; 2020).<sup>3</sup> Note that the resulting nabladot gradient of hybrid function *'* is a striated vector field with a first-order partial derivative component along the *x*-axis (continuous) and a first-order partial difference component along the *y*-axis (discrete), hence the striation of denote the first-order derivative and first-order difference with respect to *X* and *Y* (Cioffi 2014; 2017; 2019; 2020; 2020).3 Note that the resulting nabladot gradient of hybrid function 3. Mathematical methods for hybrid functions in IR Consider a *hybrid function*, *Z* = *'*(*X, Y* ), such that *'* : (*X, Y* ) *! Z 2 R*, where *X* and *Y* are real-valued continuous and discrete *independent variables*, respectively.<sup>1</sup> Nabladot analysis of a hybrid function *'*(*X, Y* ) begins by (1) clarifying the hybrid domain of *'*, specifically its substantive subdomains *x 2 X* and *y 2 Y* along each independent variable—such subdomain always being a bounded subspace of some broader mathematical domain and then, (2) specifying each variable's unit of measurement. This initial phase of analysis normally includes various is a striated vector field with a first-order partial derivative component along the *x*-axis (continuous) and a first-order partial difference component along the *y*-axis (discrete), hence the striation of the vector field's topology.4 The analytical process thus far has focused on scalar properties of the IR hybrid function under investigation. The first nabladot operation is to calculate the hybrid gradient of *Z* to discover the magnitude *and* direction of changes in *Z* as a function of changes in *X* and *Y* . The result of applying the nabladot operator (a vector operator) to scalar hybrid function *'* is a hybrid vector function Φ = *r· '* with *x*- and *y*-components. The hybrid gradient in two dimensions is the scalar vector product calculated using the new *nabladot vector operator r·* (note the dot within the nabla symbol), which is defined as follows: (Horvath & Foster 1963). This is known as a discrete Yule-Simon distribution with continuous parameter and is symptomatic of complex systems and generative processes that are far from equilibrium; otherwise, the size distribution of war alliances would be normal or Gaussian (as in the height or weight of persons). 6. Warfare and international systemic polarity (Midlarsky's law). At the systemic or macro-level of analysis, the annual frequency of warfare in the international system varies in proportion to the number of great powers in the system, known as polarity. However, the frequency of wars increases with marginally decreasing increments in systemic polarity, so this too is a hybrid,

CLAUDIO CIOFFI-REVILLA

the dependent variable of the hybrid function—a fundamental property not always obvious from

the dependent variable of the hybrid function—a fundamental property not always obvious from

The analytical process thus far has focused on scalar properties of the IR hybrid function under investigation. The first nabladot operation is to calculate the hybrid gradient of *Z* to discover the magnitude *and* direction of changes in *Z* as a function of changes in *X* and *Y* . The result of applying the nabladot operator (a vector operator) to scalar hybrid function *'* is a hybrid vector

The analytical process thus far has focused on scalar properties of the IR hybrid function under investigation. The first nabladot operation is to calculate the hybrid gradient of *Z* to discover the magnitude *and* direction of changes in *Z* as a function of changes in *X* and *Y* . The result of applying the nabladot operator (a vector operator) to scalar hybrid function *'* is a hybrid vector

simple inspection of the hybrid function under investigation.

simple inspection of the hybrid function under investigation.

CLAUDIO CIOFFI-REVILLA

CLAUDIO CIOFFI-REVILLA

the vector field's topology.<sup>4</sup>

the vector field's topology.<sup>4</sup>

graphic analyses.

graphic analyses.

variable(s).

variable(s).

10

10

The absolute and standardized norms of the hybrid gradient *r· '* are calculated next, along with corresponding graphs for investigating the resulting vector field. Each pair of plots for a vector field and corresponding norm should use identical domains to facilitate understanding through comparative analysis. Cardinal directions (N, E, S, W) are used for simple orientation in graphs. Other hybrid operations of nabladot calculus equivalent to the divergence, curl, Laplacian, Hessian, and Jacobian are subsequently calculated to shed additional (and usually new) light on the original function *Z* = *'*(*X, Y* ) through the medium of nabladot operators, each supported by additional The main results of nabladot analysis shed new light on fundamental, real-world, substantive The absolute and standardized norms of the hybrid gradient *r· '* are calculated next, along with corresponding graphs for investigating the resulting vector field. Each pair of plots for a vector field and corresponding norm should use identical domains to facilitate understanding through comparative analysis. Cardinal directions (N, E, S, W) are used for simple orientation in graphs. Other hybrid operations of nabladot calculus equivalent to the divergence, curl, Laplacian, Hessian, and Jacobian are subsequently calculated to shed additional (and usually new) light on the original function *Z* = *'*(*X, Y* ) through the medium of nabladot operators, each supported by additional The main results of nabladot analysis shed new light on fundamental, real-world, substantive graphs of *'* for visual analysis, which are typically 2D or 3D surface graphs and contour plots of the hybrid function under investigation. As we shall see, ensembles of these interrelated graphs constitute theoretical landscapes—complete with singularities, basins, escarpments, canyons, and other topographic features—that provide sometimes surprisingly faceted or nuanced explanations and deeper understanding of each of the hybrid functions. The next phase—and first properly analytical step in theoretical analysis—is to closely examine the causal effect of each independent variable on the dependent variable of interest, which is how the emergent field (dependent variable *Z*) is generated by the hybrid domain—given that *'* maps the former (causes) onto the latter (effects). This consists of two steps that examine absolute and standardized effects, respectively. First, the first-order derivative and first-order difference of hybrid function are separately calculated, graphed, and examined, to understand absolute The absolute and standardized norms of the hybrid gradient *r· ' ⌘ @x'* i + ∆*y'* j*,* (1) where, by convention, i and j denote unit vectors along *x*- and *y*-dimensions, respectively, and *@<sup>x</sup>* and ∆*<sup>y</sup>* denote the first-order derivative and first-order difference with respect to *X* and *Y* (Cioffi 2014; 2017; 2019; 2020; 2020).<sup>3</sup> Note that the resulting nabladot gradient of hybrid function *'* is a striated vector field with a first-order partial derivative component along the *x*-axis (continuous) and a first-order partial difference component along the *y*-axis (discrete), hence the striation of the vector field's topology.<sup>4</sup> The absolute and standardized norms of the hybrid gradient *r· '* are calculated next, along with corresponding graphs for investigating the resulting vector field. Each pair of plots for a vector are calculated next, along with corresponding graphs for investigating the resulting vector field. Each pair of plots for a vector field and corresponding norm should use identical domains to facilitate understanding through comparative analysis. Cardinal directions (N, E, S, W) are used for simple orientation in graphs. Other hybrid operations of nabladot calculus equivalent to the divergence, curl, Laplacian, Hessian, and Jacobian are subsequently calculated to shed additional (and usually new) light on the original function nonlinear relationship (Midlarsky 1974). Numerous other instances of international phenomena and corresponding theoretical explanations exist in international relations. Here we shall use instances 1, 3, and 6 to demonstrate how and why they are investigated through nabladot analysis, as described in the next section. 3. Mathematical methods for hybrid functions in IR Consider a *hybrid function*, *Z* = *'*(*X, Y* ), such that *'* : (*X, Y* ) *! Z 2 R*, where *X* and *Y* are real-valued continuous and discrete *independent variables*, respectively.<sup>1</sup> Nabladot analysis through the medium of nabladot operators, each supported by additional graphic analyses.

properties and features of the original hybrid function under investigation, features that remain hidden or inaccessible through other forms of analysis. Each main formal expression is accompanied by an interpretation in plain English, although this is not always possible without some loss of precision or clarity. Some results can be somewhat complicated nonlinear functions that do not further simplify; we prefer them that way rather than introducing artificial approximations which may be simpler but unrealistic or unnatural objects, unlike real IR phenomena. In most cases an ensemble of images and visual analytics (Thomas and Cook 2005; Wellin 2013) of complicated functions can add significant clarity. Among the most important substantive (and testable) results from nabladot analysis are the dominance principles mentioned above—they explain which independent variable has dominant effect on the dependent variable, a major theoretical (and arguably policy) question impossible to answer *ex ante*—as well as other characteristic phenomena of interest (e.g., discrete striations, inflection or "tipping" points, asymptotes and other singularities, constant or invariant subfields, properties and features of the original hybrid function under investigation, features that remain hidden or inaccessible through other forms of analysis. Each main formal expression is accompanied by an interpretation in plain English, although this is not always possible without some loss of precision or clarity. Some results can be somewhat complicated nonlinear functions that do not further simplify; we prefer them that way rather than introducing artificial approximations which may be simpler but unrealistic or unnatural objects, unlike real IR phenomena. In most cases an ensemble of images and visual analytics (Thomas and Cook 2005; Wellin 2013) of complicated functions can add significant clarity. Among the most important substantive (and testable) results from nabladot analysis are the dominance principles mentioned above—they explain which independent variable has dominant effect on the dependent variable, a major theoretical (and arguably policy) question impossible to answer *ex ante*—as well as other characteristic phenomena of interest (e.g., discrete striations, inflection or "tipping" points, asymptotes and other singularities, constant or invariant subfields, variations with respect to changes in *X* (continuous independent variable) and *Y* (discrete). This phase maintains the original units of measurement corresponding to each variable, since derivatives and differences are simple rates of change. Second, the point elasticity and the arc elasticity of *'*, denoted by *⌘<sup>x</sup>* and *⌘y*, respectively, are calculated to understand how patterns of variation in percentage change in each independent variable compare *independent of units of measurement* (which is what elasticity operators *⌘x*(*Z*) and *⌘y*(*Z*) are designed to investigate).<sup>2</sup> Additional graphs and visual analytics are used as well to better understand the structure and effects of elasticities—and add to the theoretical landscape of each hybrid function. This second phase results in transformed standardized dimensional space without units of measurements, making all independent variables and their direct effect on the dependent variable directly comparable. These results lead to one or more dominance principles, which are law-like statements that specify which independent variable has greatest causal effect on field and corresponding norm should use identical domains to facilitate understanding through comparative analysis. Cardinal directions (N, E, S, W) are used for simple orientation in graphs. Other hybrid operations of nabladot calculus equivalent to the divergence, curl, Laplacian, Hessian, and Jacobian are subsequently calculated to shed additional (and usually new) light on the original function *Z* = *'*(*X, Y* ) through the medium of nabladot operators, each supported by additional graphic analyses. The main results of nabladot analysis shed new light on fundamental, real-world, substantive properties and features of the original hybrid function under investigation, features that remain hidden or inaccessible through other forms of analysis. Each main formal expression is accompanied by an interpretation in plain English, although this is not always possible without some loss of precision or clarity. Some results can be somewhat complicated nonlinear functions that do not further simplify; we prefer them that way rather than introducing artificial approximations which may be simpler but unrealistic or unnatural objects, unlike real IR phenomena. In most cases an of a hybrid function *'*(*X, Y* ) begins by (1) clarifying the hybrid domain of *'*, specifically its substantive subdomains *x 2 X* and *y 2 Y* along each independent variable—such subdomain always being a bounded subspace of some broader mathematical domain and then, (2) specifying each variable's unit of measurement. This initial phase of analysis normally includes various graphs of *'* for visual analysis, which are typically 2D or 3D surface graphs and contour plots of the hybrid function under investigation. As we shall see, ensembles of these interrelated graphs constitute theoretical landscapes—complete with singularities, basins, escarpments, canyons, and other topographic features—that provide sometimes surprisingly faceted or nuanced explanations and deeper understanding of each of the hybrid functions. The next phase—and first properly analytical step in theoretical analysis—is to closely examine the causal effect of each independent variable on the dependent variable of interest, which is how the emergent field (dependent variable *Z*) is generated by the hybrid domain—given that *'* maps the former (causes) onto the latter (effects). This consists of two steps that examine absolute The main results of nabladot analysis shed new light on fundamental, realworld, substantive properties and features of the original hybrid function under investigation, features that remain hidden or inaccessible through other forms of analysis. Each main formal expression is accompanied by an interpretation in plain English, although this is not always possible without some loss of precision or clarity. Some results can be somewhat complicated nonlinear functions that do not further simplify; we prefer them that way rather than introducing artificial approximations which may be simpler but unrealistic or unnatural objects, unlike real IR phenomena. In most cases an ensemble of images and visual analytics (Thomas and Cook 2005; Wellin 2013) of complicated functions can add significant clarity. NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS 5. Size of war alliances (Horvath-Foster law). The frequency of war alliances in politico-military

and others) revealed by geometric and topological information. In addition, interesting scalar and vector fields of *'* become accessible to direct investigation through formal tools of nabladot calculus and analysis. A novel and valuable feature of this approach is that nabladot calculus provides exact results in analytical investigations where the classical infinitesimal calculus of hybrid IR functions would provide approximations with errors over the discrete domain of independent and others) revealed by geometric and topological information. In addition, interesting scalar and vector fields of *'* become accessible to direct investigation through formal tools of nabladot calculus and analysis. A novel and valuable feature of this approach is that nabladot calculus provides exact results in analytical investigations where the classical infinitesimal calculus of hybrid IR functions would provide approximations with errors over the discrete domain of independent <sup>1</sup> We shall restrict attention to *scalar* hybrid functions, although *vector* hybrid functions also arise in nabladot analysis of scalar functions, as we shall see later in section .4., just as they do in classical analysis. Surveys of mathematical methods in political science and international relations include Cioffi (1979), Ashford et al. (1993), Moore and Siegel (2013). <sup>2</sup> Economists call this "comparative statics," a phrase we shall *not* use here because *time* can be an independent variable of interest (e.g., as in Taagepera's law of empires) which—by definition—is not static. ensemble of images and visual analytics (Thomas and Cook 2005; Wellin 2013) of complicated functions can add significant clarity. Among the most important substantive (and testable) results from nabladot analysis are the dominance principles mentioned above—they explain which independent variable has dominant effect on the dependent variable, a major theoretical (and arguably policy) question impossible to answer *ex ante*—as well as other characteristic phenomena of interest (e.g., discrete striations, inflection or "tipping" points, asymptotes and other singularities, constant or invariant subfields, and others) revealed by geometric and topological information. In addition, interesting scalar and standardized effects, respectively. First, the first-order derivative and first-order difference of hybrid function are separately calculated, graphed, and examined, to understand absolute variations with respect to changes in *X* (continuous independent variable) and *Y* (discrete). This phase maintains the original units of measurement corresponding to each variable, since derivatives and differences are simple rates of change. Second, the point elasticity and the arc elasticity of *'*, denoted by *⌘<sup>x</sup>* and *⌘y*, respectively, are calculated to understand how patterns of variation in percentage change in each independent variable compare *independent of units of measurement* (which is what elasticity operators *⌘x*(*Z*) Among the most important substantive (and testable) results from nabladot analysis are the dominance principles mentioned above—they explain which independent variable has dominant effect on the dependent variable, a major theoretical (and arguably policy) question impossible to answer *ex ante*—as well as other characteristic phenomena of interest (e.g., discrete striations, inflection or "tipping" points, asymptotes and other singularities, constant or invariant subfields, and others) revealed by geometric and topological inforhistory shows a pattern that decreases with the size of the alliance. Informally, there have been many small war alliances, very few large ones, and an intermediate number in between (Horvath & Foster 1963). This is known as a discrete Yule-Simon distribution with continuous parameter and is symptomatic of complex systems and generative processes that are far from equilibrium; otherwise, the size distribution of war alliances would be normal or Gaussian (as in the height or weight of persons). 6. Warfare and international systemic polarity (Midlarsky's law). At the systemic or macro-level of analysis, the annual frequency of warfare in the international system varies in proportion to the number of great powers in the system, known as polarity. However, the frequency of wars

nabladot calculus, as in the next section. To contain notation, we shall use i and j to denote unit vectors along continuous and discrete dimensions, respectively, rather than create new unit vectors for each variable. <sup>4</sup> Use of the partial derivative with respect to *<sup>Y</sup>* (a discrete variable) instead of the partial difference—which is often used in approximations—produces a measurable error that varies in magnitude depending on the structure of *'* and values of *Y* . Measurable discrepancies between the two operators (nabladot and classical nabla) are demonstrable but beyond the present scope due to space limitations (Cioffi 2021). nabladot calculus, as in the next section. To contain notation, we shall use i and j to denote unit vectors along continuous and discrete dimensions, respectively, rather than create new unit vectors for each variable. <sup>4</sup> Use of the partial derivative with respect to *<sup>Y</sup>* (a discrete variable) instead of the partial difference—which is often used in approximations—produces a measurable error that varies in magnitude depending on the structure of *'* and values of *Y* . Measurable discrepancies between the two operators (nabladot and classical nabla) are demonstrable but beyond the present scope due to space limitations (Cioffi 2021). provides exact results in analytical investigations where the classical infinitesimal calculus of hybrid IR functions would provide approximations with errors over the discrete domain of independent variable(s). each hybrid function. This second phase results in transformed standardized dimensional space without units of measurements, making all independent variables and their direct effect on the dependent variable directly comparable. These results lead to one or more dominance principles, which are law-like statements that specify which independent variable has greatest causal effect on <sup>3</sup> IR scholars rarely consider the presence of vectors in international relations, other than metaphorically. This analysis demonstrates the rigorous analysis of vectors and vector fields in IR using formal methods from nabladot calculus, as in the next section. To contain notation, we shall use **i** and **j** to denote unit vectors along continuous and discrete dimensions, respectively, rather than create new unit vectors for each variable. Numerous other instances of international phenomena and corresponding theoretical explanations exist in international relations. Here we shall use instances 1, 3, and 6 to demonstrate how and why they are investigated through nabladot analysis, as described in the next section. 3. Mathematical methods for hybrid functions in IR

nonlinear relationship (Midlarsky 1974).

and deeper understanding of each of the hybrid functions.

<sup>3</sup> IR scholars rarely consider the presence of vectors in international relations, other than metaphorically. This analysis demonstrates the rigorous analysis of vectors and vector fields in IR using formal methods from

<sup>3</sup> IR scholars rarely consider the presence of vectors in international relations, other than metaphorically. This analysis demonstrates the rigorous analysis of vectors and vector fields in IR using formal methods from

> <sup>3</sup> IR scholars rarely consider the presence of vectors in international relations, other than metaphorically. This analysis demonstrates the rigorous analysis of vectors and vector fields in IR using formal methods from nabladot calculus, as in the next section. To contain notation, we shall use i and j to denote unit vectors along continuous and discrete dimensions, respectively, rather than create new unit vectors for each variable. <sup>4</sup> Use of the partial derivative with respect to *<sup>Y</sup>* (a discrete variable) instead of the partial difference—which is often used in approximations—produces a measurable error that varies in magnitude depending on the structure of *'* and values of *Y* . Measurable discrepancies between the two operators (nabladot and classical <sup>1</sup> We shall restrict attention to *scalar* hybrid functions, although *vector* hybrid functions also arise in nabladot analysis of scalar functions, as we shall see later in section .4., just as they do in classical analysis. Surveys of mathematical methods in political science and international relations include Cioffi (1979), Ashford et al. (1993), Moore and Siegel (2013). <sup>2</sup> Economists call this "comparative statics," a phrase we shall *not* use here because *time* can be an independent variable of interest (e.g., as in Taagepera's law of empires) which—by definition—is not static. <sup>4</sup> Use of the partial derivative with respect to *Y* (a discrete variable) instead of the partial difference—which is often used in approximations—produces a measurable error that varies in magnitude depending on the structure of Consider a *hybrid function*, *Z* = *'*(*X, Y* ), such that *'* : (*X, Y* ) *! Z 2 R*, where *X* and *Y* are real-valued continuous and discrete *independent variables*, respectively.<sup>1</sup> Nabladot analysis of a hybrid function *'*(*X, Y* ) begins by (1) clarifying the hybrid domain of *'*, specifically its substantive subdomains *x 2 X* and *y 2 Y* along each independent variable—such subdomain always being a bounded subspace of some broader mathematical domain and then, (2) specifying each variable's unit of measurement. This initial phase of analysis normally includes various and values of *Y*. Measurable discrepancies between the two operators (nabladot and classical nabla) are demonstrable but beyond the present scope due to space limitations (Cioffi 2021).

> > and differences are simple rates of change.

10

and vector fields of *'* become accessible to direct investigation through formal tools of nabladot calculus and analysis. A novel and valuable feature of this approach is that nabladot calculus

and *⌘y*(*Z*) are designed to investigate).<sup>2</sup> Additional graphs and visual analytics are used as well to better understand the structure and effects of elasticities—and add to the theoretical landscape of

increases with marginally decreasing increments in systemic polarity, so this too is a hybrid,

constitute theoretical landscapes—complete with singularities, basins, escarpments, canyons, and other topographic features—that provide sometimes surprisingly faceted or nuanced explanations

The next phase—and first properly analytical step in theoretical analysis—is to closely examine the causal effect of each independent variable on the dependent variable of interest, which is how the emergent field (dependent variable *Z*) is generated by the hybrid domain—given that *'* maps the former (causes) onto the latter (effects). This consists of two steps that examine absolute and standardized effects, respectively. First, the first-order derivative and first-order difference of hybrid function are separately calculated, graphed, and examined, to understand absolute variations with respect to changes in *X* (continuous independent variable) and *Y* (discrete). This phase maintains the original units of measurement corresponding to each variable, since derivatives

Second, the point elasticity and the arc elasticity of *'*, denoted by *⌘<sup>x</sup>* and *⌘y*, respectively, are calculated to understand how patterns of variation in percentage change in each independent variable compare *independent of units of measurement* (which is what elasticity operators *⌘x*(*Z*) and *⌘y*(*Z*) are designed to investigate).<sup>2</sup> Additional graphs and visual analytics are used as well to better understand the structure and effects of elasticities—and add to the theoretical landscape of each hybrid function. This second phase results in transformed standardized dimensional space without units of measurements, making all independent variables and their direct effect on the dependent variable directly comparable. These results lead to one or more dominance principles, which are law-like statements that specify which independent variable has greatest causal effect on <sup>1</sup> We shall restrict attention to *scalar* hybrid functions, although *vector* hybrid functions also arise in nabladot analysis of scalar functions, as we shall see later in section .4., just as they do in classical analysis. Surveys of mathematical methods in political science and international relations include Cioffi (1979), Ashford et al. (1993), Moore and Siegel (2013). <sup>2</sup> Economists call this "comparative statics," a phrase we shall *not* use here because *time* can be an independent variable of interest (e.g., as in Taagepera's law of empires) which—by definition—is not static.

9

9

9

NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS

5. Size of war alliances (Horvath-Foster law). The frequency of war alliances in politico-military history shows a pattern that decreases with the size of the alliance. Informally, there have been many small war alliances, very few large ones, and an intermediate number in between (Horvath & Foster 1963). This is known as a discrete Yule-Simon distribution with continuous parameter and is symptomatic of complex systems and generative processes that are far from equilibrium; otherwise, the size distribution of war alliances would be normal or Gaussian (as

6. Warfare and international systemic polarity (Midlarsky's law). At the systemic or macro-level of analysis, the annual frequency of warfare in the international system varies in proportion to the number of great powers in the system, known as polarity. However, the frequency of wars increases with marginally decreasing increments in systemic polarity, so this too is a hybrid,

Numerous other instances of international phenomena and corresponding theoretical explanations exist in international relations. Here we shall use instances 1, 3, and 6 to demonstrate how and

5. Size of war alliances (Horvath-Foster law). The frequency of war alliances in politico-military

been many small war alliances, very few large ones, and an intermediate number in between

NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS

following expression:

mation. In addition, interesting scalar and vector fields of are real-valued continuous and discrete *independent variables*, respectively.<sup>1</sup> Nabladot analysis of a hybrid function *'*(*X, Y* ) begins by (1) clarifying the hybrid domain of *'*, specifically its substantive subdomains *x 2 X* and *y 2 Y* along each independent variable—such subdomain always being a bounded subspace of some broader mathematical domain and then, (2) specifying each variable's unit of measurement. This initial phase of analysis normally includes various graphs of *'* for visual analysis, which are typically 2D or 3D surface graphs and contour plots of the hybrid function under investigation. As we shall see, ensembles of these interrelated graphs constitute theoretical landscapes—complete with singularities, basins, escarpments, canyons, and become accessible to direct investigation through formal tools of nabladot calculus and analysis. A novel and valuable feature of this approach is that nabladot calculus provides exact results in analytical investigations where the classical infinitesimal calculus of hybrid IR functions would provide approximations with errors over the discrete domain of independent variable(s).

in the height or weight of persons).

nonlinear relationship (Midlarsky 1974).

3. Mathematical methods for hybrid functions in IR

NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS

and 6 in section .2.—where hybrid functions play a central role in describing and explaining political

Here we shall investigate three illustrative cases of IR theories and research areas—numbered 1, 3,

Here we shall investigate three illustrative cases of IR theories and research areas—numbered 1, 3,

in all domains of international relations theory. For example, consider the event defined by the

phenomena. The scientific purpose is to deepen our understanding and provide foundations for

NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS

5. Size of war alliances (Horvath-Foster law). The frequency of war alliances in politico-military history shows a pattern that decreases with the size of the alliance. Informally, there have been many small war alliances, very few large ones, and an intermediate number in between (Horvath & Foster 1963). This is known as a discrete Yule-Simon distribution with continuous parameter and is symptomatic of complex systems and generative processes that are far from equilibrium; otherwise, the size distribution of war alliances would be normal or Gaussian (as

6. Warfare and international systemic polarity (Midlarsky's law). At the systemic or macro-level of analysis, the annual frequency of warfare in the international system varies in proportion to the number of great powers in the system, known as polarity. However, the frequency of wars increases with marginally decreasing increments in systemic polarity, so this too is a hybrid,

Numerous other instances of international phenomena and corresponding theoretical explanations exist in international relations. Here we shall use instances 1, 3, and 6 to demonstrate how and

Consider a *hybrid function*, *Z* = *'*(*X, Y* ), such that *'* : (*X, Y* ) *! Z 2 R*, where *X* and *Y*

other topographic features—that provide sometimes surprisingly faceted or nuanced explanations

phase maintains the original units of measurement corresponding to each variable, since derivatives

and 6 in section .2.—where hybrid functions play a central role in describing and explaining political

without units of measurements, making all independent variables and their direct effect on the

necessary conditions specific to the event. Such causal necessity is universal—a fundamental axiom

and 6 in section .2.—where hybrid functions play a central role in describing and explaining political

9

why they are investigated through nabladot analysis, as described in the next section.

#### and deeper understanding of each of the hybrid functions. The next phase—and first properly analytical step in theoretical analysis—is to closely examine 4. Applications to areas of international relations

the causal effect of each independent variable on the dependent variable of interest, which is how the emergent field (dependent variable *Z*) is generated by the hybrid domain—given that *'* maps the former (causes) onto the latter (effects). This consists of two steps that examine absolute and standardized effects, respectively. First, the first-order derivative and first-order difference of hybrid function are separately calculated, graphed, and examined, to understand absolute variations with respect to changes in *X* (continuous independent variable) and *Y* (discrete). This Here we shall investigate three illustrative cases of IR theories and research areas—numbered 1, 3, and 6 in section 2—where hybrid functions play a central role in describing and explaining political phenomena. The scientific purpose is to deepen our understanding and provide foundations for more advanced analysis. Each "case study" follows the analytical procedure just outlined in section 3. 4. Applications to areas of international relations Here we shall investigate three illustrative cases of IR theories and research areas—numbered 1, 3, and 6 in section .2.—where hybrid functions play a central role in describing and explaining political phenomena. The scientific purpose is to deepen our understanding and provide foundations for more advanced analysis. Each "case study" follows the analytical procedure just outlined in section NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS 4. Applications to areas of international relations Here we shall investigate three illustrative cases of IR theories and research areas—numbered 1, 3, and 6 in section .2.—where hybrid functions play a central role in describing and explaining political NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS 4. Applications to areas of international relations Here we shall investigate three illustrative cases of IR theories and research areas—numbered 1, 3, NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS 4. Applications to areas of international relations Here we shall investigate three illustrative cases of IR theories and research areas—numbered 1, 3, NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS 4. Applications to areas of international relations Here we shall investigate three illustrative cases of IR theories and research areas—numbered 1, 3, NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS 4. Applications to areas of international relations NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS 4. Applications to areas of international relations NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS 4. Applications to areas of international relations

#### and differences are simple rates of change. Second, the point elasticity and the arc elasticity of *'*, denoted by *⌘<sup>x</sup>* and *⌘y*, respectively, 4.1 Case 1: Peace and other international events 4.1 Case 1: Peace and other international events more advanced analysis. Each "case study" follows the analytical procedure just outlined in section .3.. phenomena. The scientific purpose is to deepen our understanding and provide foundations for more advanced analysis. Each "case study" follows the analytical procedure just outlined in section and 6 in section .2.—where hybrid functions play a central role in describing and explaining political phenomena. The scientific purpose is to deepen our understanding and provide foundations for phenomena. The scientific purpose is to deepen our understanding and provide foundations for more advanced analysis. Each "case study" follows the analytical procedure just outlined in section phenomena. The scientific purpose is to deepen our understanding and provide foundations for more advanced analysis. Each "case study" follows the analytical procedure just outlined in section Here we shall investigate three illustrative cases of IR theories and research areas—numbered 1, 3, 4. Applications to areas of international relations Here we shall investigate three illustrative cases of IR theories and research areas—numbered 1, 3, and 6 in section .2.—where hybrid functions play a central role in describing and explaining political phenomena. The scientific purpose is to deepen our understanding and provide foundations for and 6 in section .2.—where hybrid functions play a central role in describing and explaining political phenomena. The scientific purpose is to deepen our understanding and provide foundations for 4. Applications to areas of international relations and 6 in section .2.—where hybrid functions play a central role in describing and explaining political phenomena. The scientific purpose is to deepen our understanding and provide foundations for

Here we shall investigate three illustrative cases of IR theories and research areas—numbered 1, 3,

following expression:

in all domains of international relations theory. For example, consider the event defined by the

2. Each state may prefer to negotiate over colliding interests, before escalating to war.

All international events in the real world are "compound" because they are always produced by

are calculated to understand how patterns of variation in percentage change in each independent variable compare *independent of units of measurement* (which is what elasticity operators *⌘x*(*Z*) and *⌘y*(*Z*) are designed to investigate).<sup>2</sup> Additional graphs and visual analytics are used as well to better understand the structure and effects of elasticities—and add to the theoretical landscape of each hybrid function. This second phase results in transformed standardized dimensional space All international events in the real world are "compound" because they are always produced by necessary conditions specific to the event. Such causal necessity is universal—a fundamental axiom in all domains of international relations theory. For example, consider the event defined by the following expression: All international events in the real world are "compound" because they are always produced by necessary conditions specific to the event. Such causal necessity is universal—a fundamental axiom in all domains of international relations theory. For example, consider the event defined by the following expression: 4.1 Case 1: Peace and other international events All international events in the real world are "compound" because they are always produced by necessary conditions specific to the event. Such causal necessity is universal—a fundamental axiom NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS 4. Applications to areas of international relations .3.. 4.1 Case 1: Peace and other international events All international events in the real world are "compound" because they are always produced by more advanced analysis. Each "case study" follows the analytical procedure just outlined in section .3.. 4.1 Case 1: Peace and other international events All international events in the real world are "compound" because they are always produced by .3.. 4.1 Case 1: Peace and other international events All international events in the real world are "compound" because they are always produced by and 6 in section .2.—where hybrid functions play a central role in describing and explaining political phenomena. The scientific purpose is to deepen our understanding and provide foundations for more advanced analysis. Each "case study" follows the analytical procedure just outlined in section and 6 in section .2.—where hybrid functions play a central role in describing and explaining political phenomena. The scientific purpose is to deepen our understanding and provide foundations for more advanced analysis. Each "case study" follows the analytical procedure just outlined in section .3.. more advanced analysis. Each "case study" follows the analytical procedure just outlined in section 4.1 Case 1: Peace and other international events more advanced analysis. Each "case study" follows the analytical procedure just outlined in section .3.. 4.1 Case 1: Peace and other international events Here we shall investigate three illustrative cases of IR theories and research areas—numbered 1, 3, and 6 in section .2.—where hybrid functions play a central role in describing and explaining political phenomena. The scientific purpose is to deepen our understanding and provide foundations for more advanced analysis. Each "case study" follows the analytical procedure just outlined in section .3.. more advanced analysis. Each "case study" follows the analytical procedure just outlined in section .3.. 4.1 Case 1: Peace and other international events

dependent variable directly comparable. These results lead to one or more dominance principles, which are law-like statements that specify which independent variable has greatest causal effect on **S** *⌘* "a state of stable peace exists between two countries." (2) "a state of stable peace exists between two countries." (2) in all domains of international relations theory. For example, consider the event defined by the following expression: Here we shall investigate three illustrative cases of IR theories and research areas—numbered 1, 3, in all domains of international relations theory. For example, consider the event defined by the All international events in the real world are "compound" because they are always produced by necessary conditions specific to the event. Such causal necessity is universal—a fundamental axiom necessary conditions specific to the event. Such causal necessity is universal—a fundamental axiom in all domains of international relations theory. For example, consider the event defined by the in all domains of international relations theory. For example, consider the event defined by the 4.1 Case 1: Peace and other international events 4.1 Case 1: Peace and other international events All international events in the real world are "compound" because they are always produced by necessary conditions specific to the event. Such causal necessity is universal—a fundamental axiom All international events in the real world are "compound" because they are always produced by necessary conditions specific to the event. Such causal necessity is universal—a fundamental axiom 4.1 Case 1: Peace and other international events All international events in the real world are "compound" because they are always produced by necessary conditions specific to the event. Such causal necessity is universal—a fundamental axiom

following expression:

necessary conditions specific to the event. Such causal necessity is universal—a fundamental axiom

All international events in the real world are "compound" because they are always produced by

and 6 in section .2.—where hybrid functions play a central role in describing and explaining political

<sup>1</sup> We shall restrict attention to *scalar* hybrid functions, although *vector* hybrid functions also arise in nabladot analysis of scalar functions, as we shall see later in section .4., just as they do in classical analysis. Surveys This is a compound event because **S** requires the following set of causally necessary conditions, each of which constitutes an event by itself: This is a compound event because **S** *⌘* "a state of stable peace exists between two countries." (2) requires the following set of causally necessary conditions, each of which constitutes an event by itself: phenomena. The scientific purpose is to deepen our understanding and provide foundations for more advanced analysis. Each "case study" follows the analytical procedure just outlined in section **S** *⌘* "a state of stable peace exists between two countries." (2) **S** *⌘* "a state of stable peace exists between two countries." (2) **S** *⌘* "a state of stable peace exists between two countries." (2) necessary conditions specific to the event. Such causal necessity is universal—a fundamental axiom in all domains of international relations theory. For example, consider the event defined by the necessary conditions specific to the event. Such causal necessity is universal—a fundamental axiom in all domains of international relations theory. For example, consider the event defined by the following expression: All international events in the real world are "compound" because they are always produced by necessary conditions specific to the event. Such causal necessity is universal—a fundamental axiom in all domains of international relations theory. For example, consider the event defined by the following expression:


Specifically, every international event **E** is produced by causal *conjunction* (operator *^*) of necessary events—i.e., set-theoretic *intersection* (operator *\*) or Boolean logic *product* (operator AND). Let *{***X***i}<sup>n</sup> <sup>i</sup>*=1 denote a set of *N* necessary events that produce **E**. Causal production of **E** is The universal existence of such necessary conditions makes **S** a compound event, by definition. Specifically, every international event **E** is produced by causal *conjunction* (operator *^*) of The universal existence of such necessary conditions makes **S** *⌘* "a state of stable peace exists between two countries." (2) a compound event, by definition. 3. When states do negotiate, they may—depending on conditions—find a nonviolent resolution. The universal existence of such necessary conditions makes **S** a compound event, by definition. 3. When states do negotiate, they may—depending on conditions—find a nonviolent resolution. 3. When states do negotiate, they may—depending on conditions—find a nonviolent resolution. The universal existence of such necessary conditions makes **S** a compound event, by definition. 3. When states do negotiate, they may—depending on conditions—find a nonviolent resolution. The universal existence of such necessary conditions makes **S** a compound event, by definition. acting completely oblivious or independently of extant foreign interests. 2. Each state may prefer to negotiate over colliding interests, before escalating to war. acting completely oblivious or independently of extant foreign interests. 2. Each state may prefer to negotiate over colliding interests, before escalating to war. 3. When states do negotiate, they may—depending on conditions—find a nonviolent resolution. 3. When states do negotiate, they may—depending on conditions—find a nonviolent resolution. 1. Neither state will pursue issues deemed as highly threatening to the other, as opposed to acting completely oblivious or independently of extant foreign interests. 2. Each state may prefer to negotiate over colliding interests, before escalating to war. 3. When states do negotiate, they may—depending on conditions—find a nonviolent resolution.

specified by an *event function*, **<sup>E</sup>** : *{***X***i} !* **E**, which maps necessary events in *{***X***i}* onto **E** using causal conjunctions. The number of events in a compound event (its "size" or "conjunctivity"), is called *cardinality*, *|***E***|* = *{*1*,* 2*,* 3*, ··· , n} ⇢ N* , which is always a natural number (positive integer), so cardinality (or "event size") is always a discrete variable. For example, compound event **S** in equation 2 ("a size 3 event") is first-order conjunctive with respect to its three causally necessary events.<sup>5</sup> Formally, we can summarize these ideas through the following expression: **<sup>E</sup>** *(* (**E**) = **<sup>E</sup>**<sup>1</sup> *^* **<sup>E</sup>**<sup>2</sup> *^ ··· ^* **<sup>E</sup>***<sup>n</sup>* <sup>=</sup> ^*<sup>n</sup>* **E***i.* (3) necessary events—i.e., set-theoretic *intersection* (operator *\*) or Boolean logic *product* (operator AND). Let *{***X***i}<sup>n</sup> <sup>i</sup>*=1 denote a set of *N* necessary events that produce **E**. Causal production of **E** is specified by an *event function*, **<sup>E</sup>** : *{***X***i} !* **E**, which maps necessary events in *{***X***i}* onto **E** using causal conjunctions. The number of events in a compound event (its "size" or "conjunctivity"), is called *cardinality*, *|***E***|* = *{*1*,* 2*,* 3*, ··· , n} ⇢ N* , which is always a natural number (positive integer), so cardinality (or "event size") is always a discrete variable. For example, compound event **S** in equation 2 ("a size 3 event") is first-order conjunctive with respect to its three causally necessary events.<sup>5</sup> Formally, we can summarize these ideas through the following expression: This is a compound event because **S** requires the following set of causally necessary conditions, each of which constitutes an event by itself: 1. Neither state will pursue issues deemed as highly threatening to the other, as opposed to acting completely oblivious or independently of extant foreign interests. 2. Each state may prefer to negotiate over colliding interests, before escalating to war. 3. When states do negotiate, they may—depending on conditions—find a nonviolent resolution. Specifically, every international event Specifically, every international event **E** is produced by causal *conjunction* (operator *^*) of necessary events—i.e., set-theoretic *intersection* (operator *\*) or Boolean logic *product* (operator AND). Let *{***X***i}<sup>n</sup> <sup>i</sup>*=1 denote a set of *N* necessary events that produce **E**. Causal production of **E** is specified by an *event function*, **<sup>E</sup>** : *{***X***i} !* **E**, which maps necessary events in *{***X***i}* onto **E** using causal conjunctions. The number of events in a compound event (its "size" or "conjunctivity"), is called *cardinality*, *|***E***|* = *{*1*,* 2*,* 3*, ··· , n} ⇢ N* , which is always a natural number (positive integer), so cardinality (or "event size") is always a discrete variable. For example, compound event **S** in equation 2 ("a size 3 event") is first-order conjunctive with respect to its three causally necessary events.<sup>5</sup> Formally, we can summarize these ideas through the following expression: is produced by causal *conjunction* (operator The universal existence of such necessary conditions makes **S** a compound event, by definition. Specifically, every international event **E** is produced by causal *conjunction* (operator *^*) of necessary events—i.e., set-theoretic *intersection* (operator *\*) or Boolean logic *product* (operator *<sup>i</sup>*=1 denote a set of *N* necessary events that produce **E**. Causal production of **E** is specified by an *event function*, **<sup>E</sup>** : *{***X***i} !* **E**, which maps necessary events in *{***X***i}* onto **E** using causal conjunctions. The number of events in a compound event (its "size" or "conjunctivity"), is called *cardinality*, *|***E***|* = *{*1*,* 2*,* 3*, ··· , n} ⇢ N* , which is always a natural number (positive integer), so cardinality (or "event size") is always a discrete variable. For example, compound event **S** in equation 2 ("a size 3 event") is first-order conjunctive with respect to its three causally necessary ) of necessary events—i.e., set-theoretic *intersection* (operator Specifically, every international event **E** is produced by causal *conjunction* (operator *^*) of necessary events—i.e., set-theoretic *intersection* (operator *\*) or Boolean logic *product* (operator AND). Let *{***X***i}<sup>n</sup> <sup>i</sup>*=1 denote a set of *N* necessary events that produce **E**. Causal production of **E** is specified by an *event function*, **<sup>E</sup>** : *{***X***i} !* **E**, which maps necessary events in *{***X***i}* onto **E** using causal conjunctions. The number of events in a compound event (its "size" or "conjunctivity"), is called *cardinality*, *|***E***|* = *{*1*,* 2*,* 3*, ··· , n} ⇢ N* , which is always a natural number (positive integer), so cardinality (or "event size") is always a discrete variable. For example, compound event **S** in equation 2 ("a size 3 event") is first-order conjunctive with respect to its three causally necessary events.<sup>5</sup> Formally, we can summarize these ideas through the following expression: ) or Boolean logic *product* (operator AND). Let Specifically, every international event **E** is produced by causal *conjunction* (operator *^*) of necessary events—i.e., set-theoretic *intersection* (operator *\*) or Boolean logic *product* (operator AND). Let *{***X***i}<sup>n</sup> <sup>i</sup>*=1 denote a set of *N* necessary events that produce **E**. Causal production of **E** is specified by an *event function*, **<sup>E</sup>** : *{***X***i} !* **E**, which maps necessary events in *{***X***i}* onto **E** using causal conjunctions. The number of events in a compound event (its "size" or "conjunctivity"), is called *cardinality*, *|***E***|* = *{*1*,* 2*,* 3*, ··· , n} ⇢ N* , which is always a natural number (positive integer), so cardinality (or "event size") is always a discrete variable. For example, compound event **S** in equation 2 ("a size 3 event") is first-order conjunctive with respect to its three causally necessary events.<sup>5</sup> Formally, we can summarize these ideas through the following expression: denote a set of *N* necessary events that produce 3. When states do negotiate, they may—depending on conditions—find a nonviolent resolution. The universal existence of such necessary conditions makes **S** a compound event, by definition. Specifically, every international event **E** is produced by causal *conjunction* (operator *^*) of necessary events—i.e., set-theoretic *intersection* (operator *\*) or Boolean logic *product* (operator *<sup>i</sup>*=1 denote a set of *N* necessary events that produce **E**. Causal production of **E** is specified by an *event function*, **<sup>E</sup>** : *{***X***i} !* **E**, which maps necessary events in *{***X***i}* onto **E** using causal conjunctions. The number of events in a compound event (its "size" or "conjunctivity"), is called *cardinality*, *|***E***|* = *{*1*,* 2*,* 3*, ··· , n} ⇢ N* , which is always a natural number (positive integer), . Causal production of 3. When states do negotiate, they may—depending on conditions—find a nonviolent resolution. The universal existence of such necessary conditions makes **S** a compound event, by definition. Specifically, every international event **E** is produced by causal *conjunction* (operator *^*) of necessary events—i.e., set-theoretic *intersection* (operator *\*) or Boolean logic *product* (operator AND). Let *{***X***i}<sup>n</sup> <sup>i</sup>*=1 denote a set of *N* necessary events that produce **E**. Causal production of **E** is specified by an *event function*, **<sup>E</sup>** : *{***X***i} !* **E**, which maps necessary events in *{***X***i}* onto **E** using causal conjunctions. The number of events in a compound event (its "size" or "conjunctivity"), is called *cardinality*, *|***E***|* = *{*1*,* 2*,* 3*, ··· , n} ⇢ N* , which is always a natural number (positive integer), is specified by an *event function*, The universal existence of such necessary conditions makes **S** a compound event, by definition. Specifically, every international event **E** is produced by causal *conjunction* (operator *^*) of necessary events—i.e., set-theoretic *intersection* (operator *\*) or Boolean logic *product* (operator *<sup>i</sup>*=1 denote a set of *N* necessary events that produce **E**. Causal production of **E** is specified by an *event function*, **<sup>E</sup>** : *{***X***i} !* **E**, which maps necessary events in *{***X***i}* onto **E** using causal conjunctions. The number of events in a compound event (its "size" or "conjunctivity"), is called *cardinality*, *|***E***|* = *{*1*,* 2*,* 3*, ··· , n} ⇢ N* , which is always a natural number (positive integer), so cardinality (or "event size") is always a discrete variable. For example, compound event **S** in equation 2 ("a size 3 event") is first-order conjunctive with respect to its three causally necessary , which maps necessary events in The universal existence of such necessary conditions makes **S** a compound event, by definition. Specifically, every international event **E** is produced by causal *conjunction* (operator *^*) of necessary events—i.e., set-theoretic *intersection* (operator *\*) or Boolean logic *product* (operator AND). Let *{***X***i}<sup>n</sup> <sup>i</sup>*=1 denote a set of *N* necessary events that produce **E**. Causal production of **E** is specified by an *event function*, **<sup>E</sup>** : *{***X***i} !* **E**, which maps necessary events in *{***X***i}* onto **E** using causal conjunctions. The number of events in a compound event (its "size" or "conjunctivity"), is called *cardinality*, *|***E***|* = *{*1*,* 2*,* 3*, ··· , n} ⇢ N* , which is always a natural number (positive integer), so cardinality (or "event size") is always a discrete variable. For example, compound event **S** in equation 2 ("a size 3 event") is first-order conjunctive with respect to its three causally necessary onto 2. Each state may prefer to negotiate over colliding interests, before escalating to war. 3. When states do negotiate, they may—depending on conditions—find a nonviolent resolution. The universal existence of such necessary conditions makes **S** a compound event, by definition. Specifically, every international event **E**is produced by causal *conjunction* (operator *^*) of necessary events—i.e., set-theoretic *intersection* (operator *\*) or Boolean logic *product* (operator AND). Let *{***X***i}<sup>n</sup> <sup>i</sup>*=1 denote a set of *N* necessary events that produce **E**. Causal production of **E** is specified by an *event function*, **<sup>E</sup>** : *{***X***i} !* **E**, which maps necessary events in *{***X***i}* onto **E**using causal conjunctions. The number of events in a compound event (its "size" or "conjunctivity"), is using causal conjunctions. The number of events in a compound event (its "size" or "conjunctivity"), is called *cardinality*, The universal existence of such necessary conditions makes **S** a compound event, by definition. Specifically, every international event **E** is produced by causal *conjunction* (operator *^*) of necessary events—i.e., set-theoretic *intersection* (operator *\*) or Boolean logic *product* (operator AND). Let *{***X***i}<sup>n</sup> <sup>i</sup>*=1 denote a set of *N* necessary events that produce **E**. Causal production of **E** is specified by an *event function*, **<sup>E</sup>** : *{***X***i} !* **E**, which maps necessary events in *{***X***i}* onto **E** using causal conjunctions. The number of events in a compound event (its "size" or "conjunctivity"), is called *cardinality*, *|***E***|* = *{*1*,* 2*,* 3*, ··· , n} ⇢ N* , which is always a natural number (positive integer), so cardinality (or "event size") is always a discrete variable. For example, compound event **S** in , which is always a natural number (positive integer), so cardinality (or "event size") is always a discrete

> Next, an international event **E** has probability Pr (**E**) that is determined by the naturally uncertain occurrence of necessary conditions—a type of causation known as *probabilistic causality* (Salmon 1980; Suppes 1984; Eels 1991). Based on Kolmogorov's (1933) fundamental theorem of compound

events.<sup>5</sup> Formally, we can summarize these ideas through the following expression:

*i*=1

**<sup>E</sup>** *(* (**E**) = **<sup>E</sup>**<sup>1</sup> *^* **<sup>E</sup>**<sup>2</sup> *^ ··· ^* **<sup>E</sup>***<sup>n</sup>* <sup>=</sup> ^*<sup>n</sup>*

Next, an international event **E** has probability Pr (**E**) that is determined by the naturally uncertain occurrence of necessary conditions—a type of causation known as *probabilistic causality* (Salmon 1980; Suppes 1984; Eels 1991). Based on Kolmogorov's (1933) fundamental theorem of compound

**<sup>E</sup>** *(* (**E**) = **<sup>E</sup>**<sup>1</sup> *^***E**<sup>2</sup> *^ ··· ^* **<sup>E</sup>***<sup>n</sup>* <sup>=</sup> ^*<sup>n</sup>*

events.<sup>5</sup> Formally, we can summarize these ideas through the following expression:

Next, an international event **E** has probability Pr (**E**) that is determined by the naturally uncertain occurrence of necessary conditions—a type of causation known as *probabilistic causality* (Salmon 1980; Suppes 1984; Eels 1991). Based on Kolmogorov's (1933) fundamental theorem of compound

Pr(**E***i*)*,* (4)

Pr(**E**) = Pr(**E**1) *·* Pr(**E**2) *· ... ·* Pr(**E***n*) = <sup>Y</sup>*<sup>n</sup>*

events, the joint probability *E* of compound event **E** is given by the following expression:

where events **E***<sup>i</sup>* are independent; when they are *not* independent, *conditional probabilities* are used and, by Kolmogorov's theorem, they still multiply. Letting Pr(**E**) = *E* and Pr(**E***i*) = *pi*, since

events, the joint probability *E* of compound event **E** is given by the following expression:

Pr(**E**) = Pr(**E**1) *·* Pr(**E**2) *· ... ·* Pr(**E***n*) = <sup>Y</sup>*<sup>n</sup>*

events, the joint probability *E* of compound event **E** is given by the following expression:

<sup>5</sup> An event function is also called *indicator function*, *structure function*, or *production function*, in areas of

*i*=1

mathematics, engineering, and economics, respectively.

mathematics, engineering, and economics, respectively.

mathematics, engineering, and economics, respectively.

mathematics, engineering, and economics, respectively.

<sup>5</sup> An event function is also called *indicator function*, *structure function*, or *production function*, in areas of

where events **E***<sup>i</sup>* are independent; when they are *not* independent, *conditional probabilities* are used and, by Kolmogorov's theorem, they still multiply. Letting Pr(**E**) = *E* and Pr(**E***i*) = *pi*, since

<sup>5</sup> An event function is also called *indicator function*, *structure function*, or *production function*, in areas of

where events **E***<sup>i</sup>* are independent; when they are *not* independent, *conditional probabilities* are used and, by Kolmogorov's theorem, they still multiply. Letting Pr(**E**) = *E* and Pr(**E***i*) = *pi*, since

Pr(**E**) = Pr(**E**1) *·* Pr(**E**2) *· ... ·* Pr(**E***n*) = <sup>Y</sup>*<sup>n</sup>*

events, the joint probability *E* of compound event **E** is given by the following expression:

*i*=1

*i*=1

mathematics, engineering, and economics, respectively.

Pr(**E**) = Pr(**E**1) *·* Pr(**E**2) *· ... ·* Pr(**E***n*) = <sup>Y</sup>*<sup>n</sup>*

mathematics, engineering, and economics, respectively.

where events **E***<sup>i</sup>* are independent; when they are *not* independent, *conditional probabilities* are used and, by Kolmogorov's theorem, they still multiply. Letting Pr(**E**) = *E* and Pr(**E***i*) = *pi*, since

<sup>5</sup> An event function is also called *indicator function*, *structure function*, or *production function*, in areas of

events, the joint probability *E* of compound event **E** is given by the following expression:

Pr(**E**) = Pr(**E**1) *·* Pr(**E**2) *· ... ·* Pr(**E***n*) = <sup>Y</sup>*<sup>n</sup>*

events.<sup>5</sup> Formally, we can summarize these ideas through the following expression:

Next, an international event **<sup>E</sup>** has probability Pr (**E**) that is determined by the naturally uncertainoccurrence of necessary conditions—a type of causation known as *probabilistic causality* (Salmon 1980; Suppes 1984; Eels 1991). Based on Kolmogorov's (1933) fundamental theorem of compound

events, the joint probability *E* of compound event **E** is given by the following expression:

Pr(**E**) = Pr(**E**1) *·* Pr(**E**2) *· ... ·* Pr(**E***n*) = <sup>Y</sup>*<sup>n</sup>*

**<sup>E</sup>** *(* (**E**) = **<sup>E</sup>**<sup>1</sup> *^* **<sup>E</sup>**<sup>2</sup> *^ ··· ^* **<sup>E</sup>***<sup>n</sup>* <sup>=</sup> ^*<sup>n</sup>*

**<sup>E</sup>** *(* (**E**) = **<sup>E</sup>**<sup>1</sup> *^* **<sup>E</sup>**<sup>2</sup> *^ ··· ^* **<sup>E</sup>***<sup>n</sup>* <sup>=</sup> ^*<sup>n</sup>*

Next, an international event **E** has probability Pr (**E**) that is determined by the naturally uncertain occurrence of necessary conditions—a type of causation known as *probabilistic causality* (Salmon 1980; Suppes 1984; Eels 1991). Based on Kolmogorov's (1933) fundamental theorem of compound

11

mathematics, engineering, and economics, respectively.

<sup>5</sup> An event function is also called *indicator function*, *structure function*, or *production function*, in areas of

mathematics, engineering, and economics, respectively.

where events **E***<sup>i</sup>* are independent; when they are *not* independent, *conditional probabilities* are used and, by Kolmogorov's theorem, they still multiply. Letting Pr(**E**) = *E* and Pr(**E***i*) = *pi*, since

*i*=1

equation 2 ("a size 3 event") is first-order conjunctive with respect to its three causally necessary

events.<sup>5</sup> Formally, we can summarize these ideas through the following expression:

**<sup>E</sup>** *(* (**E**) = **<sup>E</sup>**<sup>1</sup> *^* **<sup>E</sup>**<sup>2</sup> *^ ···^* **<sup>E</sup>***<sup>n</sup>* <sup>=</sup> ^*<sup>n</sup>*

2. Each state may prefer to negotiate over colliding interests, before escalating to war.

**<sup>E</sup>** *(* (**E**) = **<sup>E</sup>**<sup>1</sup> *^* **<sup>E</sup>**<sup>2</sup> *^ ··· ^* **<sup>E</sup>***<sup>n</sup>* <sup>=</sup> ^*<sup>n</sup>*

**<sup>E</sup>** *(* (**E**) = **<sup>E</sup>**<sup>1</sup> *^* **<sup>E</sup>**<sup>2</sup> *^ ··· ^* **<sup>E</sup>***<sup>n</sup>* <sup>=</sup> ^*<sup>n</sup>*

AND). Let *{***X***i}<sup>n</sup>*

events.<sup>5</sup> Formally, we can summarize these ideas through the following expression:

**E***i.* (3)

**<sup>E</sup>** *(* (**E**) = **<sup>E</sup>**<sup>1</sup> *^* **<sup>E</sup>**<sup>2</sup> *^ ··· ^* **<sup>E</sup>***<sup>n</sup>* <sup>=</sup> ^*<sup>n</sup>*

Next, an international event **E** has probability Pr (**E**) that is determined by the naturally uncertain occurrence of necessary conditions—a type of causation known as *probabilistic causality* (Salmon 1980; Suppes 1984; Eels 1991). Based on Kolmogorov's (1933) fundamental theorem of compound

*i*=1

where events **E***<sup>i</sup>* are independent; when they are *not* independent, *conditional probabilities* are used and, by Kolmogorov's theorem, they still multiply. Letting Pr(**E**) = *E* and Pr(**E***i*) = *pi*, since

<sup>5</sup> An event function is also called *indicator function*, *structure function*, or *production function*, in areas of

*i*=1

11

mathematics, engineering, and economics, respectively.

11

events, the joint probability *E* of compound event **E** is given by the following expression:

*i*=1

*i*=1

called *cardinality*, *|***E***|* = *{*1*,* 2*,* 3*, ··· , n} ⇢ N* , which is always a natural number (positive integer), so cardinality (or "event size") is always a discrete variable. For example, compound event **S** in equation 2 ("a size 3 event") is first-order conjunctive with respect to its three causally necessary

Next, an international event **E** has probability Pr (**E**) that is determined by the naturally uncertain occurrence of necessary conditions—a type of causation known as *probabilistic causality* (Salmon 1980; Suppes 1984; Eels 1991). Based on Kolmogorov's (1933) fundamental theorem of compound

**<sup>E</sup>** *(* (**E**) = **<sup>E</sup>**<sup>1</sup> *^* **<sup>E</sup>**<sup>2</sup> *^ ··· ^* **<sup>E</sup>***<sup>n</sup>* <sup>=</sup> ^*<sup>n</sup>*

Next, an international event **E** has probability Pr (**E**) that is determined by the naturally uncertain occurrence of necessary conditions—a type of causation known as *probabilistic causality* (Salmon 1980; Suppes 1984; Eels 1991). Based on Kolmogorov's (1933) fundamental theorem of compound

**E***i.* (3)

Pr(**E**) = Pr(**E**1) *·* Pr(**E**2) *· ... ·* Pr(**E***n*) = <sup>Y</sup>*<sup>n</sup>*

Next, an international event **E** has probability Pr (**E**) that is determined by the naturally uncertain occurrence of necessary conditions—a type of causation known as *probabilistic causality* (Salmon 1980; Suppes 1984; Eels 1991). Based on Kolmogorov's (1933) fundamental theorem of compound

*i*=1

where events **E***<sup>i</sup>* are independent; when they are *not* independent, *conditional probabilities* are used and, by Kolmogorov's theorem, they still multiply. Letting Pr(**E**) = *E* and Pr(**E***i*) = *pi*, since

Pr(**E***i*)*,* (4)

events, the joint probability *E* of compound event **E** is given by the following expression:

where events **E***<sup>i</sup>* are independent; when they are *not* independent, *conditional probabilities* are used and, by Kolmogorov's theorem, they still multiply. Letting Pr(**E**) = *E* and Pr(**E***i*) = *pi*, since

Pr(**E**) = Pr(**E**1) *·* Pr(**E**2) *· ... ·* Pr(**E***n*) = <sup>Y</sup>*<sup>n</sup>*

<sup>5</sup> An event function is also called *indicator function*, *structure function*, or *production function*, in areas of

where events **E***<sup>i</sup>* are independent; when they are *not* independent, *conditional probabilities* are used and, by Kolmogorov's theorem, they still multiply. Letting Pr(**E**) = *E* and Pr(**E***i*) = *pi*, since

<sup>5</sup> An event function is also called *indicator function*, *structure function*, or *production function*, in areas of

<sup>5</sup> An event function is also called *indicator function*, *structure function*, or *production function*, in areas of

mathematics, engineering, and economics, respectively.

*i*=1

events, the joint probability *E* of compound event **E** is given by the following expression:

Next, an international event **E** has probability Pr (**E**) that is determined by the naturally uncertain occurrence of necessary conditions—a type of causation known as *probabilistic causality* (Salmon 1980; Suppes 1984; Eels 1991). Based on Kolmogorov's (1933) fundamental theorem of compound

events, the joint probability *E* of compound event **E** is given by the following expression:

Pr(**E**) = Pr(**E**1) *·* Pr(**E**2) *· ... ·* Pr(**E***n*) = <sup>Y</sup>*<sup>n</sup>*

Pr(**E**) = Pr(**E**1) *·* Pr(**E**2) *· ... ·* Pr(**E***n*) = <sup>Y</sup>*<sup>n</sup>*

where events **E***<sup>i</sup>* are independent; when they are *not* independent, *conditional probabilities* are used and, by Kolmogorov's theorem, they still multiply. Letting Pr(**E**) = *E* and Pr(**E***i*) = *pi*, since

<sup>5</sup> An event function is also called *indicator function*, *structure function*, or *production function*, in areas of

Pr(**E***i*)*,* (4)

events, the joint probability *E* of compound event **E** is given by the following expression:

*i*=1

*i*=1

*i*=1

Pr(**E**) = Pr(**E**1) *·* Pr(**E**2) *· ... ·* Pr(**E***n*) = <sup>Y</sup>*<sup>n</sup>*

*i*=1

*i*=1

*i*=1

**E***i.* (3)

**E***i.* (3)

Next, an international event **E** has probability Pr (**E**) that is determined by the naturally uncertain occurrence necessary conditions—a type of causation known as *probabilistic causality* (Salmon 1980; Suppes 1984; Eels 1991). Based on Kolmogorov's (1933) fundamental theorem of compound

*i*=1

**<sup>E</sup>** *(* (**E**) = **<sup>E</sup>**<sup>1</sup> *^* **<sup>E</sup>**<sup>2</sup> *^ ··· ^* **<sup>E</sup>***<sup>n</sup>* <sup>=</sup> ^*<sup>n</sup>*

The universal existence of such necessary conditions makes **S** a compound event, by definition. Specifically, every international event **E** is produced by causal *conjunction* (operator *^*) of necessary events—i.e., set-theoretic *intersection* (operator *\*) or Boolean logic *product* (operator

specified by an *event function*, **<sup>E</sup>** : *{***X***i} !* **E**, which maps necessary events in *{***X***i}* onto **E** using causal conjunctions. The number of events in a compound event (its "size" or "conjunctivity"), is called *cardinality*, *|***E***|* = *{*1*,* 2*,* 3*, ··· , n} ⇢ N* , which is always a natural number (positive integer), so cardinality (or "event size") is always a discrete variable. For example, compound event **S** in equation 2 ("a size 3 event") is first-order conjunctive with respect to its three causally necessary

**E***i.* (3)

**E***i.* (3)

**E***i.* (3)

*i*=1

**E***i.* (3)

*i*=1

Pr(**E***i*)*,* (4)

Pr(**E***i*)*,* (4)

where events **E***<sup>i</sup>* are independent; when they are *not* independent, *conditional probabilities* are used and, by Kolmogorov's theorem, they still multiply. Letting Pr(**E**) = *E* and Pr(**E***i*) = *pi*, since

<sup>5</sup> An event function is also called *indicator function*, *structure function*, or *production function*, in areas of

*i*=1

events.<sup>5</sup> Formally, we can summarize these ideas through the following expression:

**E***i.* (3)

*<sup>i</sup>*=1 denote a set of *N* necessary events that produce **E**. Causal production of **E** is

Pr(**E***i*)*,* (4)

*i*=1

*i*=1

11

11

11

11

11

**E***i.* (3)

Pr(**E***i*)*,* (4)

**E***i.* (3)

Pr(**E***i*)*,* (4)

11

11

11

events, the joint probability *E* of compound event **E** is given by the following expression:

Pr(**E**) = Pr(**E**1) *·* Pr(**E**2) *· ... ·* Pr(**E***n*) = <sup>Y</sup>*<sup>n</sup>*

**<sup>E</sup>** *(* (**E**) = **<sup>E</sup>**<sup>1</sup> *^* **<sup>E</sup>**<sup>2</sup> *^ ··· ^* **<sup>E</sup>***<sup>n</sup>* <sup>=</sup> ^*<sup>n</sup>*

Pr(**E***i*)*,* (4)

Pr(**E***i*)*,* (4)

Pr(**E***i*)*,* (4)

*i*=1

Pr(**E***i*)*,* (4)

Next, an international event **E** has probability Pr (**E**) that is determined by the naturally uncertain occurrence of necessary conditions—a type of causation known as *probabilistic causality* (Salmon 1980; Suppes 1984; Eels 1991). Based on Kolmogorov's (1933) fundamental theorem of compound

where events **E***<sup>i</sup>* are independent; when they are *not* independent, *conditional probabilities* are used and, by Kolmogorov's theorem, they still multiply. Letting Pr(**E**) = *E* and Pr(**E***i*) = *pi*, since

11

11

<sup>5</sup> An event function is also called *indicator function*, *structure function*, or *production function*, in areas of

**E***i.* (3)

Pr(**E***i*)*,* (4)

mathematics, engineering, and economics, respectively.

mathematics, engineering, and economics, respectively.

.3..

4. Applications to areas of international relations

4. Applications to areas of international relations

4.1 Case 1: Peace and other international events

each of which constitutes an event by itself:

.3..

following expression:

AND). Let *{***X***i}<sup>n</sup>*

Here we shall investigate three illustrative cases of IR theories and research areas—numbered 1, 3,

.3..

in all domains of international relations theory. For example, consider the event defined by the

.3..

following expression:

following expression:

acting completely oblivious or independently of extant foreign interests.

**<sup>E</sup>** *(* (**E**) = **<sup>E</sup>**<sup>1</sup> *^* **<sup>E</sup>**<sup>2</sup> *^ ··· ^* **<sup>E</sup>***<sup>n</sup>* <sup>=</sup> ^*<sup>n</sup>*

Next, an international event **E** has probability Pr (**E**) that is determined by the naturally uncertain occurrence of necessary conditions—a type of causation known as *probabilistic causality* (Salmon 1980; Suppes 1984; Eels 1991). Based on Kolmogorov's (1933) fundamental theorem of compound

where events **E***<sup>i</sup>* are independent; when they are *not* independent, *conditional probabilities* are used and, by Kolmogorov's theorem, they still multiply. Letting Pr(**E**) = *E* and Pr(**E***i*) = *pi*, since

<sup>5</sup> An event function is also called *indicator function*, *structure function*, or *production function*, in areas of

events, the joint probability *E* of compound event **E** is given by the following expression:

Pr(**E**) = Pr(**E**1) *·* Pr(**E**2) *· ... ·* Pr(**E***n*) = <sup>Y</sup>*<sup>n</sup>*

mathematics, engineering, and economics, respectively.

*i*=1

AND). Let *{***X***i}<sup>n</sup>*

AND). Let *{***X***i}<sup>n</sup>*

*i*=1

All international events in the real world are "compound" because they are always produced by

**S** *⌘* "a state of stable peace exists between two countries." (2)

*<sup>i</sup>*=1 denote a set of *N* necessary events that produce **E**. Causal production of **E** is

4.1 Case 1: Peace and other international events

4. Applications to areas of international relations

Here we shall investigate three illustrative cases of IR theories and research areas—numbered 1, 3, and 6 in section .2.—where hybrid functions play a central role in describing and explaining political phenomena. The scientific purpose is to deepen our understanding and provide foundations for more advanced analysis. Each "case study" follows the analytical procedure just outlined in section

**S** *⌘* "a state of stable peace exists between two countries." (2)

**S** *⌘* "a state of stable peace exists between two countries." (2)

NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS

Here we shall investigate three illustrative cases of IR theories and research areas—numbered 1, 3, and 6 in section .2.—where hybrid functions play a central role in describing and explaining political phenomena. The scientific purpose is to deepen our understanding and provide foundations for more advanced analysis. Each "case study" follows the analytical procedure just outlined in section

NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS

All international events in the real world are "compound" because they are always produced by necessary conditions specific to the event. Such causal necessity is universal—a fundamental axiom in all domains of international relations theory. For example, consider the event defined by the

This is a compound event because **S** requires the following set of causally necessary conditions,

Here we shall investigate three illustrative cases of IR theories and research areas—numbered 1, 3, and 6 in section .2.—where hybrid functions play a central role in describing and explaining political phenomena. The scientific purpose is to deepen our understanding and provide foundations for more advanced analysis. Each "case study" follows the analytical procedure just outlined in section

Here we shall investigate three illustrative cases of IR theories and research areas—numbered 1, 3,and 6 in section .2.—where hybrid functions play a central role in describing and explaining politicalphenomena. The scientific purpose is to deepen our understanding and provide foundations for

more advanced analysis. Each "case study" follows the analytical procedure just outlined in section

All international events in the real world are "compound" because they are always produced by necessary conditions specific to the event. Such causal necessity is universal—a fundamental axiom in all domains of international relations theory. For example, consider the event defined by the

All international events in the real world are "compound" because they are always produced by necessary conditions specific to the event. Such causal necessity is universal—a fundamental axiom in all domains of international relations theory. For example, consider the event defined by the

This is a compound event because **S** requires the following set of causally necessary conditions,

Here we shall investigate three illustrative cases of IR theories and research areas—numbered 1, 3, and 6 in section .2.—where hybrid functions play a central role in describing and explaining political phenomena. The scientific purpose is to deepen our understanding and provide foundations for more advanced analysis. Each "case study" follows the analytical procedure just outlined in section

1. Neither state will pursue issues deemed as highly threatening to the other, as opposed to

1. Neither state will pursue issues deemed as highly threatening to the other, as opposed to

This is a compound event because **S** requires the following set of causally necessary conditions,

3. When states do negotiate, they may—depending on conditions—find a nonviolent resolution.

The universal existence of such necessary conditions makes **S** a compound event, by definition. Specifically, every international event **E** is produced by causal *conjunction* (operator *^*) of necessary events—i.e., set-theoretic *intersection* (operator *\*) or Boolean logic *product* (operator

The universal existence of such necessary conditions makes **S** a compound event, by definition.

3. When states do negotiate, they may—depending on conditions—find a nonviolent resolution.

All international events in the real world are "compound" because they are always produced by necessary conditions specific to the event. Such causal necessity is universal—a fundamental axiom in all domains of international relations theory. For example, consider the event defined by the

acting completely oblivious or independently of extant foreign interests.

acting completely oblivious or independently of extant foreign interests.

4. Applications to areas of international relations

2. Each state may prefer to negotiate over colliding interests, before escalating to war.

2. Each state may prefer to negotiate over colliding interests, before escalating to war.

4.1 Case 1: Peace and other international events

NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS

NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS

variable. For example, compound event **S** *⌘* "a state of stable peace exists between two countries." (2) This is a compound event because **S** requires the following set of causally necessary conditions, in equation 2 ("a size 3 event") is firstorder conjunctive with respect to its three causally necessary events.5 Formally, we can summarize these ideas through the following expression: called *cardinality*, *|***E***|* = *{*1*,* 2*,* 3*, ··· , n} ⇢ N* , which is always a natural number (positive integer), so cardinality (or "event size") is always a discrete variable. For example, compound event **S** in equation 2 ("a size 3 event") is first-order conjunctive with respect to its three causally necessary events.<sup>5</sup> Formally, we can summarize these ideas through the following expression: each of which constitutes an event by itself: 1. Neither state will pursue issues deemed as highly threatening to the other, as opposed to acting completely oblivious or independently of extant foreign interests. each of which constitutes an event by itself: 1. Neither state will pursue issues deemed as highly threatening to the other, as opposed to acting completely oblivious or independently of extant foreign interests. **S** *⌘* "a state of stable peace exists between two countries." (2) This is a compound event because **S** requires the following set of causally necessary conditions, specified by an *event function*, **<sup>E</sup>** : *{***X***i} !* **E**, which maps necessary events in *{***X***i}* onto **E** using causal conjunctions. The number of events in a compound event (its "size" or "conjunctivity"), is called *cardinality*, *|***E***|* = *{*1*,* 2*,* 3*, ··· , n} ⇢ N* , which is always a natural number (positive integer), so cardinality (or "event size") is always a discrete variable. For example, compound event **S** in 1. Neither state will pursue issues deemed as highly threatening to the other, as opposed to acting completely oblivious or independently of extant foreign interests. necessary events—i.e., set-theoretic *intersection* (operator *\*) or Boolean logic *product* (operator *<sup>i</sup>*=1 denote a set of *N* necessary events that produce **E**. Causal production of **E** is specified by an *event function*, **<sup>E</sup>** : *{***X***i} !* **E**, which maps necessary events in *{***X***i}* onto **E** using causal conjunctions. The number of events in a compound event (its "size" or "conjunctivity"), is called *cardinality*, *|***E***|* = *{*1*,* 2*,* 3*, ··· , n} ⇢ N* , which is always a natural number (positive integer), *<sup>i</sup>* denote a *N*necessary events that **E**. **<sup>E</sup>**isspecified by *function*, *!* maps necessary *<sup>i</sup>* **<sup>E</sup>**usingcausal conjunctions. iscalled,*|,*3*,···N*which is always a natural

equation 2 ("a size 3 event") is first-order conjunctive with respect to its three causally necessary

causal conjunctions. The number of events in a compound event (its "size" or "conjunctivity"), is

This is a compound event because **S** requires the following set of causally necessary conditions,

each of which constitutes an event by itself:

.3..

4. Applications to areas of international relations

following expression:

4.1 Case 1: Peace and other international events

4. Applications to areas of international relations

4. Applications to areas of international relations

4.1 Case 1: Peace and other international events

4.1 Case 1: Peace and other international events

NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS

each of which constitutes an event by itself:

.3..

each of which constitutes an event by itself:

4.1 Case 1: Peace and other international events

4. Applications to areas of international relations

.3..

Here we shall investigate three illustrative cases of IR theories and research areas—numbered 1, 3, and 6 in section .2.—where hybrid functions play a central role in describing and explaining political phenomena. The scientific purpose is to deepen our understanding and provide foundations for more advanced analysis. Each "case study" follows the analytical procedure just outlined in section

and section describing and politicalphenomena. to deepen our understanding and provide foundations formore the analytical section.3..

Here cases of and research

OF HYBRID INTERNATIONAL RELATIONS4.

All international events in the real world are "compound" because they are always produced by necessary conditions specific to the event. Such causal necessity is universal—a fundamental axiom in all domains of international relations theory. For example, consider the event defined by the

the real produced bynecessary conditions universal—a fundamental axiomin domains of consider the defined by thefollowing

This is a compound event because **S** requires the following set of causally necessary conditions,

"a state of exists between (2)is a requires the each Neither state will threatening to opposed toacting completely oblivious or independently interests.

1. Neither state will pursue issues deemed as highly threatening to the other, as opposed to

3. When states do negotiate, they may—depending on conditions—find a nonviolent resolution.

.3..

.3..

4. Applications to areas of international relations

4.1 Case 1: Peace and other international events

acting completely oblivious or independently of extant foreign interests.

2. Each state may prefer to negotiate over colliding interests, before escalating to war.

Each before escalating to

Pr(**E**) = Pr(**E**1) *·* Pr(**E**2) *· ... ·* Pr(**E***n*) = <sup>Y</sup>*<sup>n</sup>*

*i*=1

each of which constitutes an event by itself:

.3..

following expression:

AND). Let *{***X***i}<sup>n</sup>*

CLAUDIO CIOFFI-REVILLA

equation (4) in simpler notation:

.3..

4.1Case 1:

following expression:

each of which constitutes an event by itself:

4. Applications to areas of international relations

4.1 Case 1: Peace and other international events

AND). Let *{***X***i}<sup>n</sup>*

mathematics, engineering, and economics, respectively.

event size is always a discrete or natural number (of necessary conditions).

by the steep north-south escarpment along the east edge as *P !* 1*.*0.

approximately the same underlying (*p, n*)-domain.

6 and Figures 1a and b).

12

all probabilities are continuous variables (i.e., *E* and *p<sup>i</sup>* are variables, not events), we can rewrite

*<sup>E</sup>* <sup>=</sup> *<sup>p</sup>*<sup>1</sup> *· <sup>p</sup>*<sup>2</sup> *· ... · <sup>p</sup><sup>n</sup>* <sup>=</sup> <sup>Y</sup>*<sup>n</sup>*

Note that change caused by each variable differs, as shown in Figures 1c through f. Calculating

We see immediately that changes in causal probability *P* and event size *N* have different effects on the probability of an international event *E*, besides the trivial observation that *any* change on either variable has *some* effect on *E* (since *@pE >* 0 and ∆*<sup>N</sup> E <* 0, per equations 7-8). To wit: Opposite political effects. Whereas change in causal probability *P* has proportional or positive effect on *E*, change in event size *N* has an opposite effect. This is shown by the purple deep bottom in Figure 1c versus the high red plateau in Figure 1e; both features observed over

First- and second-order effects. Whereas the contour plot of *E*(*P, N*) shows strictly concave isocontours (Figure 1b), the contours of the derivative *@pE* (Figure 1d) also show a convex, mild spur protruding on the southwestern wall of the escarpment, near *p* = 0*.*75 (third contour, between green and blue elevations), which means more complex change for small-size events. This is a second-order effect and not at all intuitive from the basic model (equation

Second-order effects caused by changes in event size. The graphs of the change in *E* with respect to event size, ∆*nE* (equation 8 and Figures 1e and f), show a pronounced ravine or canyon along the north-south direction dropping into a deep precipice at relatively high values of *P* as *N !* 2. Interestingly, in this case all the isolines have mixed concavity (low

Geometrically opposite extrema. The extreme high range of *@pE* in Figures 1c and d and the extreme low range of ∆*nE* in Figures 1e and f have opposite (or inverse) geometries with some common similarities: the former rises from a flat basin to an escarpment while the latter drops from a large plateau toward a deepening canyon that dives into a deep well. Both features are indicative of major political effects on event probability *E* caused by

*P*) and convexity (higher *P* values), which is another second-order effect.

underlying changes in causal probabilities *P* and event size *N*.

<sup>∆</sup>*n<sup>E</sup>* <sup>=</sup> *<sup>P</sup> <sup>N</sup>*+1 *<sup>−</sup> <sup>P</sup> <sup>N</sup>* (for *<sup>N</sup>*'s effect on *<sup>E</sup>*, in Figures 1e and f)*.* (8)

the partial derivative and partial difference of *E* with respect to *P* and *N*, respectively:

following expression:

NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS

following expression:

Here we shall investigate three illustrative cases of IR theories and research areas—numbered 1, 3, and 6 in section .2.—where hybrid functions play a central role in describing and explaining political phenomena. The scientific purpose is to deepen our understanding and provide foundations for more advanced analysis. Each "case study" follows the analytical procedure just outlined in section

AND). Let *{***X***i}<sup>n</sup>*

AND). Let *{***X***i}<sup>n</sup>*

The universal existence of such necessary conditions makes **S** a compound event, by definition.

Pr(**E**) = Pr(**E**1) *·* Pr(**E**2) *· ... ·* Pr(**E***n*) = <sup>Y</sup>*<sup>n</sup>*

following expression:

.3..

This is a compound event because **S** requires the following set of causally necessary conditions,

following expression:

**S** *⌘* "a state of stable peace exists between two countries." (2)

AND). Let *{***X***i}<sup>n</sup>*

4. Applications to areas of international relations

*i*=1

approximately the same underlying (*p, n*)-domain.

events.<sup>5</sup> Formally, we can summarize these ideas through the following expression:

6 and Figures 1a and b).

mathematics, engineering, and economics, respectively.

6 and Figures 1a and b).

12

12

following expression:

All international events in the real world are "compound" because they are always produced by necessary conditions specific to the event. Such causal necessity is universal—a fundamental axiom in all domains of international relations theory. For example, consider the event defined by the

.3..

**S** *⌘* "a state of stable peace exists between two countries." (2)

$$\mathbb{E} \Leftrightarrow \Psi(\mathbb{E}) = \mathbb{E}\_1 \wedge \mathbb{E}\_2 \wedge \dots \wedge \mathbb{E}\_n = \bigwedge\_{i=1}^n \mathbb{E}\_i. \tag{3}$$

2. Each state may prefer to negotiate over colliding interests, before escalating to war. 3. When states do negotiate, they may—depending on conditions—find a nonviolent resolution. The universal existence of such necessary conditions makes **S** a compound event, by definition. Specifically, every international event **E** is produced by causal *conjunction* (operator *^*) of necessary events—i.e., set-theoretic *intersection* (operator *\*) or Boolean logic *product* (operator Next, an international event **E** has probability Pr (**E**) that is determined by the naturally uncertain occurrence of necessary conditions—a type of causation known as *probabilistic causality* (Salmon 1980; Suppes 1984; Eels 1991). Based on Kolmogorov's (1933) fundamental theorem of compound events, the joint probability *E* of compound event **E** is given by the following expression: Pr(**E**) = Pr(**E**1) *·* Pr(**E**2) *· ... ·* Pr(**E***n*) = <sup>Y</sup>*<sup>n</sup>* Next, an international event The universal existence of such necessary conditions makes **S** a compound event, by definition. Specifically, every international event **E** is produced by causal *conjunction* (operator *^*) of necessary events—i.e., set-theoretic *intersection* (operator *\*) or Boolean logic *product* (operator AND). Let *{***X***i}<sup>n</sup> <sup>i</sup>*=1 denote a set of *N* necessary events that produce **E**. Causal production of **E** is specified by an *event function*, **<sup>E</sup>** : *{***X***i} !* **E**, which maps necessary events in *{***X***i}* onto **E** using causal conjunctions. The number of events in a compound event (its "size" or "conjunctivity"), is called *cardinality*, *|***E***|* = *{*1*,* 2*,* 3*, ··· , n} ⇢ N* , which is always a natural number (positive integer), has probability Pr( The universal existence of such necessary conditions makes **S** a compound event, by definition. Specifically, every international event **E** is produced by causal *conjunction* (operator *^*) of necessary events—i.e., set-theoretic *intersection* (operator *\*) or Boolean logic *product* (operator AND). Let *{***X***i}<sup>n</sup> <sup>i</sup>*=1 denote a set of *N* necessary events that produce **E**. Causal production of **E** is specified by an *event function*, **<sup>E</sup>** : *{***X***i} !* **E**, which maps necessary events in *{***X***i}* onto **E** using causal conjunctions. The number of events in a compound event (its "size" or "conjunctivity"), is called *cardinality*, *|***E***|* = *{*1*,* 2*,* 3*, ··· , n} ⇢ N* , which is always a natural number (positive integer), ) that is determined by the naturally uncertain occurrence of necessary conditions—a type of causation known as *probabilistic causality* (Salmon 1980; Suppes 1984; Eels 1991). Based on Kolmogorov's (1933) fundamental theorem of compound events, the joint probability *E* of compound event 2. Each state may prefer to negotiate over colliding interests, before escalating to war. 3. When states do negotiate, they may—depending on conditions—find a nonviolent resolution. The universal existence of such necessary conditions makes **S** a compound event, by definition. Specifically, every international event **E** is produced by causal *conjunction* (operator *^*) of is given by the following expression: *i*=1 Next, an international event **E** has probability Pr (**E**) that is determined by the naturally uncertain occurrence of necessary conditions—a type of causation known as *probabilistic causality* (Salmon 1980; Suppes 1984; Eels 1991). Based on Kolmogorov's (1933) fundamental theorem of compound events, the joint probability *E* of compound event **E** is given by the following expression: Specifically, every international event **E** is produced by causal *conjunction* (operator *^*) of necessary events—i.e., set-theoretic *intersection* (operator *\*) or Boolean logic *product* (operator *<sup>i</sup>*=1 denote a set of *N* necessary events that produce **E**. Causal production of **E** is specified by an *event function*, **<sup>E</sup>** : *{***X***i} !* **E**, which maps necessary events in *{***X***i}* onto **E** using causal conjunctions. The number of events in a compound event (its "size" or "conjunctivity"), is called *cardinality*, *|***E***|* = *{*1*,* 2*,* 3*, ··· , n} ⇢ N* , which is always a natural number (positive integer), **<sup>E</sup>** *(* (**E**) = **<sup>E</sup>**<sup>1</sup> *^* **<sup>E</sup>**<sup>2</sup> *^ ··· ^* **<sup>E</sup>***<sup>n</sup>* <sup>=</sup> ^*<sup>n</sup> i*=1 **E***i.* (3) Next, an international event **E** has probability Pr (**E**) that is determined by the naturally uncertain occurrence of necessary conditions—a type of causation known as *probabilistic causality* (Salmon 1980; Suppes 1984; Eels 1991). Based on Kolmogorov's (1933) fundamental theorem of compound (**E**1*^* **<sup>E</sup>***n*<sup>=</sup> ^ (3)Next, has probability Pr (**E**)that naturally uncertainoccurrence of causation *probabilistic causality*(Salmon 1991). Based on Kolmogorov's (1933) events, the probabilityof compound is by the

AND). Let *{***X***i}<sup>n</sup>*

specified by an *event function*, **<sup>E</sup>** : *{***X***i} !* **E**, which maps necessary events in *{***X***i}* onto **E** using causal conjunctions. The number of events in a compound event (its "size" or "conjunctivity"), is called *cardinality*, *|***E***|* = *{*1*,* 2*,* 3*, ··· , n} ⇢ N* , which is always a natural number (positive integer), *i*=1 where events **E***<sup>i</sup>* are independent; when they are *not* independent, *conditional probabilities* are so cardinality (or "event size") is always a discrete variable. For example, compound event **S** in equation 2 ("a size 3 event") is first-order conjunctive with respect to its three causally necessary events.<sup>5</sup> Formally, we can summarize these ideas through the following expression: so cardinality (or "event size") is always a discrete variable. For example, compound event **S** in equation 2 ("a size 3 event") is first-order conjunctive with respect to its three causally necessary events.<sup>5</sup> Formally, we can summarize these ideas through the following expression: *<sup>i</sup>*=1 denote a set of *N* necessary events that produce **E**. Causal production of **E** is specified by an *event function*, **<sup>E</sup>** : *{***X***i} !* **E**, which maps necessary events in *{***X***i}* onto **E** using causal conjunctions. The number of events in a compound event (its "size" or "conjunctivity"), is called *cardinality*, *|***E***|* = *{*1*,* 2*,* 3*, ··· , n} ⇢ N* , which is always a natural number (positive integer), Pr(**E**) = Pr(**E**1) *·* Pr(**E**2) *· ... ·* Pr(**E***n*) = <sup>Y</sup>*<sup>n</sup> i*=1 Pr(**E***i*)*,* (4) (4) so cardinality (or "event size") is always a discrete variable. For example, compound event **S** in equation 2 ("a size 3 event") is first-order conjunctive with respect to its three causally necessary events.<sup>5</sup> Formally, we can summarize these ideas through the following expression: events, the joint probability *E* of compound event **E** is given by the following expression: Pr(**E***i*)*,* (4) Pr(**EE**1Pr(**<sup>E</sup>** Pr(**E***n*<sup>=</sup> <sup>Y</sup>*i*(4)NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS

necessary events—i.e., set-theoretic *intersection* (operator *\*) or Boolean logic *product* (operator

equation 2 ("a size 3 event") is first-order conjunctive with respect to its three causally necessary events.<sup>5</sup> Formally, we can summarize these ideas through the following expression: **<sup>E</sup>** *(* (**E**) = **<sup>E</sup>**<sup>1</sup> *^* **<sup>E</sup>**<sup>2</sup> *^ ··· ^* **<sup>E</sup>***<sup>n</sup>* <sup>=</sup> ^*<sup>n</sup>* <sup>5</sup> An event function is also called *indicator function*, *structure function*, or *production function*, in areas of mathematics, engineering, and economics, respectively. **<sup>E</sup>** *(* (**E**) = **<sup>E</sup>**<sup>1</sup> *^* **<sup>E</sup>**<sup>2</sup> *^ ··· ^* **<sup>E</sup>***<sup>n</sup>* <sup>=</sup> ^*<sup>n</sup> i*=1 **E***i.* (3) Next, an international event **E** has probability Pr (**E**) that is determined by the naturally uncertain occurrence of necessary conditions—a type of causation known as *probabilistic causality* (Salmon **<sup>E</sup>** *(* (**E**) = **<sup>E</sup>**<sup>1</sup> *^* **<sup>E</sup>**<sup>2</sup> *^ ··· ^* **<sup>E</sup>***<sup>n</sup>* <sup>=</sup> ^*<sup>n</sup> i*=1 **E***i.* (3) Next, an international event **E** has probability Pr (**E**) that is determined by the naturally uncertain occurrence of necessary conditions—a type of causation known as *probabilistic causality* (Salmon so cardinality (or "event size") is always a discrete variable. For example, compound event **S** in equation 2 ("a size 3 event") is first-order conjunctive with respect to its three causally necessary events.<sup>5</sup> Formally, we can summarize these ideas through the following expression: **<sup>E</sup>** *(* (**E**) = **<sup>E</sup>**<sup>1</sup> *^* **<sup>E</sup>**<sup>2</sup> *^ ··· ^* **<sup>E</sup>***<sup>n</sup>* <sup>=</sup> ^*<sup>n</sup>* where events **E***<sup>i</sup>* are independent; when they are *not* independent, *conditional probabilities* are used and, by Kolmogorov's theorem, they still multiply. Letting Pr(**E**) = *E* and Pr(**E***i*) = *pi*, since <sup>5</sup> An event function is also called *indicator function*, *structure function*, or *production function*, in areas of mathematics, engineering, and economics, respectively. where events **<sup>E</sup>** *(* (**E**) = **<sup>E</sup>**<sup>1</sup> *^* **<sup>E</sup>**<sup>2</sup> *^ ··· ^* **<sup>E</sup>***<sup>n</sup>* <sup>=</sup> ^*<sup>n</sup> i*=1 **E***i.* (3) Next, an international event **E** has probability Pr (**E**) that is determined by the naturally uncertain occurrence of necessary conditions—a type of causation known as *probabilistic causality* (Salmon are independent; when they are *not* independent, *conditional probabilities* are used and, by Kolmogorov's theorem, they still multiply. Letting *i*=1 where events **E***<sup>i</sup>* are independent; when they are *not* independent, *conditional probabilities* are used and, by Kolmogorov's theorem, they still multiply. Letting Pr(**E**) = *E* and Pr(**E***i*) = *pi*, since and are *probabilities*are used **E**) = ) *pi*since*E* and *pi* are variables, not events), we can rewrite equation (4) in simpler notation: CLAUDIO CIOFFI-REVILLA all probabilities are continuous variables (i.e., *E* and *p<sup>i</sup>* are variables, not events), we can rewrite equation (4) in simpler notation: CLAUDIO CIOFFI-REVILLA all probabilities are continuous variables (i.e., *E* and *p<sup>i</sup>* are variables, not events), we can rewrite equation (4) in simpler notation: Here we shall investigate three illustrative cases of IR theories and research areas—numbered 1, 3, and 6 in section .2.—where hybrid functions play a central role in describing and explaining political phenomena. The scientific purpose is to deepen our understanding and provide foundations for more advanced analysis. Each "case study" follows the analytical procedure just outlined in section

used and, by Kolmogorov's theorem, they still multiply. Letting Pr(**E**) = *E* and Pr(**E***i*) = *pi*, since

$$\Pr(\exists) = E \text{ and } \Pr(\exists\_i) = p\_i \text{ s.inse all probabilities are continuous variables} \newline (i.e. \newline \exists \text{ and } p\_i \text{ are variables}, \newline \text{not events}), \newline \text{we can rewrite equation (4) in simplex notation:} $$

$$E = p\_1 \cdot p\_2 \cdot \dots \cdot p\_n = \prod\_{i=1}^n p\_i \tag{5}$$

$$\mathbf{P}^{\mathcal{N}}\_{\mathcal{I}} = \mathbf{P}^{\mathcal{N}}\_{\mathcal{I}} \tag{6}$$

*i*=1

*i*=1

Pr(**E**) = Pr(**E**1) *·* Pr(**E**2) *· ... ·* Pr(**E***n*) = <sup>Y</sup>*<sup>n</sup>* where events **E***<sup>i</sup>* are independent; when they are *not* independent, *conditional probabilities* are where events **E***<sup>i</sup>* are independent; when they are *not* independent, *conditional probabilities* are used and, by Kolmogorov's theorem, they still multiply. Letting Pr(**E**) = *E* and Pr(**E***i*) = *pi*, since <sup>5</sup> An event function is also called *indicator function*, *structure function*, or *production function*, in areas of mathematics, engineering, and economics, respectively. where events **E***<sup>i</sup>* are independent; when they are *not* independent, *conditional probabilities* are used and, by Kolmogorov's theorem, they still multiply. Letting Pr(**E**) = *E* and Pr(**E***i*) = *pi*, since <sup>5</sup> An event function is also called *indicator function*, *structure function*, or *production function*, in areas of mathematics, engineering, and economics, respectively. events, the joint probability *E* of compound event **E** is given by the following expression: Pr(**E**) = Pr(**E**1) *·* Pr(**E**2) *· ... ·* Pr(**E***n*) = <sup>Y</sup>*<sup>n</sup> i*=1 Pr(**E***i*)*,* (4) where events **E***<sup>i</sup>* are independent; when they are *not* independent, *conditional probabilities* are used and, by Kolmogorov's theorem, they still multiply. Letting Pr(**E**) = *E* and Pr(**E***i*) = *pi*, since <sup>5</sup> An event function is also called *indicator function*, *structure function*, or *production function*, in areas of mathematics, engineering, and economics, respectively. where *P* is the probability of causal events and *N ⌘ |***E***|* is the cardinality or event size. Note that equation 6 is a *hybrid function*, because probability is continuous on the unit interval of real numbers [0*,* 1]—which lies between (and includes) impossibility (*E* = 0) and certainty (*E* = 1*.*0)—while event size is always a discrete or natural number (of necessary conditions). The politically relevant domain of interest—in this case—is bound by 0 *P* 1 and 2 *N* / 20 along causal probability and size dimensions in the hybrid (*p, c*) domain. Specifically, equation 6 is where *P* is the probability of causal events and where *P* is the probability of causal events and *N ⌘ |***E***|* is the cardinality or event size. Note that equation 6 is a *hybrid function*, because probability is continuous on the unit interval of real numbers [0*,* 1]—which lies between (and includes) impossibility (*E* = 0) and certainty (*E* = 1*.*0)—while event size is always a discrete or natural number (of necessary conditions). The politically relevant domain of interest—in this case—is bound by 0 *P* 1 and 2 *N* / 20 is the cardinality or event size. Note that equation 6 is a *hybrid function*, because probability is continuous on the unit interval of real numbers [0,1]—which lies between (and includes) impossibility (*E* = 0) and certainty (*E* = 1.0)—while event size is always a discrete or natural number (of necessary conditions). *p<sup>i</sup>* (5) = *P <sup>N</sup> ,* (6) where *P* is the probability of causal events and *N ⌘ |***E***|* is the cardinality or event size. Note that equation 6 is a *hybrid function*, because probability is continuous on the unit interval of real numbers [0*,* 1]—which lies between (and includes) impossibility (*E* = 0) and certainty (*E* = 1*.*0)—while This is a compound event because **S** requires the following set of causally necessary conditions, each of which constitutes an event by itself: 1. Neither state will pursue issues deemed as highly threatening to the other, as opposed to acting completely oblivious or independently of extant foreign interests. 2. Each state may prefer to negotiate over colliding interests, before escalating to war.

**S** *⌘* "a state of stable peace exists between two countries." (2)

used and, by Kolmogorov's theorem, they still multiply. Letting Pr(**E**) = *E* and Pr(**E***i*) = *pi*, since <sup>5</sup> An event function is also called *indicator function*, *structure function*, or *production function*, in areas of mathematics, engineering, and economics, respectively. where events **E***<sup>i</sup>* are independent; when they are *not* independent, *conditional probabilities* are used and, by Kolmogorov's theorem, they still multiply. Letting Pr(**E**) = *E* and Pr(**E***i*) = *pi*, since <sup>5</sup> An event function is also called *indicator function*, *structure function*, or *production function*, in areas of mathematics, engineering, and economics, respectively. 11 a hybrid function linearly dependent in *P* and exponentially in *N*, so this means that changes in either variables will cause different political effects on the probability of an event. The 3D surface graph of equation 6 is in Figure 1a, which shows values of the probability *E* of an international event rising through a steep escarpment as causal probability *P* increases and cardinality *N* decreases. The contour plot in Figure 1b shows the graph looking straight down, which highlights the ample basin floor where causal probability is very low (0 *P* / 0*.*9), flanked along causal probability and size dimensions in the hybrid (*p, c*) domain. Specifically, equation 6 is a hybrid function linearly dependent in *P* and exponentially in *N*, so this means that changes in either variables will cause different political effects on the probability of an event. The 3D surface graph of equation 6 is in Figure 1a, which shows values of the probability *E* of an international event rising through a steep escarpment as causal probability *P* increases and cardinality *N* decreases. The contour plot in Figure 1b shows the graph looking straight down, The politically relevant domain of interest—in this case—is bound by 0 ≤ *P* ≤ 1 and 2 ≤ *N*  The politically relevant domain of interest—in this case—is bound by 0 *P* 1 and 2 *N* / 20 along causal probability and size dimensions in the hybrid (*p, c*) domain. Specifically, equation 6 is a hybrid function linearly dependent in *P* and exponentially in *N*, so this means that changes in either variables will cause different political effects on the probability of an event. The 3D surface graph of equation 6 is in Figure 1a, which shows values of the probability *E* of 20 along causal probability and size dimensions in the hybrid (*p*, *n*) domain. Specifically, equation 6 is a hybrid function linearly dependent in *P* and exponentially in *N*, so this means that changes in either variables will cause different political effects on the probability of an event. 3. When states do negotiate, they may—depending on conditions—find a nonviolent resolution. The universal existence of such necessary conditions makes **S** a compound event, by definition. Specifically, every international event **E** is produced by causal *conjunction* (operator *^*) of necessary events—i.e., set-theoretic *intersection* (operator *\*) or Boolean logic *product* (operator AND). Let *{***X***i}<sup>n</sup> <sup>i</sup>*=1 denote a set of *N* necessary events that produce **E**. Causal production of **E** is

by the steep north-south escarpment along the east edge as *P !* 1*.*0. Note that change caused by each variable differs, as shown in Figures 1c through f. Calculating the partial derivative and partial difference of *E* with respect to *P* and *N*, respectively: which highlights the ample basin floor where causal probability is very low (0 *P* / 0*.*9), flanked by the steep north-south escarpment along the east edge as*P !* 1*.*0. Note that change caused by each variable differs, as shown in Figures 1c through f. Calculating an international event rising through a steep escarpment as causal probability *P* increases and cardinality *N* decreases. The contour plot in Figure 1b shows the graph looking straight down, which highlights the ample basin floor where causal probability is very low (0 *P* / 0*.*9), flanked The 3D surface graph of equation 6 is in Figure 1a, which shows values of the probability *E* of an international event rising through a steep escarpment as causal probability *P* increases and cardinality *N* decreases. The contour plot specified by an *event function*, **<sup>E</sup>** : *{***X***i} !* **E**, which maps necessary events in *{***X***i}* onto **E** using causal conjunctions. The number of events in a compound event (its "size" or "conjunctivity"), is called *cardinality*, *|***E***|* = *{*1*,* 2*,* 3*, ··· , n} ⇢ N* , which is always a natural number (positive integer), so cardinality (or "event size") is always a discrete variable. For example, compound event **S** in equation 2 ("a size 3 event") is first-order conjunctive with respect to its three causally necessary

either variable has *some* effect on *E* (since *@pE >* 0 and ∆*<sup>N</sup> E <* 0, per equations 7-8). To wit: Opposite political effects. Whereas change in causal probability *P* has proportional or positive effect on *E*, change in event size *N* has an opposite effect. This is shown by the purple deep bottom in Figure 1c versus the high red plateau in Figure 1e; both features observed over

First- and second-order effects. Whereas the contour plot of *E*(*P, N*) shows strictly concave isocontours (Figure 1b), the contours of the derivative *@pE* (Figure 1d) also show a convex, mild spur protruding on the southwestern wall of the escarpment, near *p* = 0*.*75 (third contour, between green and blue elevations), which means more complex change for small-size events. This is a second-order effect and not at all intuitive from the basic model (equation

*i*=1

Second-order effects caused by changes in event size. The graphs of the change in *E* with respect to event size, ∆*nE* (equation 8 and Figures 1e and f), show a pronounced ravine or canyon along the north-south direction dropping into a deep precipice at relatively high values of *P* as *N !* 2. Interestingly, in this case all the isolines have mixed concavity (low

Geometrically opposite extrema. The extreme high range of *@pE* in Figures 1c and d and the extreme low range of ∆*nE* in Figures 1e and f have opposite (or inverse) geometries with some common similarities: the former rises from a flat basin to an escarpment while the latter drops from a large plateau toward a deepening canyon that dives into a deep well. Both features are indicative of major political effects on event probability *E* caused by

underlying changes in causal probabilities *P* and event size *N*.

*P*) and convexity (higher *P* values), which is another second-order effect.

*P*) and convexity (higher *P* values), which is another second-order effect.

approximately the same underlying (*p, n*)-domain.

where events **E***<sup>i</sup>* are independent; when they are *not* independent, *conditional probabilities* are used and, by Kolmogorov's theorem, they still multiply. Letting Pr(**E**) = *E* and Pr(**E***i*) = *pi*, since <sup>5</sup> An event function is also called *indicator function*, *structure function*, or *production function*, in areas of

Next, an international event **E** has probability Pr (**E**) that is determined by the naturally uncertain occurrence of necessary conditions—a type of causation known as *probabilistic causality* (Salmon 1980; Suppes 1984; Eels 1991). Based on Kolmogorov's (1933) fundamental theorem of compound events, the joint probability *E* of compound event **E** is given by the following expression:

Pr(**E**) = Pr(**E**1) *·* Pr(**E**2) *· ... ·* Pr(**E***n*) = <sup>Y</sup>*<sup>n</sup>*

underlying changes in causal probabilities *P* and event size *N*.

*@pE* = *NP <sup>N</sup>−*<sup>1</sup> (for *P*'s effect on *E*, in Figures 1c and d) (7)

the partial derivative and partial difference of *E* with respect to *P* and *N*, respectively:

on the probability of an international event *E*, besides the trivial observation that *any* change on either variable has *some* effect on *E* (since *@pE >* 0 and ∆*<sup>N</sup> E <* 0, per equations 7-8). To wit: Opposite political effects. Whereas change in causal probability *P* has proportional or positive effect on *E*, change in event size *N* has an opposite effect. This is shown by the purple deep bottom in Figure 1c versus the high red plateau in Figure 1e; both features observed over

Pr(**E***i*)*,* (4)

11

First- and second-order effects. Whereas the contour plot of *E*(*P, N*) shows strictly concave isocontours (Figure 1b), the contours of the derivative *@pE* (Figure 1d) also show a convex, mild spur protruding on the southwestern wall of the escarpment, near *p* = 0*.*75 (third contour, between green and blue elevations), which means more complex change for small-size events. This is a second-order effect and not at all intuitive from the basic model (equation

Second-order effects caused by changes in event size. The graphs of the change in *E* with respect to event size, ∆*nE* (equation 8 and Figures 1e and f), show a pronounced ravine or canyon along the north-south direction dropping into a deep precipice at relatively high values of *P* as *N !* 2. Interestingly, in this case all the isolines have mixed concavity (low

Geometrically opposite extrema. The extreme high range of *@pE* in Figures 1c and d and the extreme low range of ∆*nE* in Figures 1e and f have opposite (or inverse) geometries with some common similarities: the former rises from a flat basin to an escarpment while the latter drops from a large plateau toward a deepening canyon that dives into a deep well. Both features are indicative of major political effects on event probability *E* caused by

NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS

Here we shall investigate three illustrative cases of IR theories and research areas—numbered 1, 3, and 6 in section .2.—where hybrid functions play a central role in describing and explaining political phenomena. The scientific purpose is to deepen our understanding and provide foundations for more advanced analysis. Each "case study" follows the analytical procedure just outlined in section

NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS

All international events in the real world are "compound" because they are always produced by necessary conditions specific to the event. Such causal necessity is universal—a fundamental axiom

**S** *⌘* "a state of stable peace exists between two countries." (2)

*<sup>i</sup>*=1 denote a set of *N* necessary events that produce **E**. Causal production of **E** is

*i*=1

Pr(**E***i*)*,* (4)

**E***i.* (3)

*i*=1

11

11

Pr(**E***i*)*,* (4)

11

*i*=1

11

**E***i.* (3)

Pr(**E***i*)*,* (4)

11

11

so cardinality (or "event size") is always a discrete variable. For example, compound event **S** in

<sup>∆</sup>*n<sup>E</sup>* <sup>=</sup> *<sup>P</sup> <sup>N</sup>*+1 *<sup>−</sup> <sup>P</sup> <sup>N</sup>* (for *<sup>N</sup>*'s effect on *<sup>E</sup>*, in Figures 1e and f)*.* (8) *@pE* = *NP <sup>N</sup>−*<sup>1</sup> (for *P*'s effect on *E*, in Figures 1c and d) (7) <sup>∆</sup>*n<sup>E</sup>* <sup>=</sup> *<sup>P</sup> <sup>N</sup>*+1 *<sup>−</sup> <sup>P</sup> <sup>N</sup>* (for *<sup>N</sup>*'s effect on *<sup>E</sup>*, in Figures 1e and f)*.* (8) *@pE* = *NP <sup>N</sup>−*<sup>1</sup> (for *P*'s effect on *E*, in Figures 1c and d) (7) <sup>5</sup> An event function **<sup>E</sup>** *(* (**E**) = **<sup>E</sup>**<sup>1</sup> *^* **<sup>E</sup>**<sup>2</sup> *^ ··· ^* **<sup>E</sup>***<sup>n</sup>* <sup>=</sup> ^*<sup>n</sup> i*=1 **E***i.* (3) is also called *indicator function*, *structure function*, or *production function*, in areas of mathematics, engineering, and economics, respectively.

CLAUDIO CIOFFI-REVILLA

all probabilities are continuous variables (i.e., *E* and *p<sup>i</sup>* are variables, not events), we can rewrite

probabilities are continuous variables (i.e.,*<sup>E</sup> <sup>p</sup><sup>i</sup>* are variables, not we can rewriteequation (4) in simpler notation:

*<sup>E</sup>* <sup>=</sup> *<sup>p</sup>*<sup>1</sup> *· <sup>p</sup>*<sup>2</sup> *· ... · <sup>p</sup><sup>n</sup>* <sup>=</sup> <sup>Y</sup>*<sup>n</sup>*

*<sup>E</sup> <sup>p</sup>*<sup>1</sup> *· <sup>p</sup>*2*· ...·p<sup>n</sup>* <sup>=</sup> <sup>Y</sup>*<sup>n</sup>*

where *P* is the probability of causal events and *N ⌘ |***E***|* is the cardinality or event size. Note that equation 6 is a *hybrid function*, because probability is continuous on the unit interval of real numbers [0*,* 1]—which lies between (and includes) impossibility (*E* = 0) and certainty (*E* = 1*.*0)—while

where*<sup>P</sup>* is the probability of causal and *⌘ <sup>|</sup>***E***<sup>|</sup>* is the cardinality or event size. Note thatequation is a*hybrid function*, because probability is continuous on the unit interval of numbers

along causal probability and size dimensions in the hybrid (*p, c*) domain. Specifically, equation 6 is

equation (4) in simpler notation:

CLAUDIO CIOFFI-REVILLA

equation (4) in simpler notation:

all probabilities are continuous variables (i.e., *E* and *p<sup>i</sup>* are variables, not events), we can rewrite

*i*=1

*i*=1

*<sup>E</sup>* <sup>=</sup> *<sup>p</sup>*<sup>1</sup> *· <sup>p</sup>*<sup>2</sup> *· ... · <sup>p</sup><sup>n</sup>* <sup>=</sup> <sup>Y</sup>*<sup>n</sup>*

equation 6 is a *hybrid function*, because probability is continuous on the unit interval of real numbers

cardinality *N* decreases. The contour plot in Figure 1b shows the graph looking straight down,

*i*=1

= *P <sup>N</sup> ,* (6)

*P <sup>N</sup> ,* (6)

either variables will cause different political effects on the probability of an event.

event size is always a discrete or natural number (of necessary conditions).

= *P <sup>N</sup> ,* (6)

where *P* is the probability of causal events and *N ⌘ |***E***|* is the cardinality or event size. Note that equation 6 is a *hybrid function*, because probability is continuous on the unit interval of real numbers [0*,* 1]—which lies between (and includes) impossibility (*E* = 0) and certainty (*E* = 1*.*0)—while

*p<sup>i</sup>* (5)

CLAUDIO CIOFFI-REVILLA

*p<sup>i</sup>* (5)

all probabilities are continuous variables (i.e., *E* and *p<sup>i</sup>* are variables, not events), we can rewrite

*<sup>E</sup>* <sup>=</sup> *<sup>p</sup>*<sup>1</sup> *· <sup>p</sup>*<sup>2</sup> *· ... · <sup>p</sup><sup>n</sup>* <sup>=</sup> <sup>Y</sup>*<sup>n</sup>*

event size is always a discrete or natural number (of necessary conditions).

CLAUDIO CIOFFI-REVILLA

CLAUDIO CIOFFI-REVILLA

CLAUDIO CIOFFI-REVILLA

equation (4) in simpler notation:

equation (4) in simpler notation:

in Figure 1b shows the graph looking straight down, which highlights the ample basin fl oor where causal probability is very low (0≤ *P*  [0*,* 1]—which lies between (and includes) impossibility (*E* = 0) and certainty (*E* = 1*.*0)—while event size is always a discrete or natural number (of necessary conditions). The politically relevant domain of interest—in this case—is bound by 0 *P* 1 and 2 *N* / 20 along causal probability and size dimensions in the hybrid (*p, c*) domain. Specifically, equation 6 is ≤0.9), fl anked by the steep north-south escarpment along the east edge as P→1.0. The 3D surface graph of equation 6 is in Figure 1a, which shows values of the probability *E* of an international event rising through a steep escarpment as causal probability *P* increases and cardinality *N* decreases. The contour plot in Figure 1b shows the graph looking straight down, The 3D surface graph of equation 6 is in Figure 1a, which shows values of the probability *E* of an international event rising through a steep escarpment as causal probability *P* increases and cardinality *N* decreases. The contour plot in Figure 1b shows the graph looking straight down, a hybrid function linearly dependent in *P* and exponentially in *N*, so this means that changes in either variables will cause different political effects on the probability of an event. The 3D surface graph of equation 6 is in Figure 1a, which shows values of the probability *E* of a hybrid function linearly dependent in *<sup>P</sup>* and exponentially in*<sup>N</sup>*, so this means that changes ineither variables will cause different political of an event. The 3D surface graph of equation 6is in Figure 1a, which shows values probability*E* of = *P <sup>N</sup> ,* (6) where *P* is the probability of causal events and *N ⌘ |***E***|* is the cardinality or event size. Note that equation 6 is a *hybrid function*, because probability is continuous on the unit interval of real numbers CLAUDIO CIOFFI-REVILLA

either variables will cause different political effects on the probability of an event.

The politically relevant domain of interest—in this case—is bound by 0 *P* 1 and 2 *N* /20

*p<sup>i</sup>* (5)

The politically relevant domain of interest—in this case—is bound by 0 *P* 1 and 2 *N* / 20

equation (4) in simpler notation:

where *P* is the probability of causal events and *N ⌘ |***E***|* is the cardinality or event size. Note that equation 6 is a *hybrid function*, because probability is continuous on the unit interval of real numbers [0*,* 1]—which lies between (and includes) impossibility (*E* = 0) and certainty (*E* = 1*.*0)—while

*i*=1

all probabilities are continuous variables (i.e., *E* and *p<sup>i</sup>* are variables, not events), we can rewrite

*<sup>E</sup>* <sup>=</sup> *<sup>p</sup>*<sup>1</sup> *· <sup>p</sup>*<sup>2</sup> *· ... · <sup>p</sup><sup>n</sup>* <sup>=</sup> <sup>Y</sup>*<sup>n</sup>*

*i*=1

= *P <sup>N</sup> ,* (6)

= *P <sup>N</sup> ,* (6)

*pi*(5)

*p<sup>i</sup>* (5)

all probabilities are continuous variables (i.e., *E* and *p<sup>i</sup>* are variables, not events), we can rewrite

*<sup>E</sup>* <sup>=</sup> *<sup>p</sup>*<sup>1</sup> *· <sup>p</sup>*<sup>2</sup> *· ... · <sup>p</sup><sup>n</sup>* <sup>=</sup> <sup>Y</sup>*<sup>n</sup>*

along causal probability and size dimensions in the hybrid (*p, c*) domain. Specifically, equation 6 is

all probabilities are continuous variables (i.e., *E* and *p<sup>i</sup>* are variables, not events), we can rewrite

an international event rising through a steep escarpment as causal probability *P* increases and

*i*=1

*i*=1

with some common similarities: the former rises from a flat basin to an escarpment while the latter drops from a large plateau toward a deepening canyon that dives into a deep well. Both features are indicative of major political effects on event probability *E* caused by

effect on *E*, change in event size *N* has an opposite effect. This is shown by the purple deep bottom in Figure 1c versus the high red plateau in Figure 1e; both features observed over

contour, between green and blue elevations), which means more complex change for small-size events. This is a second-order effect and not at all intuitive from the basic model (equation

First- and second-order effects. Whereas the contour plot of *E*(*P, N*) shows strictly concave isocontours (Figure 1b), the contours of the derivative *@pE* (Figure 1d) also show a convex, mild spur protruding on the southwestern wall of the escarpment, near *p* = 0*.*75 (third contour, between green and blue elevations), which means more complex change for small-size events. This is a second-order effect and not at all intuitive from the basic model (equation

Second-order effects caused by changes in event size. The graphs of the change in *E* with respect to event size, ∆*nE* (equation 8 and Figures 1e and f), show a pronounced ravine or canyon along the north-south direction dropping into a deep precipice at relatively high values of *P* as *N !* 2. Interestingly, in this case all the isolines have mixed concavity (low

Geometrically opposite extrema. The extreme high range of *@pE* in Figures 1c and d and the extreme low range of ∆*nE* in Figures 1e and f have opposite (or inverse) geometries with some common similarities: the former rises from a flat basin to an escarpment while the latter drops from a large plateau toward a deepening canyon that dives into a deep well. Both features are indicative of major political effects on event probability *E* caused by

underlying changes in causal probabilities *P* and event size *N*.

Second-order effects caused by changes in event size. The graphs of the change in *E* with respect to event size, ∆*nE* (equation 8 and Figures 1e and f), show a pronounced ravine or canyon along the north-south direction dropping into a deep precipice at relatively high values of *P* as *N !* 2. Interestingly, in this case all the isolines have mixed concavity (low

Geometrically opposite extrema. The extreme high range of *@pE* in Figures 1c and d and the extreme low range of ∆*nE* in Figures 1e and f have opposite (or inverse) geometries with some common similarities: the former rises from a flat basin to an escarpment while the latter drops from a large plateau toward a deepening canyon that dives into a deep well. Both features are indicative of major political effects on event probability *E* caused by

*P*) and convexity (higher *P* values), which is another second-order effect.

underlying changes in causal probabilities *P* and event size *N*.

*P*) and convexity (higher *P* values), which is another second-order effect.

approximately the same underlying (*p, n*)-domain.

underlying changes in causal probabilities *P* and event size *N*.

underlying changes in causal probabilities *P* and event size *N*.

Second-order effects caused by changes in event size. The graphs of the change in *E* with respect to event size, ∆*nE* (equation 8 and Figures 1e and f), show a pronounced ravine or canyon along the north-south direction dropping into a deep precipice at relatively high values of *P* as *N !* 2. Interestingly, in this case all the isolines have mixed concavity (low

isocontours (Figure 1b), the contours of the derivative *@pE* (Figure 1d) also show a convex, mild spur protruding on the southwestern wall of the escarpment, near *p* = 0*.*75 (third contour, between green and blue elevations), which means more complex change for small-size events. This is a second-order effect and not at all intuitive from the basic model (equation

Geometrically opposite extrema. The extreme high range of *@pE* in Figures 1c and d and the extreme low range of ∆*nE* in Figures 1e and f have opposite (or inverse) geometries with some common similarities: the former rises from a flat basin to an escarpment while the latter drops from a large plateau toward a deepening canyon that dives into a deep well. Both features are indicative of major political effects on event probability *E* caused by

*P*) and convexity (higher *P* values), which is another second-order effect.

*i*=1

*p<sup>i</sup>* (5)

*p<sup>i</sup>* (5)

*p<sup>i</sup>* (5)

*p<sup>i</sup>* (5)

a hybrid function linearly dependent in *P* and exponentially in *N*, so this means that changes in either variables will cause different political effects on the probability of an event. The 3D surface graph of equation 6 is in Figure 1a, which shows values of the probability *E* of an international event rising through a steep escarpment as causal probability *P* increases and Note that change caused by each variable diff ers, as shown in Figures 1c through f. Calculating the partial derivative and partial diff erence of *E* with respect to *P* and *N*, respectively: which highlights the ample basin floor where causal probability is very low (0 *P* / 0*.*9), flanked by the steep north-south escarpment along the east edge as *P !* 1*.*0. Note that change caused by each variable differs, as shown in Figures 1c through f. Calculating the partial derivative and partial difference of *E* with respect to *P* and *N*, respectively: which highlights the ample basin floor where causal probability is very low (0 *P* / 0*.*9), flanked by the steep north-south escarpment along the east edge as *P !* 1*.*0. Note that change caused by each variable differs, as shown in Figures 1c through f. Calculating the partial derivative and partial difference of *E* with respect to *P* and *N*, respectively: an international event rising through a steep escarpment as causal probability *P* increases and cardinality *<sup>N</sup>* decreases. The contour plot in Figure 1b shows the graph looking straight down,which highlights the ample basin floor where causal probability is very low (<sup>0</sup> *<sup>P</sup>* / <sup>0</sup>*.*9), flanked by the steep north-south escarpment along the east edge as *P !* 1*.*0. an international event rising through a steep escarpment as causal probability *<sup>P</sup>* increases andcardinality decreases. The contour in Figure 1b shows the graph looking straight down,which ample basin floor where causal probability is very low (<sup>0</sup>*<sup>P</sup>* /*.*9), flanked steep north-south escarpment along the east edge *P*1*.*0. [0*,* 1]—which lies between (and includes) impossibility (*E* = 0) and certainty (*E* = 1*.*0)—while event size is always a discrete or natural number (of necessary conditions). The politically relevant domain of interest—in this case—is bound by 0 *P* 1 and 2 *N* / 20 all probabilities are continuous variables (i.e., *E* and *p<sup>i</sup>* are variables, not events), we can rewrite equation (4) in simpler notation: CLAUDIO CIOFFI-REVILLA

which highlights the ample basin floor where causal probability is very low (0 *P* / 0*.*9), flanked by the steep north-south escarpment along the east edge as *P !* 1*.*0. Note that change caused by each variable differs, as shown in Figures 1c through f. Calculating the partial derivative and partial difference of *E* with respect to *P* and *N*, respectively: *@pE* = *NP <sup>N</sup>−*<sup>1</sup> (for *P*'s effect on *E*, in Figures 1c and d) (7) <sup>∆</sup>*n<sup>E</sup>* <sup>=</sup> *<sup>P</sup> <sup>N</sup>*+1 *<sup>−</sup> <sup>P</sup> <sup>N</sup>* (for *<sup>N</sup>*'s effect on *<sup>E</sup>*, in Figures 1e and f)*.* (8) *@pE* = *NP <sup>N</sup>−*<sup>1</sup> (for *P*'s effect on *E*, in Figures 1c and d) (7) <sup>∆</sup>*n<sup>E</sup>* <sup>=</sup> *<sup>P</sup> <sup>N</sup>*+1 *<sup>−</sup>P<sup>N</sup>* (for *<sup>N</sup>*'s effect on *<sup>E</sup>*, in Figures 1e and f)*.* (8) (7) (8) Note that change caused by each variable differs, as shown in Figures 1c through f. Calculating the partial derivative and partial difference of *E* with respect to *P* and *N*, respectively: *@pE* = *NP <sup>N</sup>−*<sup>1</sup> (for *P*'s effect on *E*, in Figures 1c and d) (7) Note that change caused by each variable differs, as shown in Figures 1c through f. Calculating the partial derivative and partial difference of *E* with respect to*P* and*N*, respectively: *@pE*= *NP <sup>N</sup>−*<sup>1</sup> (for*P*'s effect on *E*, in Figures1c and d) (7) a hybrid function linearly dependent in *P* and exponentially in *N*, so this means that changes in either variables will cause different political effects on the probability of an event. The 3D surface graph of equation 6 is in Figure 1a, which shows values of the probability *E* of *<sup>E</sup>* <sup>=</sup> *<sup>p</sup>*<sup>1</sup> *· <sup>p</sup>*<sup>2</sup> *· ... · <sup>p</sup><sup>n</sup>* <sup>=</sup> <sup>Y</sup>*<sup>n</sup>* equation (4) in simpler notation: *<sup>E</sup>* <sup>=</sup> *<sup>p</sup>*<sup>1</sup> *· <sup>p</sup>*<sup>2</sup> *· ... · <sup>p</sup><sup>n</sup>* <sup>=</sup> <sup>Y</sup>*<sup>n</sup>* CLAUDIO CIOFFI-REVILLA

*@pE* = *NP <sup>N</sup>−*<sup>1</sup> (for *P*'s effect on *E*, in Figures 1c and d) (7) <sup>∆</sup>*n<sup>E</sup>* <sup>=</sup> *<sup>P</sup> <sup>N</sup>*+1 *<sup>−</sup> <sup>P</sup> <sup>N</sup>* (for *<sup>N</sup>*'s effect on *<sup>E</sup>*, in Figures 1e and f)*.* (8) We see immediately that changes in causal probability *P* and event size *N* have different effects on the probability of an international event *E*, besides the trivial observation that *any* change on We see immediately that changes in causal probability *P* and event size *N* have different effects on the probability of an international event *E*, besides the trivial observation that *any* change on either variable has *some* effect on *E* (since *@pE >* 0 and ∆*<sup>N</sup> E <* 0, per equations 7-8). To wit: Opposite political effects. Whereas change in causal probability *P* has proportional or positive We see immediately that changes in causal probability *P* and event size *N* have different effects on the probability of an international event *E*, besides the trivial observation that *any* change on either variable has *some* effect on *E* (since *@pE >* 0 and ∆*<sup>N</sup> E <* 0, per equations 7-8). To wit: Opposite political effects. Whereas change in causal probability *P* has proportional or positive We see immediately that changes in causal probability *P* and event size *N* have diff erent eff ects on the probability of an international event *E*, besides the trivial observation that *any* change on either variable has *some* eff ect on *E* (since <sup>∆</sup>*n<sup>E</sup>* <sup>=</sup> *<sup>P</sup> <sup>N</sup>*+1 *<sup>−</sup> <sup>P</sup> <sup>N</sup>* (for *<sup>N</sup>*'s effect on *<sup>E</sup>*, in Figures 1e and f)*.* (8) We see immediately that changes in causal probability *P* and event size *N* have different effects on the probability of an international event *E*, besides the trivial observation that *any* change on either variable has *some* effect on *E* (since *@pE >* 0 and ∆*<sup>N</sup> E <* 0, per equations 7-8). To wit: ≠ and <sup>∆</sup>*nE*<sup>=</sup> *<sup>P</sup><sup>N</sup>*+1*−<sup>P</sup> <sup>N</sup>* (foreffect on*E*in Figures and f)*.* (8) We immediately that changes in probability *P* and event size *N* have different on the probability of an international event*E*, besides the trivial observation that *any* change oneither variable has *some* effect on *<sup>E</sup>* (since *@pE >* <sup>0</sup> and <sup>∆</sup>*NE <* per equations 7-8). To wit: cardinality *N* decreases. The contour plot in Figure 1b shows the graph looking straight down, which highlights the ample basin floor where causal probability is very low (0 *P* / 0*.*9), flanked by the steep north-south escarpment along the east edge as *P !* 1*.*0. Note that change caused by each variable differs, as shown in Figures 1c through f. Calculating the partial derivative and partial difference of *E* with respect to *P* and *N*, respectively: = *P <sup>N</sup> ,* (6) where *P* is the probability of causal events and *N ⌘ |***E***|* is the cardinality or event size. Note that equation 6 is a *hybrid function*, because probability is continuous on the unit interval of real numbers [0*,* 1]—which lies between (and includes) impossibility (*E* = 0) and certainty (*E* = 1*.*0)—while *i*=1 = *P <sup>N</sup> ,* (6) where *P* is the probability of causal events and *N ⌘ |***E***|* is the cardinality or event size. Note that equation 6 is a *hybrid function*, because probability is continuous on the unit interval of real numbers all probabilities are continuous variables (i.e., *E* and *p<sup>i</sup>* are variables, not events), we can rewrite equation (4) in simpler notation: *<sup>E</sup>* <sup>=</sup> *<sup>p</sup>*<sup>1</sup> *· <sup>p</sup>*<sup>2</sup> *· ... · <sup>p</sup><sup>n</sup>* <sup>=</sup> <sup>Y</sup>*<sup>n</sup>*

either variable has *some* effect on *E* (since *@pE >* 0 and ∆*<sup>N</sup> E <* 0, per equations 7-8). To wit: Opposite political effects. Whereas change in causal probability *P* has proportional or positive effect on *E*, change in event size *N* has an opposite effect. This is shown by the purple deep bottom in Figure 1c versus the high red plateau in Figure 1e; both features observed over approximately the same underlying (*p, n*)-domain. effect on *E*, change in event size *N* has an opposite effect. This is shown by the purple deep bottom in Figure 1c versus the high red plateau in Figure 1e; both features observed over approximately the same underlying (*p, n*)-domain. First- and second-order effects. Whereas the contour plot of *E*(*P, N*) shows strictly concave isocontours (Figure 1b), the contours of the derivative *@pE* (Figure 1d) also show a convex, effect on *E*, change in event size *N* has an opposite effect. This is shown by the purple deep bottom in Figure 1c versus the high red plateau in Figure 1e; both features observed over approximately the same underlying (*p, n*)-domain. First- and second-order effects. Whereas the contour plot of *E*(*P, N*)shows strictly concave isocontours (Figure 1b), the contours of the derivative *@pE* (Figure1d) also show a convex, Opposite political effects. Whereas change in causal probability *P* has proportional or positive effect on *E*, change in event size *N* has an opposite effect. This is shown by the purple deep bottom in Figure 1c versus the high red plateau in Figure 1e; both features observed over approximately the same underlying (*p, n*)-domain. Opposite political effects. Whereas change in causal probability *<sup>P</sup>* has proportional or effect on *<sup>E</sup>*, change in event size *<sup>N</sup>* has an effect. This shown by the purple deep bottom in Figure 1c versus the high red plateau in Figure 1e; both features observed over approximately the same underlying (*p, n*)-domain.**Opposite political eff ects.** Whereas change in causal probability *P* has proportional or positive eff ect on *E*, change in event size *N* has an opposite eff ect. Th is is shown by the purple deep bott om in Figure 1c versus the high red plateau in Figure 1e; both features observed over approximately the same underlying (*p*, *n*)-domain. *@pE* = *NP <sup>N</sup>−*<sup>1</sup> (for *P*'s effect on *E*, in Figures 1c and d) (7) <sup>∆</sup>*n<sup>E</sup>* <sup>=</sup> *<sup>P</sup> <sup>N</sup>*+1 *<sup>−</sup> <sup>P</sup> <sup>N</sup>* (for *<sup>N</sup>*'s effect on *<sup>E</sup>*, in Figures 1e and f)*.* (8) We see immediately that changes in causal probability *P* and event size *N* have different effects event size is always a discrete or natural number (of necessary conditions). The politically relevant domain of interest—in this case—is bound by 0 *P* 1 and 2 *N* / 20 along causal probability and size dimensions in the hybrid (*p, c*) domain. Specifically, equation 6 is a hybrid function linearly dependent in *P* and exponentially in *N*, so this means that changes in either variables will cause different political effects on the probability of an event. The 3D surface graph of equation 6 is in Figure 1a, which shows values of the probability *E* of [0*,* 1]—which lies between (and includes) impossibility (*E* = 0) and certainty (*E* = 1*.*0)—while event size is always a discrete or natural number (of necessary conditions). The politically relevant domain of interest—in this case—is bound by 0 *P* 1 and 2 *N* / 20 along causal probability and size dimensions in the hybrid (*p, c*) domain. Specifically, equation 6 is a hybrid function linearly dependent in *P* and exponentially in *N*, so this means that changes in = *P <sup>N</sup> ,* (6) where *P* is the probability of causal events and *N ⌘ |***E***|* is the cardinality or event size. Note that equation 6 is a *hybrid function*, because probability is continuous on the unit interval of real numbers

First- and second-order effects. Whereas the contour plot of *E*(*P, N*) shows strictly concave isocontours (Figure 1b), the contours of the derivative *@pE* (Figure 1d) also show a convex, mild spur protruding on the southwestern wall of the escarpment, near *p* = 0*.*75 (third contour, between green and blue elevations), which means more complex change for small-size events. This is a second-order effect and not at all intuitive from the basic model (equation 6 and Figures 1a and b). Second-order effects caused by changes in event size. The graphs of the change in *E* with respect to event size, ∆*nE* (equation 8 and Figures 1e and f), show a pronounced ravine mild spur protruding on the southwestern wall of the escarpment, near *p* = 0*.*75 (third contour, between green and blue elevations), which means more complex change for small-size events. This is a second-order effect and not at all intuitive from the basic model (equation 6 and Figures 1a and b). Second-order effects caused by changes in event size. The graphs of the change in *E* with respect to event size, ∆*nE* (equation 8 and Figures 1e and f), show a pronounced ravine or canyon along the north-south direction dropping into a deep precipice at relatively high mild spur protruding on the southwestern wall of the escarpment, near *p* = 0*.*75 (third contour, between green and blue elevations), which means more complex change for small-size events. This is a second-order effect and not at all intuitive from the basic model (equation 6 and Figures 1a and b). Second-order effects caused by changes in event size. The graphs of the change in *E* with respect to event size, ∆*nE* (equation 8 and Figures 1e and f), show a pronounced ravine or canyon along the north-south direction dropping into a deep precipice at relatively high First- and second-order effects. Whereas the contour plot of *E*(*P, N*) shows strictly concave isocontours (Figure 1b), the contours of the derivative *@pE* (Figure 1d) also show a convex, mild spur protruding on the southwestern wall of the escarpment, near *p* = 0*.*75 (third contour, between green and blue elevations), which means more complex change for small-size events. This is a second-order effect and not at all intuitive from the basic model (equation Second-order effects caused by changes in event size. The graphs of the change in *E* with First- and second-order effects. Whereas the contour plot *<sup>E</sup>*(*P, N*) shows strictly concave1b), the contours of derivative *@E* (Figure 1d) also show a convex,spur protruding on the southwestern wall of the escarpment, near*<sup>p</sup> .*<sup>75</sup> green and blue elevations), which means more complex change for small-sizeThis is a second-order effect and not at all intuitive from basic model Second-order effects caused by changes in event The graphs of the change in *E* 12<sup>≠</sup> , per equations 7-8). To wit: **First- and second-order eff ects***.* Whereas the contour plot of *E*(*P*, *N*) shows strictly concave isocontours (Figure 1b), the contours of the derivative on the probability of an international event *E*, besides the trivial observation that *any* change on either variable has *some* effect on *E* (since *@pE >* 0 and ∆*<sup>N</sup> E <* 0, per equations 7-8). To wit: Opposite political effects. Whereas change in causal probability *P* has proportional or positive effect on *E*, change in event size *N* has an opposite effect. This is shown by the purple deep bottom in Figure 1c versus the high red plateau in Figure 1e; both features observed over approximately the same underlying (*p, n*)-domain. (Figure 1d) also show a convex, mild spur protruding on the southwestern wall of the escarpment, near *p* = 0.75 (third contour, between green and blue elevations), which means more complex change for small-size events. Th is is a second-order eff ect and not at all intuitive from the basic model (equation 6 and Figures 1a and b). an international event rising through a steep escarpment as causal probability *P* increases and cardinality *N* decreases. The contour plot in Figure 1b shows the graph looking straight down, which highlights the ample basin floor where causal probability is very low (0 *P* / 0*.*9), flanked by the steep north-south escarpment along the east edge as *P !* 1*.*0. Note that change caused by each variable differs, as shown in Figures 1c through f. Calculating the partial derivative and partial difference of *E* with respect to *P* and *N*, respectively: either variables will cause different political effects on the probability of an event. The 3D surface graph of equation 6 is in Figure 1a, which shows values of the probability *E* of an international event rising through a steep escarpment as causal probability *P* increases and cardinality *N* decreases. The contour plot in Figure 1b shows the graph looking straight down, which highlights the ample basin floor where causal probability is very low (0 *P* / 0*.*9), flanked by the steep north-south escarpment along the east edge as *P !* 1*.*0. Note that change caused by each variable differs, as shown in Figures 1c through f. Calculating the partial derivative and partial difference of *E* with respect to *P* and *N*, respectively: [0*,* 1]—which lies between (and includes) impossibility (*E* = 0) and certainty (*E* = 1*.*0)—while event size is always a discrete or natural number (of necessary conditions). The politically relevant domain of interest—in this case—is bound by 0 *P* 1 and 2 *N* / 20 along causal probability and size dimensions in the hybrid (*p, c*) domain. Specifically, equation 6 is a hybrid function linearly dependent in *P* and exponentially in *N*, so this means that changes in either variables will cause different political effects on the probability of an event. The 3D surface graph of equation 6 is in Figure 1a, which shows values of the probability *E* of an international event rising through a steep escarpment as causal probability *P* increases and

or canyon along the north-south direction dropping into a deep precipice at relatively high values of *P* as *N !* 2. Interestingly, in this case all the isolines have mixed concavity (low *P*) and convexity (higher *P* values), which is another second-order effect. Geometrically opposite extrema. The extreme high range of *@pE* in Figures 1c and d and the extreme low range of ∆*nE* in Figures 1e and f have opposite (or inverse) geometries with some common similarities: the former rises from a flat basin to an escarpment while values of *P* as *N !* 2. Interestingly, in this case all the isolines have mixed concavity (low *P*) and convexity (higher *P* values), which is another second-order effect. Geometrically opposite extrema. The extreme high range of *@pE* in Figures 1c and d and the extreme low range of ∆*nE* in Figures 1e and f have opposite (or inverse) geometries with some common similarities: the former rises from a flat basin to an escarpment while values of *P* as *N !* 2. Interestingly, in this case all the isolines have mixed concavity (low *P*) and convexity (higher *P* values), which is another second-order effect. Geometrically opposite extrema. The extreme high range of *@pE* in Figures 1c and d and the extreme low range of ∆*nE* in Figures 1e and f have opposite (or inverse) geometries with some common similarities: the former rises from a flat basin to an escarpment while respect to event size, ∆*nE* (equation 8 and Figures 1e and f), show a pronounced ravine or canyon along the north-south direction dropping into a deep precipice at relatively high values of *P* as *N !* 2. Interestingly, in this case all the isolines have mixed concavity (low *P*) and convexity (higher *P* values), which is another second-order effect. Geometrically opposite extrema. The extreme high range of *@pE* in Figures 1c and d and the extreme low range of ∆*nE* in Figures 1e and f have opposite (or inverse) geometries to event size, <sup>∆</sup>*n<sup>E</sup>* (equation and Figures1e and f), show a or canyon along the north-south direction dropping into a deep precipice at relatively high values of*<sup>P</sup>* as *<sup>N</sup> !* . Interestingly, in this case all the isolines have mixed concavity (low*P*) and convexity (higher *<sup>P</sup>* values), which another second-order effect. opposite extrema. *<sup>p</sup><sup>E</sup>* in Figures 1c and d andlow range of<sup>∆</sup>*E*in Figures 1e have opposite (or First- and second-order effects. Whereas the contour plot of *E*(*P, N*) shows strictly concave isocontours (Figure 1b), the contours of the derivative *@pE* (Figure 1d) also show a convex, mild spur protruding on the southwestern wall of the escarpment, near *p* = 0*.*75 (third contour, between green and blue elevations), which means more complex change for small-size events. This is a second-order effect and not at all intuitive from the basic model (equation 6 and Figures 1a and b). **Second-order eff ects caused by changes in event size.**Th e graphs of the change in *E* with respect to event size, *@pE* = *NP <sup>N</sup>−*<sup>1</sup> (for *P*'s effect on *E*, in Figures 1c and d) (7) <sup>∆</sup>*n<sup>E</sup>* <sup>=</sup> *<sup>P</sup> <sup>N</sup>*+1 *<sup>−</sup> <sup>P</sup> <sup>N</sup>* (for *<sup>N</sup>*'s effect on *<sup>E</sup>*, in Figures 1e and f)*.* (8) We see immediately that changes in causal probability *P* and event size *N* have different effects on the probability of an international event *E*, besides the trivial observation that *any* change on either variable has *some* effect on *E* (since *@pE >* 0 and ∆*<sup>N</sup> E <* 0, per equations 7-8). To wit: (equation 8 and Figures 1e and f), show a pronounced ravine or canyon along the north-south direction dropping into a deep precipice at relatively high values of *P* as *N* → 2. Interestingly, in this case all the isolines have mixed concavity (low *P*) and convexity (higher *P* values), which is another second-order eff ect. *@pE* = *NP <sup>N</sup>−*<sup>1</sup> (for *P*'s effect on *E*, in Figures 1c and d) (7) <sup>∆</sup>*n<sup>E</sup>* <sup>=</sup> *<sup>P</sup> <sup>N</sup>*+1 *<sup>−</sup> <sup>P</sup> <sup>N</sup>* (for *<sup>N</sup>*'s effect on *<sup>E</sup>*, in Figures 1e and f)*.* (8) We see immediately that changes in causal probability *P* and event size *N* have different effects on the probability of an international event *E*, besides the trivial observation that *any* change on cardinality *N* decreases. The contour plot in Figure 1b shows the graph looking straight down, which highlights the ample basin floor where causal probability is very low (0 *P* / 0*.*9), flanked by the steep north-south escarpment along the east edge as *P !* 1*.*0. Note that change caused by each variable differs, as shown in Figures 1c through f. Calculating the partial derivative and partial difference of *E* with respect to *P* and *N*, respectively:

the latter drops from a large plateau toward a deepening canyon that dives into a deep well. Both features are indicative of major political effects on event probability *E* caused by underlying changes in causal probabilities *P* and event size *N*. the latter drops from a large plateau toward a deepening canyon that dives into a deep well. Both features are indicative of major political effects on event probability *E* caused by underlying changes in causal probabilities *P* and event size *N*. the latter drops from a large plateau toward a deepening canyon that dives into a deep well. Both features are indicative of major political effects on event probability *E* caused by underlying changes in causal probabilities *P* and event size *N*. with some common similarities: the former rises from a flat basin to an escarpment while the latter drops from a large plateau toward a deepening canyon that dives into a deep well. Both features are indicative of major political effects on event probability *E* caused by underlying changes in causal probabilities *P* and event size *N*. with some common similarities: the former rises from a flat basin to an escarpment latter drops from a large plateau toward deepening canyon that dives into deepwell. Both features are indicative of major political effects on probability *<sup>E</sup>*caused by underlying changes in probabilities*P* and event size *N*.Second-order effects caused by changes in event size. The graphs of the change in *E* with respect to event size, ∆*nE* (equation 8 and Figures 1e and f), show a pronounced ravine or canyon along the north-south direction dropping into a deep precipice at relatively high values of *P* as *N !* 2. Interestingly, in this case all the isolines have mixed concavity (low *P*) and convexity (higher *P* values), which is another second-order effect. Geometrically opposite extrema. The extreme high range of *@pE* in Figures 1c and d and the extreme low range of ∆*nE* in Figures 1e and f have opposite (or inverse) geometries Opposite political effects. Whereas change in causal probability *P* has proportional or positive effect on *E*, change in event size *N*has an opposite effect. This is shown by the purple deep bottom in Figure 1c versus the high red plateau in Figure 1e; both features observed over approximately the same underlying (*p, n*)-domain. First- and second-order effects. Whereas the contour plot of *E*(*P, N*) shows strictly concave isocontours (Figure 1b), the contours of the derivative *@pE* (Figure 1d) also show a convex, mild spur protruding on the southwestern wall of the escarpment, near *p* = 0*.*75 (third **Geometrically opposite extrema.** Th e extreme high range of either variable has *some* effect on *E* (since *@pE >* 0 and ∆*<sup>N</sup> E <* 0, per equations 7-8). To wit: Opposite political effects. Whereas change in causal probability *P* has proportional or positive effect on *E*, change in event size *N* has an opposite effect. This is shown by the purple deep bottom in Figure 1c versus the high red plateau in Figure 1e; both features observed over approximately the same underlying (*p, n*)-domain. First- and second-order effects. Whereas the contour plot of *E*(*P, N*) shows strictly concave in Figures 1c and d and the extreme low range of *@pE* = *NP <sup>N</sup>−*<sup>1</sup> (for *P*'s effect on *E*, in Figures 1c and d) (7) <sup>∆</sup>*n<sup>E</sup>* <sup>=</sup> *<sup>P</sup> <sup>N</sup>*+1 *<sup>−</sup> <sup>P</sup> <sup>N</sup>* (for *<sup>N</sup>*'s effect on *<sup>E</sup>*, in Figures 1e and f)*.* (8) We see immediately that changes in causal probability *P* and event size *N* have different effects on the probability of an international event *E*, besides the trivial observation that *any* change on either variable has *some* effect on *E* (since *@pE >* 0 and ∆*<sup>N</sup> E <* 0, per equations 7-8). To wit: Opposite political effects. Whereas change in causal probability *P* has proportional or positive in Figures 1e and f have opposite (or inverse) geometries with some common similarities: the former rises from a fl at basin to an escarpment while the latt er drops from a large plateau toward a deepening canyon that dives into a deep well. Both features are indicative of major political eff ects on event probability *E* caused by underlying changes in causal probabilities *P* and event size *N*.

6 and Figures 1a and b).

6 and Figures 1a and b).

6 and Figures 1a and b).

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6 and Figures 1a and b).

and Figures 1a and b).

NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS

conditions for its occurrence. (a) 3D surface of hybrid function *E* = *P <sup>N</sup>* ; (b) contour plot of (a); (c) 3D surface of *@pE*, the first-order partial derivative of *E*(*P, N*) with respect to *P*; (d) contour plot of (c); (e) 3D surface of ∆*nE*, the first-order partial difference of *E*(*P, N*) with respect to *N*; (f) contour plot of (e). 13 Figure 1. Probability of an international event *E*(*P*, *N*) as a hybrid function of the probability *P* of *N* necessary conditions for its occurrence. (a) 3D surface of hybrid function *E* = *PN*; (b) contour plot of (a); (c) 3D surface of We see immediately that changes in causal probability *P* and event size *N* have different effects on the probability of an international event *E*, besides the trivial observation that *any* change on either variable has *some* effect on *E* (since *@pE >* 0 and ∆*<sup>N</sup> E <* 0, per equations 7-8). To wit: Opposite political effects. Whereas change in causal probability *P* has proportional or positive effect on *E*, change in event size *N* has an opposite effect. This is shown by the purple deep , the fi rst-order partial derivative of *E*(*P*, *N*) with respect to *P*; (d) contour plot of (c); (e) 3D surface of the partial derivative and partial difference of *E* with respect to *P* and *N*, respectively: *@pE* = *NP <sup>N</sup>−*<sup>1</sup> (for *P*'s effect on *E*, in Figures 1c and d) (7) <sup>∆</sup>*n<sup>E</sup>* <sup>=</sup> *<sup>P</sup> <sup>N</sup>*+1 *<sup>−</sup> <sup>P</sup> <sup>N</sup>* (for *<sup>N</sup>*'s effect on *<sup>E</sup>*, in Figures 1e and f)*.* (8) We see immediately that changes in causal probability *P* and event size *N* have different effects , the fi rst-order partial diff erence of *E*(*P*, *N*) with respect to *N*; (f) contour plot of (e).

approximately the same underlying (*p, n*)-domain.

6 and Figures 1a and b).

12

Figura 1. Probability of an international event *E*(*P, N*) as a hybrid function of the probability *P* of *N* necessary

6 and Figures 1a and b).

by the steep north-south escarpment along the east edge as *P !* 1*.*0.

approximately the same underlying (*p, n*)-domain.

*<sup>E</sup>* <sup>=</sup> *<sup>p</sup>*<sup>1</sup> *· <sup>p</sup>*<sup>2</sup> *· ... · <sup>p</sup><sup>n</sup>* <sup>=</sup> <sup>Y</sup>*<sup>n</sup>*

Note that change caused by each variable differs, as shown in Figures 1c through f. Calculating

on the probability of an international event *E*, besides the trivial observation that *any* change on either variable has *some* effect on *E* (since *@pE >* 0 and ∆*<sup>N</sup> E <* 0, per equations 7-8). To wit:

First- and second-order effects. Whereas the contour plot of *E*(*P, N*) shows strictly concave isocontours (Figure 1b), the contours of the derivative *@pE* (Figure 1d) also show a convex, mild spur protruding on the southwestern wall of the escarpment, near *p* = 0*.*75 (third contour, between green and blue elevations), which means more complex change for small-size events. This is a second-order effect and not at all intuitive from the basic model (equation

Second-order effects caused by changes in event size. The graphs of the change in *E* with respect to event size, ∆*nE* (equation 8 and Figures 1e and f), show a pronounced ravine or canyon along the north-south direction dropping into a deep precipice at relatively high values of *P* as *N !* 2. Interestingly, in this case all the isolines have mixed concavity (low

Geometrically opposite extrema. The extreme high range of *@pE* in Figures 1c and d and the extreme low range of ∆*nE* in Figures 1e and f have opposite (or inverse) geometries with some common similarities: the former rises from a flat basin to an escarpment while the latter drops from a large plateau toward a deepening canyon that dives into a deep well. Both features are indicative of major political effects on event probability *E* caused by

*P*) and convexity (higher *P* values), which is another second-order effect.

underlying changes in causal probabilities *P* and event size *N*.

effect on *E*, change in event size *N* has an opposite effect. This is shown by the purple deep bottom in Figure 1c versus the high red plateau in Figure 1e; both features observed over

*i*=1

= *P <sup>N</sup> ,* (6)

*p<sup>i</sup>* (5)

<sup>∆</sup>*n<sup>E</sup>* <sup>=</sup> *<sup>P</sup> <sup>N</sup>*+1 *<sup>−</sup> <sup>P</sup> <sup>N</sup>* (for *<sup>N</sup>*'s effect on *<sup>E</sup>*, in Figures 1e and f)*.* (8)

bottom in Figure 1c versus the high red plateau in Figure 1e; both features observed over

isocontours (Figure 1b), the contours of the derivative *@pE* (Figure 1d) also show a convex, mild spur protruding on the southwestern wall of the escarpment, near *p* = 0*.*75 (third contour, between green and blue elevations), which means more complex change for small-size events. This is a second-order effect and not at all intuitive from the basic model (equation

Second-order effects caused by changes in event size. The graphs of the change in *E* with respect to event size, ∆*nE* (equation 8 and Figures 1e and f), show a pronounced ravine or canyon along the north-south direction dropping into a deep precipice at relatively high values of *P* as *N !* 2. Interestingly, in this case all the isolines have mixed concavity (low

Geometrically opposite extrema. The extreme high range of *@pE* in Figures 1c and d and the extreme low range of ∆*nE* in Figures 1e and f have opposite (or inverse) geometries with some common similarities: the former rises from a flat basin to an escarpment while the latter drops from a large plateau toward a deepening canyon that dives into a deep well. Both features are indicative of major political effects on event probability *E* caused by

12

*P*) and convexity (higher *P* values), which is another second-order effect.

underlying changes in causal probabilities *P* and event size *N*.

Th ese fi rst insights begin to shed light on the nature of international events as a function of their necessary conditions. However, causal probability *P* and event size *N* are measured in diff erent units (probability and number of events, respectively), so absolute rates of change (equations 7 and 8) fail to explain which variable has dominant eff ect on *E*. To solve this problem we obtain and analyze *standardized* (or percentage) rates of change in *E* with respect to *P* and *N* (called elasticities), as in Figure 2. Calculating the point elasticity and the arc elasticity of *E* with respect to *P* and *N*, respectively: CLAUDIO CIOFFI-REVILLA These first insights begin to shed light on the nature of international events as a function of their necessary conditions. However, causal probability *P* and event size *N* are measured in different units (probability and number of events, respectively), so absolute rates of change (equations 7 and 8) fail to explain which variable has dominant effect on *E*. To solve this problem we obtain and analyze *standardized* (or percentage) rates of change in *E* with respect to *P* and *N* (called elasticities), as in Figure 2. Calculating the point elasticity and the arc elasticity of *E* with respect to *P* and *N*, respectively: CLAUDIO CIOFFI-REVILLA These first insights begin to shed light on the nature of international events as a function of their necessary conditions. However, causal probability *P* and event size *N* are measured in different units (probability and number of events, respectively), so absolute rates of change (equations 7 and 8) fail to explain which variable has dominant effect on *E*. To solve this problem we obtain and analyze *standardized* (or percentage) rates of change in *E* with respect to *P* and*N* (called elasticities), as in Figure 2. Calculating the point elasticity and the arc elasticity of *E* with respect to *P* and*N*, respectively: CLAUDIO CIOFFI-REVILLA These first insights begin to shed light on the nature of international events as a function of their necessary conditions. However, causal probability *P* and event size *N* are measured in different units (probability and number of events, respectively), so absolute rates of change (equations 7 and 8) fail to explain which variable has dominant effect on *E*. To solve this problem we obtain and analyze *standardized* (or percentage) rates of change in *E* with respect to *P* and *N* (called elasticities), as in Figure 2. Calculating the point elasticity and the arc elasticity of *E* with respect to *P* and *N*, respectively: CLAUDIO CIOFFI-REVILLA These first insights begin to shed light on the nature of international events as a function of their necessary conditions. However, causal probability *P* and event size *N* are measured in different units (probability and number of events, respectively), so absolute rates of change (equations 7 and 8) fail to explain which variable has dominant effect on *E*. To solve this problem These first insights begin to shed light on the nature of international events as a function of their necessary conditions. However, causal probability *P* and event size *N* are measured in different units (probability and number of events, respectively), so absolute rates of change (equations 7 and 8) fail to explain which variable has dominant effect on *E*. To solve this problem

*⌘pE* = *N* (for *P*'s percentage effect on *E*, in Figures 1c and d) (9) *⌘nE* = (*P −* 1)*N* (for *N*'s percentage effect on *E*, in Figures 1e and f)*.* (10) *⌘pE* = *N* (for *P*'s percentage effect on *E*, in Figures 1c and d) (9) *⌘nE* = (*P −* 1)*N* (for *N*'s percentage effect on *E*, in Figures 1e and f)*.* (10) *⌘pE* = *N* (for *P*'s percentage effect on *E*, in Figures 1c and d) (9) *⌘nE* = (*P −* 1)*N* (for *N*'s percentage effect on *E*, in Figures 1e and f)*.* (10) we obtain and analyze *standardized* (or percentage) rates of change in *E* with respect to *P* and *N* (called elasticities), as in Figure 2. Calculating the point elasticity and the arc elasticity of *E* with respect to *P* and *N*, respectively: we obtain and analyze *standardized* (or percentage) rates of change in *E* with respect to *P* and *N* (called elasticities), as in Figure 2. Calculating the point elasticity and the arc elasticity of *E* with respect to *P* and *N*, respectively: CLAUDIO CIOFFI-REVILLA

We see immediately that these standardized effects on *E* are very different from the earlier absolute, unit-based effects (equations 7 and 8). We see immediately that these standardized effects on *E* are very different from the earlier absolute, unit-based effects (equations 7 and 8). We see immediately that these standardized effects on *E* are very different from the earlier absolute, unit-based effects (equations 7 and 8). We see immediately that these standardized eff ects on *E* are very diff erent from the earlier absolute, unit-based eff ects (equations 7 and 8). *⌘pE* = *N* (for *P*'s percentage effect on *E*, in Figures 1c and d) (9) *⌘pE* = *N* (for *P*'s percentage effect on *E*, in Figures 1c and d) (9) all probabilities are continuous variables (i.e., *E* and *p<sup>i</sup>* are variables, not events), we can rewrite equation (4) in simpler notation:

The point elasticity—percentage change in the probability of an international event with respect to percentage change in *P*—is constant, as in Figures 2a and b, meaning that a percentage variation in probability of an international event *E* relative to a percentage variation in causal probability *P* is determined solely by event size *N* and is independent of *P*. By contrast, arc elasticity—percentage change in *E* with respect to percentage change in *N*—is linear in both independent variables, as in Figures 2c and d, resulting in a nonlinear scalar field as evidenced by the contour plot. Comparing the two elasticities (equations 9 and 10) answers the universal question concerning which of the two causal variables has the dominant or greater effect on the probability of an The point elasticity—percentage change in the probability of an international event with respect to percentage change in *P*—is constant, as in Figures 2a and b, meaning that a percentage variation in probability of an international event *E* relative to a percentage variation in causal probability *P* is determined solely by event size *N* and is independent of *P*. By contrast, arc elasticity—percentage change in *E* with respect to percentage change in *N*—is linear in both independent variables, as in Figures 2c and d, resulting in a nonlinear scalar field as evidenced by the contour plot. Comparing the two elasticities (equations 9 and 10) answers the universal question concerning which of the two causal variables has the dominant or greater effect on the probability of an The point elasticity—percentage change in the probability of an international event with respect to percentage change in *P*—is constant, as in Figures 2a and b, meaning that a percentage variation in probability of an international event *E* relative to a percentage variation in causal probability *P* is determined solely by event size *N* and is independent of *P*. By contrast, arc elasticity—percentage change in *E* with respect to percentage change in *N*—is linear in both independent variables, as in Figures 2c and d, resulting in a nonlinear scalar field as evidenced by Comparing the two elasticities (equations 9 and 10) answers the universal question concerning which of the two causal variables has the dominant or greater effect on the probability of an Th e point elasticity—percentage change in the probability of an international event with respect to percentage change in *P*—is constant, as in Figures 2a and b, meaning that a percentage variation in probability of an international event *E* relative to a percentage variation in causal probability *P* is determined solely by event size *N* and is independent of *P*. By contrast, arc elasticity—percentage change in *E* with respect to percentage change in *N*—is linear in both independent variables, as in Figures 2c and d, resulting in a nonlinear scalar fi eld as evidenced by the contour plot. CLAUDIO CIOFFI-REVILLA These first insights begin to shed light on the nature of international events as a function of their necessary conditions. However, causal probability *P* and event size *N* are measured in different units (probability and number of events, respectively), so absolute rates of change (equations 7 and 8) fail to explain which variable has dominant effect on *E*. To solve this problem we obtain and analyze *standardized* (or percentage) rates of change in *E* with respect to *P* and *N* CLAUDIO CIOFFI-REVILLA These first insights begin to shed light on the nature of international events as a function of their necessary conditions. However, causal probability *P* and event size *N* are measured in different units (probability and number of events, respectively), so absolute rates of change (equations 7 and 8) fail to explain which variable has dominant effect on *E*. To solve this problem we obtain and analyze *standardized* (or percentage) rates of change in *E* with respect to *P* and *N* (called elasticities), as in Figure 2. Calculating the point elasticity and the arc elasticity of *E* with *⌘nE* = (*P −* 1)*N* (for *N*'s percentage effect on *E*, in Figures 1e and f)*.* (10) We see immediately that these standardized effects on *E* are very different from the earlier absolute, unit-based effects (equations 7 and 8). The point elasticity—percentage change in the probability of an international event with respect to percentage change in *P*—is constant, as in Figures 2a and b, meaning that a percentage variation in probability of an international event *E* relative to a percentage variation in causal probability *P* is determined solely by event size *N* and is independent of *P*. By contrast, arc elasticity—percentage change in *E* with respect to percentage change in *N*—is linear in both *⌘nE* = (*P −* 1)*N* (for *N*'s percentage effect on *E*, in Figures 1e and f)*.* (10) We see immediately that these standardized effects on *E* are very different from the earlier absolute, unit-based effects (equations 7 and 8). The point elasticity—percentage change in the probability of an international event with respect to percentage change in *P*—is constant, as in Figures 2a and b, meaning that a percentage variation in probability of an international event *E* relative to a percentage variation in causal probability *P* is determined solely by event size *N* and is independent of *P*. By contrast, arc elasticity—percentage change in *E* with respect to percentage change in *N*—is linear in both *<sup>E</sup>* <sup>=</sup> *<sup>p</sup>*<sup>1</sup> *· <sup>p</sup>*<sup>2</sup> *· ... · <sup>p</sup><sup>n</sup>* <sup>=</sup> <sup>Y</sup>*<sup>n</sup> i*=1 *p<sup>i</sup>* (5) = *P <sup>N</sup> ,* (6) where *P* is the probability of causal events and *N ⌘ |***E***|* is the cardinality or event size. Note that equation 6 is a *hybrid function*, because probability is continuous on the unit interval of real numbers [0*,* 1]—which lies between (and includes) impossibility (*E* = 0) and certainty (*E* = 1*.*0)—while

international event. Since *N >* (*P −* 1)*N*, this means that point elasticity *⌘<sup>p</sup>* is greater than the arc elasticity *⌘n*. Therefore, *E* is more sensitive to change in *P* (probability of necessary causal events) than to change in cardinality *N*—a result that may be called *dominance principle for international events*. Although their effects differ, in reality both variables have joint, concurrent effects on international events. The joint effects of *P* and *N* on *E* can be better understood by calculating the gradient of *E* with respect to both variables using the nabladot operator, as follows: international event. Since *N >* (*P −* 1)*N*, this means that point elasticity *⌘<sup>p</sup>* is greater than the arc elasticity *⌘n*. Therefore, *E* is more sensitive to change in *P* (probability of necessary causal events) than to change in cardinality *N*—a result that may be called *dominance principle for international events*. Although their effects differ, in reality both variables have joint, concurrent effects on international events. The joint effects of *P* and *N*on *E* can be better understood by calculating the gradient of *E* with respect to both variables using the nabladot operator, as follows: international event. Since *N >* (*P −* 1)*N*, this means that point elasticity *⌘<sup>p</sup>* is greater than the arc elasticity *⌘n*. Therefore, *E* is more sensitive to change in *P* (probability of necessary causal events) than to change in cardinality *N*—a result that may be called *dominance principle for international events*. Although their effects differ, in reality both variables have joint, concurrent effects on international events. The joint effects of *P* and *N* on *E* can be better understood by calculating the gradient of *E* with respect to both variables using the nabladot operator, as follows: Comparing the two elasticities (equations 9 and 10) answers the universal question concerning which of the two causal variables has the dominant or greater effect on the probability of an international event. Since *N*>(*P*–1)*N*, this means that point elasticity (called elasticities), as in Figure 2. Calculating the point elasticity and the arc elasticity of *E* with respect to *P* and *N*, respectively: *⌘pE* = *N* (for *P*'s percentage effect on *E*, in Figures 1c and d) (9) *⌘nE* = (*P −* 1)*N* (for *N*'s percentage effect on *E*, in Figures 1e and f)*.* (10) is greater than the arc elasticity respect to *P* and *N*, respectively: *⌘pE* = *N* (for *P*'s percentage effect on *E*, in Figures 1c and d) (9) *⌘nE* = (*P −* 1)*N* (for *N*'s percentage effect on *E*, in Figures 1e and f)*.* (10) We see immediately that these standardized effects on *E* are very different from the earlier absolute, . Th erefore, *E* is more sensitive to change in *P* (probability of necessary causal events) than to change in cardinality *N*—a result that may be called *dominance principle for international events*. independent variables, as in Figures 2c and d, resulting in a nonlinear scalar field as evidenced by the contour plot. Comparing the two elasticities (equations 9 and 10) answers the universal question concerning which of the two causal variables has the dominant or greater effect on the probability of an international event. Since *N >* (*P −* 1)*N*, this means that point elasticity *⌘<sup>p</sup>* is greater than the arc elasticity *⌘n*. Therefore, *E* is more sensitive to change in *P* (probability of necessary causal events) than to change in cardinality *N*—a result that may be called *dominance principle for* independent variables, as in Figures 2c and d, resulting in a nonlinear scalar field as evidenced by Comparing the two elasticities (equations 9 and 10) answers the universal question concerning which of the two causal variables has the dominant or greater effect on the probability of an international event. Since *N >* (*P −* 1)*N*, this means that point elasticity *⌘<sup>p</sup>* is greater than the arc elasticity *⌘n*. Therefore, *E* is more sensitive to change in *P* (probability of necessary causal events) than to change in cardinality *N*—a result that may be called *dominance principle for* event size is always a discrete or natural number (of necessary conditions). The politically relevant domain of interest—in this case—is bound by 0 *P* 1 and 2 *N* / 20 along causal probability and size dimensions in the hybrid (*p, c*) domain. Specifically, equation 6 is a hybrid function linearly dependent in *P* and exponentially in *N*, so this means that changes in either variables will cause different political effects on the probability of an event. The 3D surface graph of equation 6 is in Figure 1a, which shows values of the probability *E* of

*<sup>r</sup>· <sup>E</sup>* <sup>=</sup> *@p<sup>P</sup> <sup>N</sup>* <sup>i</sup> <sup>+</sup> <sup>∆</sup>*n<sup>P</sup> <sup>N</sup>* <sup>j</sup> (11) <sup>=</sup> *NP*(*N−*1) <sup>i</sup> + (*<sup>P</sup> <sup>−</sup>* 1)*<sup>P</sup> <sup>N</sup>* <sup>j</sup>*,* (12) *<sup>r</sup>· <sup>E</sup>* <sup>=</sup> *@p<sup>P</sup> <sup>N</sup>* <sup>i</sup> <sup>+</sup> <sup>∆</sup>*nP<sup>N</sup>* <sup>j</sup> (11) <sup>=</sup> *NP*(*N−*1) <sup>i</sup> + (*<sup>P</sup> <sup>−</sup>*1)*<sup>P</sup> <sup>N</sup>* <sup>j</sup>*,* (12) *<sup>r</sup>· <sup>E</sup>* <sup>=</sup> *@p<sup>P</sup> <sup>N</sup>* <sup>i</sup><sup>+</sup> <sup>∆</sup>*n<sup>P</sup> <sup>N</sup>* <sup>j</sup>(11) <sup>=</sup> *NP*(*N−*1) <sup>i</sup> + (*<sup>P</sup> <sup>−</sup>* 1)*<sup>P</sup> <sup>N</sup>* <sup>j</sup>*,* (12) We see immediately that these standardized effects on *E* are very different from the earlier absolute, unit-based effects (equations 7 and 8). The point elasticity—percentage change in the probability of an international event with respect to percentage change in *P*—is constant, as in Figures 2a and b, meaning that a percentage variation in probability of an international event *E* relative to a percentage variation in causal unit-based effects (equations 7 and 8). The point elasticity—percentage change in the probability of an international event with respect to percentage change in *P*—is constant, as in Figures 2a and b, meaning that a percentage variation in probability of an international event *E* relative to a percentage variation in causal probability *P* is determined solely by event size *N* and is independent of *P*. By contrast, arc Although their eff ects diff er, in reality both variables have joint, concurrent eff ects on international events. Th e joint eff ects of *P* and *N* on *E* can be bett er understood by calculating the gradient of *E* with respect to both variables using the nabladot operator, as follows: *international events*. Although their effects differ, in reality both variables have joint, concurrent effects on international events. The joint effects of *P* and *N* on *E* can be better understood by calculating the gradient of *E* with respect to both variables using the nabladot operator, as follows: Although their effects differ, in reality both variables have joint, concurrent effects on international events. The joint effects of *P* and *N* on *E* can be better understood by calculating the gradient of *E* with respect to both variables using the nabladot operator, as follows: an international event rising through a steep escarpment as causal probability *P* increases and cardinality *N* decreases. The contour plot in Figure 1b shows the graph looking straight down, which highlights the ample basin floor where causal probability is very low (0 *P* / 0*.*9), flanked by the steep north-south escarpment along the east edge as *P !* 1*.*0. Note that change caused by each variable differs, as shown in Figures 1c through f. Calculating

which is a two-dimensional vector function E = (*P, N*). The emergent vector field of this hybrid

which is a two-dimensional vector function E = (*P, N*). The emergent vector field of this hybrid

which is a two-dimensional vector function E = (*P, N*). The emergent vector field of this hybrid

$$\nabla E = \partial\_p P^N \mathbf{i} + \Delta\_n P^N \mathbf{j} \tag{11}$$

function, is shown in Figure 2f. We see immediately that norm *|*E*|* is very similar to *@pE* in Figures 1e and d, which is not an intuitive result or obvious insight that could be obtained from function, is shown in Figure 2f. We see immediately that norm *|*E*|* is very similar to *@pE* in Figures 1e and d, which is not an intuitive result or obvious insight that could be obtained from function, is shown in Figure 2f. We see immediately that norm *|*E*|* is very similar to *@pE* in Figures 1e and d, which is not an intuitive result or obvious insight that could be obtained from elasticity—percentage change in *E* with respect to percentage change in *N*—is linear in both independent variables, as in Figures 2c and d, resulting in a nonlinear scalar field as evidenced by the contour plot. <sup>=</sup> *NP*(*N−*1) <sup>i</sup> + (*<sup>P</sup> <sup>−</sup>* 1)*<sup>P</sup> <sup>N</sup>* <sup>j</sup>*,* (12) <sup>=</sup> *NP*(*N−*1) <sup>i</sup> + (*<sup>P</sup> <sup>−</sup>* 1)*<sup>P</sup> <sup>N</sup>* <sup>j</sup>*,* (12) *@pE* = *NP <sup>N</sup>−*<sup>1</sup> (for *P*'s effect on *E*, in Figures 1c and d) (7)

which is a two-dimensional vector function E = (*P, N*). The emergent vector field of this hybrid gradient is in Figure 2e and corresponding vector magnitude or norm *|*E*|*(*P, N*), which is a scalar function, is shown in Figure 2f. We see immediately that norm *|*E*|* is very similar to *@pE* in Figures 1e and d, which is not an intuitive result or obvious insight that could be obtained from casual comparison between the two dissimilar equations for the arc elasticity (equation 8) and the

magnitude of E.

Second-order effects caused by changes in event size. The graphs of the change in *E* with respect to event size, ∆*nE* (equation 8 and Figures 1e and f), show a pronounced ravine or canyon along the north-south direction dropping into a deep precipice at relatively high values of *P* as *N !* 2. Interestingly, in this case all the isolines have mixed concavity (low

mild spur protruding on the southwestern wall of the escarpment, near *p* = 0*.*75 (third contour, between green and blue elevations), which means more complex change for small-size events. This is a second-order effect and not at all intuitive from the basic model (equation

the next section.

Geometrically opposite extrema. The extreme high range of *@pE* in Figures 1c and d and the extreme low range of ∆*nE* in Figures 1e and f have opposite (or inverse) geometries with some common similarities: the former rises from a flat basin to an escarpment while the latter drops from a large plateau toward a deepening canyon that dives into a deep well. Both features are indicative of major political effects on event probability *E* caused by

14

This concludes the first analysis of our three "case studies." The next two are presented in slightly abbreviated form to omit some procedural repetitions while maintaining the method outlined in section .3.. Analysis of the probability of international events in this first case leads us to investigate the probability of war onset—a special class of international event of fundamental historical significance since early antiquity and of theoretical interest (at least) since Thucydides—in

*<sup>r</sup>· <sup>E</sup>* <sup>=</sup> *@p<sup>P</sup> <sup>N</sup>* <sup>i</sup> <sup>+</sup> <sup>∆</sup>*n<sup>P</sup> <sup>N</sup>* <sup>j</sup> (11) <sup>=</sup> *NP*(*N−*1) <sup>i</sup> + (*<sup>P</sup> <sup>−</sup>* 1)*<sup>P</sup> <sup>N</sup>* <sup>j</sup>*,* (12)

which is a two-dimensional vector function E = (*P, N*). The emergent vector field of this hybrid gradient is in Figure 2e and corresponding vector magnitude or norm *|*E*|*(*P, N*), which is a scalar function, is shown in Figure 2f. We see immediately that norm *|*E*|* is very similar to *@pE* in Figures 1e and d, which is not an intuitive result or obvious insight that could be obtained from casual comparison between the two dissimilar equations for the arc elasticity (equation 8) and the

This concludes the first analysis of our three "case studies." The next two are presented in slightly abbreviated form to omit some procedural repetitions while maintaining the method outlined in section .3.. Analysis of the probability of international events in this first case leads us to investigate the probability of war onset—a special class of international event of fundamental historical significance since early antiquity and of theoretical interest (at least) since Thucydides—in

<sup>=</sup> *NP*(*N−*1) <sup>i</sup> + (*<sup>P</sup> <sup>−</sup>* 1)*<sup>P</sup> <sup>N</sup>* <sup>j</sup>*,* (12)

casual comparison between the two dissimilar equations for the arc elasticity (equation 8) and the magnitude of E. This concludes the first analysis of our three "case studies." The next two are presented in slightly abbreviated form to omit some procedural repetitions while maintaining the method outlined in section .3.. Analysis of the probability of international events in this first case leads us to investigate the probability of war onset—a special class of international event of fundamental historical significance since early antiquity and of theoretical interest (at least) since Thucydides—in the next section. casual comparison between the two dissimilar equations for the arc elasticity (equation 8) and the magnitude of E. This concludes the first analysis of our three "case studies." The next two are presented in slightly abbreviated form to omit some procedural repetitions while maintaining the method outlined in section .3.. Analysis of the probability of international events in this first case leads us to investigate the probability of war onset—a special class of international event of fundamental historical significance since early antiquity and of theoretical interest (at least) since Thucydides—in the next section. casual comparison between the two dissimilar equations for the arc elasticity (equation 8) and the This concludes the first analysis of our three "case studies." The next two are presented in slightly abbreviated form to omit some procedural repetitions while maintaining the method outlined in section .3.. Analysis of the probability of international events in this first case leads us to investigate the probability of war onset—a special class of international event of fundamental historical significance since early antiquity and of theoretical interest (at least) since Thucydides—in the contour plot. Comparing the two elasticities (equations 9 and 10) answers the universal question concerning which of the two causal variables has the dominant or greater effect on the probability of an international event. Since *N >* (*P −* 1)*N*, this means that point elasticity *⌘<sup>p</sup>* is greater than the arc elasticity *⌘n*. Therefore, *E* is more sensitive to change in *P* (probability of necessary causal events) than to change in cardinality *N*—a result that may be called *dominance principle for international events*. Although their effects differ, in reality both variables have joint, concurrent effects on inter-Comparing the two elasticities (equations 9 and 10) answers the universal question concerning which of the two causal variables has the dominant or greater effect on the probability of an international event. Since *N >* (*P −* 1)*N*, this means that point elasticity *⌘<sup>p</sup>* is greater than the arc elasticity *⌘n*. Therefore, *E* is more sensitive to change in *P* (probability of necessary causal events) than to change in cardinality *N*—a result that may be called *dominance principle for international events*. Although their effects differ, in reality both variables have joint, concurrent effects on international events. The joint effects of *P* and *N* on *E* can be better understood by calculating the which is a two-dimensional vector function E = (*P, N*). The emergent vector field of this hybrid gradient is in Figure 2e and corresponding vector magnitude or norm *|*E*|*(*P, N*), which is a scalar function, is shown in Figure 2f. We see immediately that norm *|*E*|* is very similar to *@pE* in Figures 1e and d, which is not an intuitive result or obvious insight that could be obtained from casual comparison between the two dissimilar equations for the arc elasticity (equation 8) and the magnitude of E. This concludes the first analysis of our three "case studies." The next two are presented in which is a two-dimensional vector function E = (*P, N*). The emergent vector field of this hybrid gradient is in Figure 2e and corresponding vector magnitude or norm *|*E*|*(*P, N*), which is a scalar function, is shown in Figure 2f. We see immediately that norm *|*E*|* is very similar to *@pE* in Figures 1e and d, which is not an intuitive result or obvious insight that could be obtained from casual comparison between the two dissimilar equations for the arc elasticity (equation 8) and the This concludes the first analysis of our three "case studies." The next two are presented in which is a two-dimensional vector function **E** = **ψ**(*P*, *N*). Th e emergent vector fi eld of this hybrid gradient is in Figure 2e and corresponding vector magnitude or norm |**E**|(*P*, *N*), which is a scalar function, is shown in Figure 2f. We see immediately that norm |**E**| is very similar to <sup>∆</sup>*n<sup>E</sup>* <sup>=</sup> *<sup>P</sup> <sup>N</sup>*+1 *<sup>−</sup> <sup>P</sup> <sup>N</sup>* (for *<sup>N</sup>*'s effect on *<sup>E</sup>*, in Figures 1e and f)*.* (8) We see immediately that changes in causal probability *P* and event size *N* have different effects on the probability of an international event *E*, besides the trivial observation that *any* change on either variable has *some* effect on *E* (since *@pE >* 0 and ∆*<sup>N</sup> E <* 0, per equations 7-8). To wit: Opposite political effects. Whereas change in causal probability *P* has proportional or positive effect on *E*, change in event size *N* has an opposite effect. This is shown by the purple deep bottom in Figure 1c versus the high red plateau in Figure 1e; both features observed over in Figures 1e and d, which is not an intuitive result or obvious insight that could be obtained from casual comparison between the two dissimilar equations for the arc elasticity (equation 8) and the magnitude of **E**.

national events. The joint effects of *P* and *N* on *E* can be better understood by calculating the gradient of *E* with respect to both variables using the nabladot operator, as follows: gradient of *E* with respect to both variables using the nabladot operator, as follows: *<sup>r</sup>· <sup>E</sup>* <sup>=</sup> *@p<sup>P</sup> <sup>N</sup>* <sup>i</sup> <sup>+</sup> <sup>∆</sup>*n<sup>P</sup> <sup>N</sup>* <sup>j</sup> (11) slightly abbreviated form to omit some procedural repetitions while maintaining the method outlined in section .3.. Analysis of the probability of international events in this first case leads us to investigate the probability of war onset—a special class of international event of fundamental slightly abbreviated form to omit some procedural repetitions while maintaining the method outlined in section .3.. Analysis of the probability of international events in this first case leads us to investigate the probability of war onset—a special class of international event of fundamental approximately the same underlying (*p, n*)-domain. Th is concludes the fi rst analysis of our three "case studies." Th e next two are presented in slightly abbreviated form to omit some procedural repetitions while

*P*) and convexity (higher *P* values), which is another second-order effect.

underlying changes in causal probabilities *P* and event size *N*.

magnitude of E.

6 and Figures 1a and b).

the next section.

14

14

14

12

the contour plot.

the contour plot.

*international events*.

CLAUDIO CIOFFI-REVILLA

magnitude of E.

the next section.

magnitude of E.

14

the next section.

14

14

maintaining the method outlined in section 3. Analysis of the probability of international events in this fi rst case leads us to investigate the probability of war onset—a special class of international event of fundamental historical signifi cance since early antiquity and of theoretical interest (at least) since Th ucydides—in the next section. NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS

Figura 2. Elasticities and gradient of the probability of an international event *E*. (a) 3D surface of point elasticity *⌘p*(*E*) respect to causal probability *P*; (b) contour plot of (a); (c) 3D surface of arc elasticity *⌘n*(*E*) with respect to number of necessary conditions *N*; (d) contour plot of (c); (e) vector field of the dot-gradient vector function *r· E*; (f) contour plot of (e). 15 Figure 2. Elasticities and gradient of the probability of an international event *E*. (a) 3D surface of point elasticity *⌘pE* = *N* (for *P*'s percentage effect on *E*, in Figures 1c and d) (9) *⌘nE* = (*P −* 1)*N* (for *N*'s percentage effect on *E*, in Figures 1e and f)*.* (10) We see immediately that these standardized effects on *E* are very different from the earlier absolute, (*E*) respect to causal probability *P*; (b) contour plot of (a); (c) 3D surface of arc elasticity *⌘pE* = *N* (for *P*'s percentage effect on *E*, in Figures 1c and d) (9) *⌘nE* = (*P −* 1)*N* (for *N*'s percentage effect on *E*, in Figures 1e and f)*.* (10) We see immediately that these standardized effects on *E* are very different from the earlier absolute, unit-based effects (equations 7 and 8). (E) with respect to number of necessary conditions *N*; (d) contour plot of (c); (e) vector fi eld of the dot-gradient vector function Figura 2. Elasticities and gradient of the probability of an international event *E*. (a) 3D surface of point elasticity *⌘p*(*E*) respect to causal probability *P*; (b) contour plot of (a); (c) 3D surface of arc elasticity *⌘n*(*E*) with respect to number of necessary conditions *N*; (d) contour plot of (c); (e) vector field of the dot-gradient vector function *r· E*; (f) contour plot of (e). ; (f) contour plot of (e).

unit-based effects (equations 7 and 8).

the contour plot.

the contour plot.

*international events*.

*international events*.

magnitude of E.

magnitude of E.

the next section.

the next section.

14

14

0

5

10

15

20

1

2

3

*<sup>r</sup>· <sup>E</sup>* <sup>=</sup> *@p<sup>P</sup> <sup>N</sup>* <sup>i</sup> <sup>+</sup> <sup>∆</sup>*n<sup>P</sup> <sup>N</sup>* <sup>j</sup> (11) <sup>=</sup> *NP*(*N−*1) <sup>i</sup> + (*<sup>P</sup> <sup>−</sup>* 1)*<sup>P</sup> <sup>N</sup>* <sup>j</sup>*,* (12)

*<sup>r</sup>· <sup>E</sup>* <sup>=</sup> *@p<sup>P</sup> <sup>N</sup>* <sup>i</sup> <sup>+</sup> <sup>∆</sup>*n<sup>P</sup> <sup>N</sup>* <sup>j</sup> (11) <sup>=</sup> *NP*(*N−*1) <sup>i</sup> + (*<sup>P</sup> <sup>−</sup>* 1)*<sup>P</sup> <sup>N</sup>* <sup>j</sup>*,* (12)

NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.2


> 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

> > 15

0.4

0.6

0.8

(*a*) (*b*)

5

5

10

15

20

10

15

20

respect to percentage change in *P*—is constant, as in Figures 2a and b, meaning that a percentage variation in probability of an international event *E* relative to a percentage variation in causal probability *P* is determined solely by event size *N* and is independent of *P*. By contrast, arc elasticity—percentage change in *E* with respect to percentage change in *N*—is linear in both independent variables, as in Figures 2c and d, resulting in a nonlinear scalar field as evidenced by

variation in probability of an international event *E* relative to a percentage variation in causal probability *P* is determined solely by event size *N* and is independent of *P*. By contrast, arc elasticity—percentage change in *E* with respect to percentage change in *N*—is linear in both independent variables, as in Figures 2c and d, resulting in a nonlinear scalar field as evidenced by

Comparing the two elasticities (equations 9 and 10) answers the universal question concerning which of the two causal variables has the dominant or greater effect on the probability of an international event. Since *N >* (*P −* 1)*N*, this means that point elasticity *⌘<sup>p</sup>* is greater than the arc elasticity *⌘n*. Therefore, *E* is more sensitive to change in *P* (probability of necessary causal events) than to change in cardinality *N*—a result that may be called *dominance principle for*

Although their effects differ, in reality both variables have joint, concurrent effects on international events. The joint effects of *P* and *N* on *E* can be better understood by calculating the

Comparing the two elasticities (equations 9 and 10) answers the universal question concerning which of the two causal variables has the dominant or greater effect on the probability of an international event. Since *N >* (*P −* 1)*N*, this means that point elasticity *⌘<sup>p</sup>* is greater than the arc elasticity *⌘n*. Therefore, *E* is more sensitive to change in *P* (probability of necessary causal events) than to change in cardinality *N*—a result that may be called *dominance principle for*

Although their effects differ, in reality both variables have joint, concurrent effects on international events. The joint effects of *P* and *N* on *E* can be better understood by calculating the

which is a two-dimensional vector function E = (*P, N*). The emergent vector field of this hybrid gradient is in Figure 2e and corresponding vector magnitude or norm *|*E*|*(*P, N*), which is a scalar function, is shown in Figure 2f. We see immediately that norm *|*E*|* is very similar to *@pE* in Figures 1e and d, which is not an intuitive result or obvious insight that could be obtained from casual comparison between the two dissimilar equations for the arc elasticity (equation 8) and the

which is a two-dimensional vector function E = (*P, N*). The emergent vector field of this hybrid gradient is in Figure 2e and corresponding vector magnitude or norm *|*E*|*(*P, N*), which is a scalar function, is shown in Figure 2f. We see immediately that norm *|*E*|* is very similar to *@pE* in Figures 1e and d, which is not an intuitive result or obvious insight that could be obtained from casual comparison between the two dissimilar equations for the arc elasticity (equation 8) and the

This concludes the first analysis of our three "case studies." The next two are presented in slightly abbreviated form to omit some procedural repetitions while maintaining the method outlined in section .3.. Analysis of the probability of international events in this first case leads us to investigate the probability of war onset—a special class of international event of fundamental historical significance since early antiquity and of theoretical interest (at least) since Thucydides—in

gradient of *E* with respect to both variables using the nabladot operator, as follows:

This concludes the first analysis of our three "case studies." The next two are presented in slightly abbreviated form to omit some procedural repetitions while maintaining the method outlined in section .3.. Analysis of the probability of international events in this first case leads us to investigate the probability of war onset—a special class of international event of fundamental historical significance since early antiquity and of theoretical interest (at least) since Thucydides—in

gradient of *E* with respect to both variables using the nabladot operator, as follows:

CLAUDIO CIOFFI-REVILLA

CLAUDIO CIOFFI-REVILLA

#### 4.2 Case 2: Crisis dynamics and onset of war

Explaining the outbreak of international war as caused by a prior crisis escalation process—which includes challenging and resistance moves, bargaining, signaling, and other events, as opposed to some other causal mechanism—was first proposed by Quincy Wright (Wright 1942, 1271, fn. 38) and later extended and generalized by Glenn H. Snyder (Snyder Diesing 1977, 13–7 et *passim*). The frame of reference here is the inter-state relational level of analysis and the specific *explanandum* is the probability *W* of a state being at war over a period of time, called an epoch in probability theory.6 During an epoch a country experiences a number *C* of crises (defined as episodes during which hostilities may occur), each individual crisis having probability *ω* of escalating to war. As a result, the probability of no war over *C* crises is (1 – *ω*)*C* (by Kolmogorov's theorem), so a country's epochal probability of war is given by all probabilities are continuous variables (i.e., *E* and *p<sup>i</sup>* are variables, not events), we can rewrite equation (4) in simpler notation: *<sup>E</sup>* <sup>=</sup> *<sup>p</sup>*<sup>1</sup> *· <sup>p</sup>*<sup>2</sup> *· ... · <sup>p</sup><sup>n</sup>* <sup>=</sup> <sup>Y</sup>*<sup>n</sup> i*=1 *p<sup>i</sup>* (5) CLAUDIO CIOFFI-REVILLA 4.2 Case 2: Crisis dynamics and onset of war Explaining the outbreak of international war as caused by a prior crisis escalation process—which includes challenging and resistance moves, bargaining, signaling, and other events, as opposed to some other causal mechanism—was first proposed by Quincy Wright (Wright 1942, 1271, fn. 38) and later extended and generalized by Glenn H. Snyder (Snyder & Diesing 1977, 13–17 et passim). The frame of reference here is the inter-state relational level of analysis and the specific *explanandum* is the probability *W* of a state being at war over a period of time, called an epoch CLAUDIO CIOFFI-REVILLA 4.2 Case 2: Crisis dynamics and onset of war Explaining the outbreak of international war as caused by a prior crisis escalation process—which includes challenging and resistance moves, bargaining, signaling, and other events, as opposed to some other causal mechanism—was first proposed by Quincy Wright (Wright 1942, 1271, fn. 38) and later extended and generalized by Glenn H. Snyder (Snyder & Diesing 1977, 13–17 et passim). The frame of reference here is the inter-state relational level of analysis and the specific *explanandum* is the probability *W* of a state being at war over a period of time, called an epoch 4.2 Case 2: Crisis dynamics and onset of war Explaining the outbreak of international war as caused by a prior crisis escalation process—which includes challenging and resistance moves, bargaining, signaling, and other events, as opposed to some other causal mechanism—was first proposed by Quincy Wright (Wright 1942, 1271, fn. 38) and later extended and generalized by Glenn H. Snyder (Snyder & Diesing 1977, 13–17 et passim). The frame of reference here is the inter-state relational level of analysis and the specific *explanandum* is the probability *W* of a state being at war over a period of time, called an epoch

$$W = 1 - (1 - \omega)^{\mathbb{C}} \tag{13}$$

which is a bivariate nonlinear hybrid function, where *ω* is continuous over the closed unit probability interval [0, 1] and *C ≥ 2* is discrete. The case when *C =* 1 (a single crisis during an entire epoch) is trivial, since *W(ω*, 1) = *ω*, as is easily shown. where *P* is the probability of causal events and *N ⌘ |***E***|* is the cardinality or event size. Note that equation 6 is a *hybrid function*, because probability is continuous on the unit interval of real numbers [0*,* 1]—which lies between (and includes) impossibility (*E* = 0) and certainty (*E* = 1*.*0)—while event size is always a discrete or natural number (of necessary conditions). escalating to war. As a result, the probability of no war over *<sup>C</sup>* crises is (1 *<sup>−</sup> !*)*<sup>C</sup>* (by Kolmogorov's theorem), so a country's epochal probability of war is given by *W* = 1 *−* (1 *− !*) *<sup>C</sup> ,* (13) escalating to war. As a result, the probability of no war over *<sup>C</sup>* crises is (1 *<sup>−</sup> !*)*<sup>C</sup>* (by Kolmogorov's theorem), so a country's epochal probability of war is given by *W* = 1 *−* (1 *− !*) *<sup>C</sup> ,*(13) escalating to war. As a result, the probability of no war over *<sup>C</sup>* crises is (1 *<sup>−</sup> !*)*<sup>C</sup>* (by Kolmogorov's theorem), so a country's epochal probability of war is given by *W* = 1 *−* (1 *− !*) *<sup>C</sup> ,* (13)

The politically relevant domain is bound by 0 ≤ *ω ≤* 1 and *2 ≤ C*  The politically relevant domain of interest—in this case—is bound by 0 *P* 1 and 2 *N* / 20 along causal probability and size dimensions in the hybrid (*p, c*) domain. Specifically, equation 6 is 20 (same as before, both equations being functions of compound events). which is a bivariate nonlinear hybrid function, where *!* is continuous over the closed unit probability which is a bivariate nonlinear hybrid function, where *!* is continuous over the closed unit probability which is a bivariate nonlinear hybrid function, where *!* is continuous over the closed unit probability

a hybrid function linearly dependent in *P* and exponentially in *N*, so this means that changes in either variables will cause different political effects on the probability of an event. The 3D surface graph of equation 6 is in Figure 1a, which shows values of the probability *E* of an international event rising through a steep escarpment as causal probability *P* increases and cardinality *N* decreases. The contour plot in Figure 1b shows the graph looking straight down, The 3D surface graph of *W*(*ω*, *C*) is in Figure 3a, which shows the probability of war rising rapidly to a maximal plateau as escalation probability *ω* and *C* increase. The contour plot in Figure 3b looks straight down, which highlights the broad plateau where war probability converges to 1, flanked by the steep northsouth escarpment along the west edge as *ω* → 0. interval [0*,* 1] and *C ≥* 2 is discrete. The case when *C* = 1 (a single crisis during an entire epoch) is trivial, since *W*(*!,* 1) = *!*, as is easily shown. The politically relevant domain is bound by 0 *!* 1 and 2 *C* / 20 (same as before, both equations being functions of compound events). The 3D surface graph of *W*(*!, C*) is in Figure 3a, which shows the probability of war rising rapidly to a maximal plateau as escalation probability *!* and *C* increase. The contour plot in interval [0*,* 1] and *C ≥* 2 is discrete. The case when *C* = 1 (a single crisis during an entire epoch) is trivial, since *W*(*!,* 1) = *!*, as is easily shown. The politically relevant domain is bound by 0 *!* 1 and 2 *C* / 20 (same as before, both equations being functions of compound events). The 3D surface graph of *W*(*!, C*) is in Figure 3a, which shows the probability of war rising rapidly to a maximal plateau as escalation probability *!* and *C* increase. The contour plot in interval [0*,* 1] and *C ≥* 2 is discrete. The case when *C* = 1 (a single crisis during an entire epoch) is trivial, since *W*(*!,* 1) = *!*, as is easily shown. The politically relevant domain is bound by 0 *!* 1 and2 *C*/ 20(same as before, both equations being functions of compound events). The 3D surface graph of *W*(*!, C*) is in Figure3a, which shows the probability of war rising rapidly to a maximal plateau as escalation probability *!* and *C* increase. The contour plot in

which highlights the ample basin floor where causal probability is very low (0 *P* / 0*.*9), flanked by the steep north-south escarpment along the east edge as *P !* 1*.*0. Note that change caused by each variable differs, as shown in Figures 1c through f. Calculating the partial derivative and partial difference of *E* with respect to *P* and *N*, respectively: Each variable increases *W* in a different way, as shown in Figures 3c through f. Calculating the partial derivative and partial difference of *W* with respect to *ω* and *C*, respectively: Figure 3b looks straight down, which highlights the broad plateau where war probability converges to 1, flanked by the steep north-south escarpment along the west edge as *! !* 0. Each variable increases *W* in a different way, as shown in Figures 3c through f. Calculating the partial derivative and partial difference of *W* with respect to *!* and *C*, respectively: Figure 3b looks straight down, which highlights the broad plateau where war probability converges to 1, flanked by the steep north-south escarpment along the west edge as *! !* 0. Each variable increases *W* in a different way, as shown in Figures 3c through f. Calculating the partial derivative and partial difference of *W* with respect to *!* and *C*, respectively: Figure 3b looks straight down, which highlights the broad plateau where war probability converges to 1, flanked by the steep north-south escarpment along the west edge as *! !* 0. Each variable increases *W* in a different way, as shown in Figures 3c through f. Calculating the partial derivative and partial difference of *W* with respect to *!* and *C*, respectively:

$$\partial\_{\omega} W = C (1 - \omega)^{C - 1} \quad \text{ (for } \omega \text{'s effect on } W \text{, in Figsres 3c and d)} \quad \text{(14)}$$

$$\Delta\_c W = \omega (1 - \omega)^C \quad \text{ (for } C \text{'s effect on } W \text{, in Figsres 3c and f)}. \quad \text{(15)}$$

We see immediately that changes in causal probability *P* and event size *N* have different effects on the probability of an international event *E*, besides the trivial observation that *any* change on either variable has *some* effect on *E* (since *@pE >* 0 and ∆*<sup>N</sup> E <* 0, per equations 7-8). To wit: war probability *W*, although both functions are positively valued: Congruent political effects. Change in either escalation probability *!* or number of crises *C* has a direct effect on *W*, as shown by strictly positive values of the graphs of derivatives and war probability *W*, although both functions are positively valued: Congruent political effects. Change in either escalation probability *!* or number of crises *C* has a direct effect on *W*, as shown by strictly positive values of the graphs of derivatives and war probability *W*, although both functions are positively valued: Congruent political effects. Change in either escalation probability *!* or number of crises *C* has a direct effect on *W*, as shown by strictly positive values of the graphs of derivatives and Changes in escalation probability *ω* and number of crises *C* have clearly different effects on epochal war probability *W*, although both functions are positively valued:

Changes in escalation probability *!* and number of crises *C* have clearly different effects on epochal

Changes in escalation probability *!* and number of crises *C* have clearly different effects on epochal

Changes in escalation probability *!* and number of crises *C* have clearly different effects on epochal

Opposite political effects. Whereas change in causal probability *P* has proportional or positive effect on *E*, change in event size *N* has an opposite effect. This is shown by the purple deep bottom in Figure 1c versus the high red plateau in Figure 1e; both features observed over approximately the same underlying (*p, n*)-domain. differences in Figure 3c through f. First- and second-order effects. Whereas the contour plot of *W*(*!, C*) shows strictly concave isocontours (Figure 3b), contours of the derivative *@!W* (Figure 3d) also show a convex, mild differences in Figure 3c through f. First- and second-order effects. Whereas the contour plot of *W*(*!, C*) shows strictly concave isocontours (Figure 3b), contours of the derivative *@!W* (Figure 3d) also show a convex, mild differences in Figure 3c through f. First- and second-order effects. Whereas the contour plot of *W*(*!, C*) shows strictly concave isocontours (Figure 3b), contours of the derivative *@!W*(Figure 3d) also show a convex, mild **Congruent political effects.** Change in either escalation probability *ω* or number of crises *C* has a direct effect on *W*, as shown by strictly positive values of the graphs of derivatives and differences in Figure 3c through f.

spur protruding on the southwestern wall of the escarpment, near *p* = 0*.*2, a second-order

spur protruding on the southwestern wall of the escarpment, near *p* = 0*.*2, a second-order

low values of *!* as *C* increases away from the minimal value of 2. All the isolines show mixed

Geometrically opposite spurs. The mild *C*-grown spur shown by *@!W* in Figures 3c and d and the more pronounced *!*-grown spur in ∆*CW* in Figures 3e and f have a type of reflective symmetry: the former protrudes from the *C* boundary at low values of *!* and *C*, while the latter extends from the *!*-boundary from low values (*p ⇡* 0*.*2) toward high values of *C*. Both features are indicative of nonlinear effects on war probability *W* caused by underlying

low values of *!* as *C* increases away from the minimal value of 2. All the isolines show mixed

Geometrically opposite spurs. The mild *C*-grown spur shown by *@!W* in Figures 3c and d and the more pronounced *!*-grown spur in ∆*CW* in Figures 3e and f have a type of reflective symmetry: the former protrudes from the *C* boundary at low values of *!* and *C*, while the latter extends from the *!*-boundary from low values (*p ⇡* 0*.*2) toward high values of *C*. Both features are indicative of nonlinear effects on war probability *W* caused by underlying

<sup>6</sup> While Wright formulated the high-level crisis-probability framework, Snyder completed it by providing the

spur protruding on the southwestern wall of the escarpment, near *p* = 0*.*2, a second-order

low values of *!* as *C* increases away from the minimal value of 2. All the isolines show mixed

First- and second-order effects. Whereas the contour plot of *E*(*P, N*) shows strictly concave

Second-order effects caused by changes in event size. The graphs of the change in *E* with respect to event size, ∆*nE* (equation 8 and Figures 1e and f), show a pronounced ravine or canyon along the north-south direction dropping into a deep precipice at relatively high values of *P* as *N !* 2. Interestingly, in this case all the isolines have mixed concavity (low

concavity and convexity, which is another second-order effect.

concavity and convexity, which is another second-order effect.

Geometrically opposite spurs. The mild *C*-grown spur shown by *@!W* in Figures 3c and d and the more pronounced *!*-grown spur in ∆*CW* in Figures 3e and f have a type of reflective symmetry: the former protrudes from the *C* boundary at low values of *!* and *C*, while the latter extends from the *!*-boundary from low values (*p ⇡* 0*.*2) toward high values of *C*. Both features are indicative of nonlinear effects on war probability *W* caused by underlying

probabilistic causal mechanism (Suppes 1984) within the crisis branching process.

<sup>6</sup> While Wright formulated the high-level crisis-probability framework, Snyder completed it by providing the

<sup>6</sup> While Wright formulated the high-level crisis-probability framework, Snyder completed it by providing the

Geometrically opposite extrema. The extreme high range of *@pE* in Figures 1c and d and the extreme low range of ∆*nE* in Figures 1e and f have opposite (or inverse) geometries with some common similarities: the former rises from a flat basin to an escarpment while the latter drops from a large plateau toward a deepening canyon that dives into a deep well. Both features are indicative of major political effects on event probability *E* caused by

probabilistic causal mechanism (Suppes 1984) within the crisis branching process.

changes in crisis escalation *!* and number of crises *C*.

*P*) and convexity (higher *P* values), which is another second-order effect.

concavity and convexity, which is another second-order effect.

changes in crisis escalation *!* and number of crises *C*.

probabilistic causal mechanism (Suppes 1984) within the crisis branching process.

underlying changes in causal probabilities *P* and event size *N*.

changes in crisis escalation *!* and number of crises *C*.

16

16

16

12

isocontours (Figure 1b), the contours of the derivative *@pE* (Figure 1d) also show a convex, mild spur protruding on the southwestern wall of the escarpment, near *p* = 0*.*75 (third contour, between green and blue elevations), which means more complex change for small-size effect in opposite direction to the previous case (Figure 1d) and, again, not apparent from the basic model (equation 13 and Figures 3a and b). Second-order effects caused by changes in number of crises. The graphs of the change effect in opposite direction to the previous case (Figure 1d) and, again, not apparent from the basic model (equation 13 and Figures 3a and b). Second-order effects caused by changes in number of crises. The graphs of the change effect in opposite direction to the previous case (Figure 1d) and, again, not apparent from the basic model (equation 13 and Figures 3a and b). Second-order effects caused by changes in number of crises. The graphs of the change <sup>6</sup> While Wright formulated the high-level crisis-probability framework, Snyder completed it by providing the probabilistic causal mechanism (Suppes 1984) within the crisis branching process.

CLAUDIO CIOFFI-REVILLA

CLAUDIO CIOFFI-REVILLA

4.2 Case 2: Crisis dynamics and onset of war

is trivial, since *W*(*!,* 1) = *!*, as is easily shown.

4.2 Case 2: Crisis dynamics and onset of war

is trivial, since *W*(*!,* 1) = *!*, as is easily shown.

CLAUDIO CIOFFI-REVILLA

4.2 Case 2: Crisis dynamics and onset of war

theorem), so a country's epochal probability of war is given by

CLAUDIO CIOFFI-REVILLA

equations being functions of compound events).

theorem), so a country's epochal probability of war is given by

*W* = 1 *−* (1 *− !*)

which is a bivariate nonlinear hybrid function, where *!* is continuous over the closed unit probability interval [0*,* 1] and *C ≥* 2 is discrete. The case when *C* = 1 (a single crisis during an entire epoch)

The politically relevant domain is bound by 0 *!* 1 and 2 *C* / 20 (same as before, both

theorem), so a country's epochal probability of war is given by

Explaining the outbreak of international war as caused by a prior crisis escalation process—which includes challenging and resistance moves, bargaining, signaling, and other events, as opposed to some other causal mechanism—was first proposed by Quincy Wright (Wright 1942, 1271, fn. 38) and later extended and generalized by Glenn H. Snyder (Snyder & Diesing 1977, 13–17 et passim). The frame of reference here is the inter-state relational level of analysis and the specific *explanandum* is the probability *W* of a state being at war over a period of time, called an epoch in probability theory.<sup>6</sup> During an epoch a country experiences a number *C* of crises (defined as episodes during which hostilities may occur), each individual crisis having probability *!* of escalating to war. As a result, the probability of no war over *<sup>C</sup>* crises is (1 *<sup>−</sup> !*)*<sup>C</sup>* (by Kolmogorov's

4.2 Case 2: Crisis dynamics and onset of war

Explaining the outbreak of international war as caused by a prior crisis escalation process—which includes challenging and resistance moves, bargaining, signaling, and other events, as opposed to some other causal mechanism—was first proposed by Quincy Wright (Wright 1942, 1271, fn. 38) and later extended and generalized by Glenn H. Snyder (Snyder & Diesing 1977, 13–17 et passim). The frame of reference here is the inter-state relational level of analysis and the specific *explanandum* is the probability *W* of a state being at war over a period of time, called an epoch in probability theory.<sup>6</sup> During an epoch a country experiences a number *C* of crises (defined as episodes during which hostilities may occur), each individual crisis having probability *!* of escalating to war. As a result, the probability of no war over *<sup>C</sup>* crises is (1 *<sup>−</sup> !*)*<sup>C</sup>* (by Kolmogorov's

*W* = 1 *−* (1 *− !*)

which is a bivariate nonlinear hybrid function, where *!* is continuous over the closed unit probability interval [0*,* 1] and *C ≥* 2 is discrete. The case when *C* = 1 (a single crisis during an entire epoch)

The politically relevant domain is bound by 0 *!* 1 and 2 *C* / 20 (same as before, both

Explaining the outbreak of international war as caused by a prior crisis escalation process—which includes challenging and resistance moves, bargaining, signaling, and other events, as opposed to some other causal mechanism—was first proposed by Quincy Wright (Wright 1942, 1271, fn. 38) and later extended and generalized by Glenn H. Snyder (Snyder & Diesing 1977, 13–17 et passim). The frame of reference here is the inter-state relational level of analysis and the specific *explanandum* is the probability *W* of a state being at war over a period of time, called an epoch in probability theory.<sup>6</sup> During an epoch a country experiences a number *C* of crises (defined as episodes during which hostilities may occur), each individual crisis having probability *!* of escalating to war. As a result, the probability of no war over *<sup>C</sup>* crises is (1 *<sup>−</sup> !*)*<sup>C</sup>* (by Kolmogorov's

*<sup>C</sup> ,* (13)

*W* = 1 *−* (1 *− !*)

Explaining the outbreak of international war as caused by a prior crisis escalation process—which includes challenging and resistance moves, bargaining, signaling, and other events, as opposed to some other causal mechanism—was first proposed by Quincy Wright (Wright 1942, 1271, fn. 38) and later extended and generalized by Glenn H. Snyder (Snyder & Diesing 1977, 13–17 et passim). The frame of reference here is the inter-state relational level of analysis and the specific *explanandum* is the probability *W* of a state being at war over a period of time, called an epoch in probability theory.<sup>6</sup> During an epoch a country experiences a number *C* of crises (defined as episodes during which hostilities may occur), each individual crisis having probability *!* of escalating to war. As a result, the probability of no war over *<sup>C</sup>* crises is (1 *<sup>−</sup> !*)*<sup>C</sup>* (by Kolmogorov's

The 3D surface graph of *W*(*!, C*) is in Figure 3a, which shows the probability of war rising rapidly to a maximal plateau as escalation probability *!* and *C* increase. The contour plot in Figure 3b looks straight down, which highlights the broad plateau where war probability converges

theorem), so a country's epochal probability of war is given by

Each variable increases *W* in a different way, as shown in Figures 3c through f. Calculating

*W* = 1 *−* (1 *− !*)

which is a bivariate nonlinear hybrid function, where *!* is continuous over the closed unit probability

*<sup>C</sup> ,* (13)

*<sup>C</sup>−*<sup>1</sup> (for *!*'s effect on *W*, in Figures 3c and d) (14)

*<sup>C</sup>−*<sup>1</sup> (for *!*'s effect on *W*, in Figures 3c and d) (14)

*<sup>C</sup>* (for *C*'s effect on *W*, in Figures 3e and f)*.* (15)

*<sup>C</sup>* (for *C*'s effect on *W*, in Figures 3e and f)*.* (15)

*<sup>C</sup> ,* (13)

*<sup>C</sup> ,* (13)

**First- and second-order effects.**Whereas the contour plot of *W*(*ω*, *C*) shows strictly concave isocontours (Figure 3b), contours of the derivative the partial derivative and partial difference of *W* with respect to *!* and *C*, respectively: *@!W* = *C*(1 *− !*) *<sup>C</sup>−*<sup>1</sup> (for *!*'s effect on *W*, in Figures 3c and d) (14) ∆*cW* = *!*(1 *− !*) *<sup>C</sup>* (for *C*'s effect on *W*, in Figures 3e and f)*.* (15) Changes in escalation probability *!* and number of crises *C* have clearly different effects on epochal war probability *W*, although both functions are positively valued: Congruent political effects. Change in either escalation probability *!* or number of crises *C* has a direct effect on *W*, as shown by strictly positive values of the graphs of derivatives and differences in Figure 3c through f. First- and second-order effects. Whereas the contour plot of *W*(*!, C*) shows strictly concave isocontours (Figure 3b), contours of the derivative *@!W* (Figure 3d) also show a convex, mild spur protruding on the southwestern wall of the escarpment, near *p* = 0*.*2, a second-order effect in opposite direction to the previous case (Figure 1d) and, again, not apparent from the basic model (equation 13 and Figures 3a and b). Second-order effects caused by changes in number of crises. The graphs of the change in *E* with respect to number of crises, ∆*cW* (equation 14 and Figures 3e and f), show a pronounced spur along the north-south direction descending from a high value of *W* along (Figure 3d) also show a convex, mild spur protruding on the southwestern wall of the escarpment, near *p* = 0.2, a second-order effect in opposite direction to the previous case (Figure 1d) and, again, not apparent from the basic model (equation 13 and Figures 3a and b). **Second-order effects caused by changes in number of crises.** The graphs of the change in *E* with respect to number of crises, rapidly to a maximal plateau as escalation probability *!* and *C* increase. The contour plot in Figure 3b looks straight down, which highlights the broad plateau where war probability converges to 1, flanked by the steep north-south escarpment along the west edge as *! !* 0. Each variable increases *W* in a different way, as shown in Figures 3c through f. Calculating the partial derivative and partial difference of *W* with respect to *!* and *C*, respectively: *@!W* = *C*(1 *− !*) *<sup>C</sup>−*<sup>1</sup> (for *!*'s effect on *W*, in Figures 3c and d) (14) ∆*cW* = *!*(1 *− !*) *<sup>C</sup>* (for *C*'s effect on *W*, in Figures 3e and f)*.* (15) Changes in escalation probability *!* and number of crises *C* have clearly different effects on epochal war probability *W*, although both functions are positively valued: Congruent political effects. Change in either escalation probability *!* or number of crises *C* has a direct effect on *W*, as shown by strictly positive values of the graphs of derivatives and differences in Figure 3c through f. First- and second-order effects. Whereas the contour plot of *W*(*!, C*) shows strictly concave isocontours (Figure 3b), contours of the derivative *@!W* (Figure 3d) also show a convex, mild spur protruding on the southwestern wall of the escarpment, near *p* = 0*.*2, a second-order effect in opposite direction to the previous case (Figure 1d) and, again, not apparent from (equation 14 and Figures 3e and f), show a pronounced spur along the north-south direction descending from a high value of *W* along low values of *ω* as *C* increases away from the minimal value of 2. All the isolines show mixed concavity and convexity, which is another second-order effect. **Geometrically opposite spurs.** The mild *C*-grown spur shown by which is a bivariate nonlinear hybrid function, where *!* is continuous over the closed unit probability interval [0*,* 1] and *C ≥* 2 is discrete. The case when *C* = 1 (a single crisis during an entire epoch) is trivial, since *W*(*!,* 1) = *!*, as is easily shown. The politically relevant domain is bound by 0 *!* 1 and 2 *C* / 20 (same as before, both equations being functions of compound events). The 3D surface graph of *W*(*!, C*) is in Figure 3a, which shows the probability of war rising rapidly to a maximal plateau as escalation probability *!* and *C* increase. The contour plot in Figure 3b looks straight down, which highlights the broad plateau where war probability converges to 1, flanked by the steep north-south escarpment along the west edge as *! !* 0. Each variable increases *W* in a different way, as shown in Figures 3c through f. Calculating the partial derivative and partial difference of *W* with respect to *!* and *C*, respectively: *@!W* = *C*(1 *− !*) ∆*cW* = *!*(1 *− !*) Changes in escalation probability *!* and number of crises *C* have clearly different effects on epochal war probability *W*, although both functions are positively valued: Congruent political effects. Change in either escalation probability *!* or number of crises *C* has a direct effect on *W*, as shown by strictly positive values of the graphs of derivatives and in Figures 3c and d and the more pronounced *ω*-grown spur in interval [0*,* 1] and *C ≥* 2 is discrete. The case when *C* = 1 (a single crisis during an entire epoch) is trivial, since *W*(*!,* 1) = *!*, as is easily shown. The politically relevant domain is bound by 0 *!* 1 and 2 *C* / 20 (same as before, both equations being functions of compound events). The 3D surface graph of *W*(*!, C*) is in Figure 3a, which shows the probability of war rising rapidly to a maximal plateau as escalation probability *!* and *C* increase. The contour plot in Figure 3b looks straight down, which highlights the broad plateau where war probability converges to 1, flanked by the steep north-south escarpment along the west edge as *! !* 0. Each variable increases *W* in a different way, as shown in Figures 3c through f. Calculating the partial derivative and partial difference of *W* with respect to *!* and *C*, respectively: *@!W* = *C*(1 *− !*) ∆*cW* = *!*(1 *− !*) Changes in escalation probability *!* and number of crises *C* have clearly different effects on epochal war probability *W*, although both functions are positively valued: Congruent political effects. Change in either escalation probability *!* or number of crises *C* has a direct effect on *W*, as shown by strictly positive values of the graphs of derivatives and in Figures 3e and f have a type of reflective symmetry: the former protrudes from the *C* boundary at low values of *ω* and *C*, while the latter extends from the *ω*-boundary from low values (*p ≈* 0.2) toward high values of *C*. Both features are indicative of nonlinear effects on war probability *W* caused by underlying changes in crisis escalation *ω* and number of crises *C*. CLAUDIO CIOFFI-REVILLA

low values of *!* as *C* increases away from the minimal value of 2. All the isolines show mixed concavity and convexity, which is another second-order effect. Geometrically opposite spurs. The mild *C*-grown spur shown by *@!W* in Figures 3c and d and the more pronounced *!*-grown spur in ∆*CW* in Figures 3e and f have a type of reflective symmetry: the former protrudes from the *C* boundary at low values of *!* and *C*, while the latter extends from the *!*-boundary from low values (*p ⇡* 0*.*2) toward high values of *C*. Both features are indicative of nonlinear effects on war probability *W* caused by underlying the basic model (equation 13 and Figures 3a and b). Second-order effects caused by changes in number of crises. The graphs of the change in *E* with respect to number of crises, ∆*cW* (equation 14 and Figures 3e and f), show a pronounced spur along the north-south direction descending from a high value of *W* along low values of *!* as *C* increases away from the minimal value of 2. All the isolines show mixed concavity and convexity, which is another second-order effect. Geometrically opposite spurs. The mild *C*-grown spur shown by *@!W* in Figures 3c and d differences in Figure 3c through f. First- and second-order effects. Whereas the contour plot of *W*(*!, C*) shows strictly concave isocontours (Figure 3b), contours of the derivative *@!W* (Figure 3d) also show a convex, mild spur protruding on the southwestern wall of the escarpment, near *p* = 0*.*2, a second-order effect in opposite direction to the previous case (Figure 1d) and, again, not apparent from the basic model (equation 13 and Figures 3a and b). Second-order effects caused by changes in number of crises. The graphs of the change differences in Figure 3c through f. First- and second-order effects. Whereas the contour plot of *W*(*!, C*) shows strictly concave isocontours (Figure 3b), contours of the derivative *@!W* (Figure 3d) also show a convex, mild spur protruding on the southwestern wall of the escarpment, near *p* = 0*.*2, a second-order effect in opposite direction to the previous case (Figure 1d) and, again, not apparent from the basic model (equation 13 and Figures 3a and b). Second-order effects caused by changes in number of crises. The graphs of the change Having obtained some initial insights on the nature of war probability as a function of crisis dynamics, we now investigate the Wright-Snyder hybrid model through standardized variables not based on units and, therefore, enable direct comparisons of causal effects. In this case we shall proceed by obtaining and analyzing elasticities of *W* with respect to *ω* and *C*, as shown in Figure 4. Calculation of the point elasticity and the arc elasticity of *W* with respect to *ω* and *C* yields the following set of hybrid equations: Having obtained some initial insights on the nature of war probability as a function of crisis dynamics, we now investigate the Wright-Snyder hybrid model through standardized variables not based on units and, therefore, enable direct comparisons of causal effects. In this case we shall proceed by obtaining and analyzing elasticities of *W* with respect to *!* and *C*, as shown in Figure 4. Calculation of the point elasticity and the arc elasticity of *W* with respect to *!* and *C* yields the following set of hybrid equations: Having obtained some initial insights on the nature of war probability as a function of crisis dynamics, we now investigate the Wright-Snyder hybrid model through standardized variables not based on units and, therefore, enable direct comparisons of causal effects. In this case we shall proceed by obtaining and analyzing elasticities of *W* with respect to *!* and *C*, as shown in Figure 4. Calculation of the point elasticity and the arc elasticity of *W* with respect to *!* and *C* yields the following set of hybrid equations:

$$\eta\_{\omega}W = \frac{C\omega(1-\omega)^{C-1}}{1-(1-\omega)^{C}} \qquad \begin{array}{l} \text{(for }\omega\text{'s percentage effect on W/in Figure)} \\ \text{4a and b} \end{array} \quad (16)$$

$$\eta\_{\psi}W = C\omega(\frac{1}{1-(1-\omega)^{C}}-1) \qquad \begin{array}{l} \text{(for }C\text{'s percentage effect on W/in Figure)} \\ \text{4c and d)} \end{array} \quad (17)$$

16 <sup>6</sup> While Wright formulated the high-level crisis-probability framework, Snyder completed it by providing the probabilistic causal mechanism (Suppes 1984) within the crisis branching process. 16 and the more pronounced *!*-grown spur in ∆*CW* in Figures 3e and f have a type of reflective symmetry: the former protrudes from the *C* boundary at low values of *!* and *C*, while the latter extends from the *!*-boundary from low values (*p ⇡* 0*.*2) toward high values of *C*. Both features are indicative of nonlinear effects on war probability *W* caused by underlying changes in crisis escalation *!* and number of crises *C*. <sup>6</sup> While Wright formulated the high-level crisis-probability framework, Snyder completed it by providing the probabilistic causal mechanism (Suppes 1984) within the crisis branching process. 16 and the more pronounced *!*-grown spur in ∆*CW* in Figures 3e and f have a type of reflective symmetry: the former protrudes from the *C* boundary at low values of *!* and *C*, while the latter extends from the *!*-boundary from low values (*p ⇡* 0*.*2) toward high values of *C*. Both features are indicative of nonlinear effects on war probability *W* caused by underlying changes in crisis escalation *!* and number of crises *C*. <sup>6</sup> While Wright formulated the high-level crisis-probability framework, Snyder completed it by providing the probabilistic causal mechanism (Suppes 1984) within the crisis branching process. 16 Based on these equations we see again that these standardized effects on *W* are quite different from the absolute, unit-based effects uncovered earlier (equations 14 and 15 and associated figures). Recalling the meaning of elasticities, here, point elasticity stands for percentage change in the probability of war onset with respect to percentage change in crisis escalation probability *!*, while arc elasticity is the percentage change in epochal war probability *W* with respect to percentage change in number of crises *C* during the epoch. In this case both elasticities are rather complicated rational hybrid functions, including denominators that are exponential in *C* (from the standardizing transformation), as shown in Figures 4a through d. While formal analysis is feasible, visual analytics of graphs reveal numerous interesting features. Figures 4a and b both show that point elasticity is high (i.e., war probability is strongly affected by escalation probability) at low *!* values and highest at lowest values of both *!* and *C* (red levels of the escarpment). This means that the risk of war changes most when crises are few and Based on these equations we see again that these standardized effects on *W* are quite different from the absolute, unit-based effects uncovered earlier (equations 14 and 15 and associated figures). Recalling the meaning of elasticities, here, point elasticity stands for percentage change in the probability of war onset with respect to percentage change in crisis escalation probability *!*, while arc elasticity is the percentage change in epochal war probability *W* with respect to percentage change in number of crises *C* during the epoch. In this case both elasticities are rather complicated rational hybrid functions, including denominators that are exponential in *C* (from the standardizing transformation), as shown in Figures 4a through d. While formal analysis is feasible, visual analytics of graphs reveal numerous interesting features. Figures 4a and b both show that point elasticity is high (i.e., war probability is strongly affected by escalation probability) at low *!* values and highest at lowest values of both *!* and *C* (red levels of the escarpment). This means that the risk of war changes most when crises are few and Based on these equations we see again that these standardized effects on *W* are quite different from the absolute, unit-based effects uncovered earlier (equations 14 and 15 and associated figures). Recalling the meaning of elasticities, here, point elasticity stands for percentage change in the probability of war onset with respect to percentage change in crisis escalation probability *ω*, while arc elasticity is the percentage change in epochal war probability *W* with respect to percentage change in number of crises *C* during the epoch. In this case both elasticities are rather complicated rational hybrid functions, including denominators that are exponential in *C* (from the standardizing transformation), as shown in Figures 4a through d. While formal analysis is feasible, visual analytics of graphs reveal numerous interesting features.

escalation probability low. By contrast, at the blue-green levels, war probability *W* is less sensitive to such instabilities (greater number of crises and higher escalation probability). A rather subtle and surprising feature occurs along the front edge of the 3D surface in Figure 4, where point elasticity for all values of *!* at the minimal boundary of *C* = 2 is convex (bulging up), whereas elsewhere (away from the front edge of the surface) point elasticity is strictly concave escalation probability low. By contrast, at the blue-green levels, war probability *W* is less sensitive to such instabilities (greater number of crises and higher escalation probability). A rather subtle and surprising feature occurs along the front edge of the 3D surface in Figure 4, where point elasticity for all values of *!* at the minimal boundary of *C* = 2 is convex (bulging up), whereas elsewhere (away from the front edge of the surface) point elasticity is strictly concave Figures 4a and b both show that point elasticity is high (i.e., war probability is strongly affected by escalation probability) at low *ω* values and highest at lowest values of both *ω* and *C* (red levels of the escarpment). This means that the risk of war changes most when crises are few and escalation probability low. By

convexity in this case being nonexistent (even at low *C* levels along the front edge).

1 *−* (1 *− !*)

1 *−* (1 *− !*)

crises surpasses the first single digits.

convexity in this case being nonexistent (even at low *C* levels along the front edge).

using the nabladot operator, as follows:

1 *−* (1 *− !*)

⇥

= *C*(1 *− !*)

function, is in Figure 4f.

and epochal war probability (OR-based disjunctive), respectively.

*r· W* = *@!*

18

crises surpasses the first single digits.

using the nabladot operator, as follows:

function, is in Figure 4f.

18

CLAUDIO CIOFFI-REVILLA

*r· W* = *@!*

*<sup>C</sup>−*<sup>1</sup> <sup>i</sup> <sup>+</sup> *!*(1 *<sup>−</sup> !*)

which is a two-dimensional vector function W = (*!, C*). The resulting vector field of this hybrid gradient is in Figure 4e and corresponding vector magnitude or norm *|*W*|*(*!, C*), now a scalar

We see from these results that both probability vector fields and norms *|*W*|* and *|*E*|*, in Figures 2e and f and Figures 4e and f, resemble each other—a surprising result from comparative analysis based on perfect bilateral vertical symmetries around the *P* = *!* = 0*.*5 axis. This is another political property not apparent from simple inspection of the basic models but consistent with formal fundamental symmetry and equivalence between causal logic conjunction and disjunction (De Morgan's laws) associated with the probability of international events (AND-based conjunctive)

*<sup>C</sup>* ⇤

two (*C ≥* 3). Concavity in point elasticity of *W* with respect to *!* accelerates as the number of

By contrast, arc elasticity of war probability *W* is strictly concave, as seen in Figures 4c and d,

Comparing the two elasticities (equations 16 and 17) yields the following dominance principle: epochal probability of war *W* is more sensitive to change in escalation probability *!* than to change in number of crises *C*, because point elasticity *⌘!* is greater than arc elasticity *⌘c*. The different but joint effects of escalation probability *!* and number of crises *C* on epochal war probability *W* are analyzed and understood by calculating the gradient of *W* with respect to both variables

⇥

i + ∆*<sup>C</sup>* ⇥

= *C*(1 *− !*)

and epochal war probability (OR-based disjunctive), respectively.

two (*C ≥* 3). Concavity in point elasticity of *W* with respect to *!* accelerates as the number of

By contrast, arc elasticity of war probability *W* is strictly concave, as seen in Figures 4c and d,

Comparing the two elasticities (equations 16 and 17) yields the following dominance principle: epochal probability of war *W* is more sensitive to change in escalation probability *!* than to change in number of crises *C*, because point elasticity *⌘!* is greater than arc elasticity *⌘c*. The different but joint effects of escalation probability *!* and number of crises *C* on epochal war probability *W* are analyzed and understood by calculating the gradient of *W* with respect to both variables

*<sup>C</sup>* ⇤

*<sup>C</sup>−*<sup>1</sup> <sup>i</sup> <sup>+</sup> *!*(1 *<sup>−</sup> !*)

which is a two-dimensional vector function W = (*!, C*). The resulting vector field of this hybrid gradient is in Figure 4e and corresponding vector magnitude or norm *|*W*|*(*!, C*), now a scalar

We see from these results that both probability vector fields and norms *|*W*|* and *|*E*|*, in Figures 2e and f and Figures 4e and f, resemble each other—a surprising result from comparative analysis based on perfect bilateral vertical symmetries around the *P* = *!* = 0*.*5 axis. This is another political property not apparent from simple inspection of the basic models but consistent with formal fundamental symmetry and equivalence between causal logic conjunction and disjunction (De Morgan's laws) associated with the probability of international events (AND-based conjunctive)

i + ∆*<sup>C</sup>* ⇥

*<sup>C</sup>* j*,* (19)

*<sup>C</sup>* ⇤

1 *−* (1 *− !*)

j (18)

*<sup>C</sup>* ⇤

*<sup>C</sup>* j*,* (19)

j (18)

NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS

Figura 3. Epochal probability of war *W*(*!, C*) as a hybrid function of crisis escalation probability *!* and number of crises *C*: computational imagery from visualization analytics. (a) 3D surface of the hybrid function *<sup>W</sup>* = (1 *<sup>−</sup> !*)*<sup>C</sup>* ; (b) contour plot of (a); (c) 3D surface of *@!W*, the first-order partial derivative of *<sup>W</sup>*(*!, C*) with respect to *!*; (d) contour plot of (c); (e) 3D surface of ∆*c*, the first-order partial difference of *W*(*!, C*) with respect to *C*; (f) contour plot of (e). 17 Figure 3. Epochal probability of war *W*(ω, C) as a hybrid function of crisis escalation probability ω and number of crises *C*: computational imagery from visualization analytics. (a) 3D surface of the hybrid function *W* = (1 − ω)*<sup>C</sup>*; (b) contour plot of (a); (c) 3D surface of to 1, flanked by the steep north-south escarpment along the west edge as *! !* 0. Each variable increases *W* in a different way, as shown in Figures 3c through f. Calculating the partial derivative and partial difference of *W* with respect to *!* and *C*, respectively: *@!W* = *C*(1 *− !*) *<sup>C</sup>−*<sup>1</sup> (for *!*'s effect on *W*, in Figures 3c and d) (14) ∆*cW* = *!*(1 *− !*) *<sup>C</sup>* (for *C*'s effect on *W*, in Figures 3e and f)*.* (15) Changes in escalation probability *!* and number of crises *C* have clearly different effects on epochal , the fi rst-order partial derivative of *W*(ω, *C*) with respect to ω; (d) contour plot of (c). (e) 3D surface of δ*<sup>c</sup>* , the fi rst-order partial diff erence of *W*(*ω*, *C*) with respect to *C*; (f) contour plot of (e).

war probability *W*, although both functions are positively valued:

the basic model (equation 13 and Figures 3a and b).

changes in crisis escalation *!* and number of crises *C*.

concavity and convexity, which is another second-order effect.

probabilistic causal mechanism (Suppes 1984) within the crisis branching process.

differences in Figure 3c through f.

Figure 3b looks straight down, which highlights the broad plateau where war probability converges

Congruent political effects. Change in either escalation probability *!* or number of crises *C* has a direct effect on *W*, as shown by strictly positive values of the graphs of derivatives and

First- and second-order effects. Whereas the contour plot of *W*(*!, C*) shows strictly concave isocontours (Figure 3b), contours of the derivative *@!W* (Figure 3d) also show a convex, mild spur protruding on the southwestern wall of the escarpment, near *p* = 0*.*2, a second-order effect in opposite direction to the previous case (Figure 1d) and, again, not apparent from

Second-order effects caused by changes in number of crises. The graphs of the change in *E* with respect to number of crises, ∆*cW* (equation 14 and Figures 3e and f), show a pronounced spur along the north-south direction descending from a high value of *W* along low values of *!* as *C* increases away from the minimal value of 2. All the isolines show mixed

Geometrically opposite spurs. The mild *C*-grown spur shown by *@!W* in Figures 3c and d and the more pronounced *!*-grown spur in ∆*CW* in Figures 3e and f have a type of reflective symmetry: the former protrudes from the *C* boundary at low values of *!* and *C*, while the latter extends from the *!*-boundary from low values (*p ⇡* 0*.*2) toward high values of *C*. Both features are indicative of nonlinear effects on war probability *W* caused by underlying

<sup>6</sup> While Wright formulated the high-level crisis-probability framework, Snyder completed it by providing the

16

Based on these equations we see again that these standardized effects on *W* are quite different from the absolute, unit-based effects uncovered earlier (equations 14 and 15 and associated figures). Recalling the meaning of elasticities, here, point elasticity stands for percentage change in the probability of war onset with respect to percentage change in crisis escalation probability

<sup>1</sup> *<sup>−</sup>* (1 *<sup>−</sup> !*)*<sup>C</sup>* (for *!*'s percentage effect on *<sup>W</sup>*, in Figures 4a and b) (16)

<sup>1</sup> *<sup>−</sup>* (1 *<sup>−</sup> !*)*<sup>C</sup> <sup>−</sup>* 1) (for *<sup>C</sup>*'s percentage effect on *<sup>W</sup>*, in Figures 4c and d)*.* (17)

Having obtained some initial insights on the nature of war probability as a function of crisis dynamics, we now investigate the Wright-Snyder hybrid model through standardized variables not based on units and, therefore, enable direct comparisons of causal effects. In this case we shall proceed by obtaining and analyzing elasticities of *W* with respect to *!* and *C*, as shown in Figure 4. Calculation of the point elasticity and the arc elasticity of *W* with respect to *!* and *C* yields

Having obtained some initial insights on the nature of war probability as a function of crisis dynamics, we now investigate the Wright-Snyder hybrid model through standardized variables not based on units and, therefore, enable direct comparisons of causal effects. In this case we shall proceed by obtaining and analyzing elasticities of *W* with respect to *!* and *C*, as shown in Figure 4. Calculation of the point elasticity and the arc elasticity of *W* with respect to *!* and *C* yields

Based on these equations we see again that these standardized effects on *W* are quite different from the absolute, unit-based effects uncovered earlier (equations 14 and 15 and associated figures). Recalling the meaning of elasticities, here, point elasticity stands for percentage change in the probability of war onset with respect to percentage change in crisis escalation probability

<sup>1</sup> *<sup>−</sup>* (1 *<sup>−</sup> !*)*<sup>C</sup>* (for *!*'s percentage effect on *<sup>W</sup>*, in Figures 4a and b) (16)

<sup>1</sup> *<sup>−</sup>* (1 *<sup>−</sup> !*)*<sup>C</sup> <sup>−</sup>* 1) (for *<sup>C</sup>*'s percentage effect on *<sup>W</sup>*, in Figures 4c and d)*.* (17)

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the following set of hybrid equations:

visual analytics of graphs reveal numerous interesting features.

*r· W* = *@!*

⇥

= *C*(1 *− !*)

1 *−* (1 *− !*)

*<sup>C</sup>* ⇤

*<sup>C</sup>−*<sup>1</sup> <sup>i</sup> <sup>+</sup> *!*(1 *<sup>−</sup> !*)

which is a two-dimensional vector function W = (*!, C*). The resulting vector field of this hybrid

i + ∆*<sup>C</sup>* ⇥

1 *−* (1 *− !*)

*<sup>C</sup>* ⇤

*<sup>C</sup>* j*,* (19)

j (18)

*⌘!<sup>W</sup>* <sup>=</sup> *C!*(1 *<sup>−</sup> !*)*<sup>C</sup>−*<sup>1</sup>

*⌘c<sup>W</sup>* <sup>=</sup> *C!*( <sup>1</sup>

Having obtained some initial insights on the nature of war probability as a function of crisis dynamics, we now investigate the Wright-Snyder hybrid model through standardized variables not based on units and, therefore, enable direct comparisons of causal effects. In this case we shall proceed by obtaining and analyzing elasticities of *W* with respect to *!* and *C*, as shown in Figure 4. Calculation of the point elasticity and the arc elasticity of *W* with respect to *!* and *C* yields

Based on these equations we see again that these standardized effects on *W* are quite different from the absolute, unit-based effects uncovered earlier (equations 14 and 15 and associated figures). Recalling the meaning of elasticities, here, point elasticity stands for percentage change in the probability of war onset with respect to percentage change in crisis escalation probability *!*, while arc elasticity is the percentage change in epochal war probability *W* with respect to percentage change in number of crises *C* during the epoch. In this case both elasticities are rather complicated rational hybrid functions, including denominators that are exponential in *C* (from the standardizing transformation), as shown in Figures 4a through d. While formal analysis is feasible,

by escalation probability) at low *!* values and highest at lowest values of both *!* and *C* (red

<sup>1</sup> *<sup>−</sup>* (1 *<sup>−</sup> !*)*<sup>C</sup>* (for *!*'s percentage effect on *<sup>W</sup>*, in Figures 4a and b) (16)

<sup>1</sup> *<sup>−</sup>* (1 *<sup>−</sup> !*)*<sup>C</sup> <sup>−</sup>* 1) (for *<sup>C</sup>*'s percentage effect on *<sup>W</sup>*, in Figures 4c and d)*.* (17)

contrast, at the blue-green levels, war probability *W* is less sensitive to such instabilities (greater number of crises and higher escalation probability). levels of the escarpment). This means that the risk of war changes most when crises are few and escalation probability low. By contrast, at the blue-green levels, war probability *W* is less sensitive to such instabilities (greater number of crises and higher escalation probability). complicated rational hybrid functions, including denominators that are exponential in *C* (from the standardizing transformation), as shown in Figures 4a through d. While formal analysis is feasible, visual analytics of graphs reveal numerous interesting features. complicated rational hybrid functions, including denominators that are exponential in *C* (from the standardizing transformation), as shown in Figures 4a through d. While formal analysis is feasible, visual analytics of graphs reveal numerous interesting features.

CLAUDIO CIOFFI-REVILLA

the following set of hybrid equations:

*⌘!<sup>W</sup>* <sup>=</sup> *C!*(1 *<sup>−</sup> !*)*<sup>C</sup>−*<sup>1</sup>

*⌘c<sup>W</sup>* <sup>=</sup> *C!*( <sup>1</sup>

CLAUDIO CIOFFI-REVILLA

the following set of hybrid equations:

*r· W* = *@!*

⇥

*⌘!<sup>W</sup>* <sup>=</sup> *C!*(1 *<sup>−</sup> !*)*<sup>C</sup>−*<sup>1</sup>

*⌘c<sup>W</sup>* <sup>=</sup> *C!*( <sup>1</sup>

18

18

A rather subtle and surprising feature occurs along the front edge of the 3D surface in Figure 4, where point elasticity for all values of *ω* at the minimal boundary of *C =* 2 is convex (bulging up), whereas elsewhere (away from the front edge of the surface) point elasticity is strictly concave ( A rather subtle and surprising feature occurs along the front edge of the 3D surface in Figure 4, where point elasticity for all values of *!* at the minimal boundary of *C* = 2 is convex (bulging up), whereas elsewhere (away from the front edge of the surface) point elasticity is strictly concave (*@⌘ <* 0). This makes minimal epochs with only two crises rather special, which is not immediately intuitive. This particular property vanishes in all epochs with a multiplicity of crises beyond just two (*C ≥* 3). Concavity in point elasticity of *W* with respect to *!* accelerates as the number of crises surpasses the first single digits. By contrast, arc elasticity of war probability *W* is strictly concave, as seen in Figures 4c and d, < 0). This makes minimal epochs with only two crises rather special, which is not immediately intuitive. This particular property vanishes in all epochs with a multiplicity of crises beyond just two (*C* ≥ 3). Concavity in point elasticity of *W* with respect to *ω* accelerates as the number of crises surpasses the first single digits. Figures 4a and b both show that point elasticity is high (i.e., war probability is strongly affected by escalation probability) at low *!* values and highest at lowest values of both *!* and *C* (red levels of the escarpment). This means that the risk of war changes most when crises are few and escalation probability low. By contrast, at the blue-green levels, war probability *W* is less sensitive to such instabilities (greater number of crises and higher escalation probability). A rather subtle and surprising feature occurs along the front edge of the 3D surface in Figure 4, where point elasticity for all values of *!* at the minimal boundary of *C* = 2 is convex (bulging up), whereas elsewhere (away from the front edge of the surface) point elasticity is strictly concave Figures 4a and b both show that point elasticity is high (i.e., war probability is strongly affected by escalation probability) at low *!* values and highest at lowest values of both *!* and *C* (red levels of the escarpment). This means that the risk of war changes most when crises are few and escalation probability low. By contrast, at the blue-green levels, war probability *W* is less sensitive to such instabilities (greater number of crises and higher escalation probability). A rather subtle and surprising feature occurs along the front edge of the 3D surface in Figure 4, where point elasticity for all values of *!* at the minimal boundary of *C* = 2 is convex (bulging up), whereas elsewhere (away from the front edge of the surface) point elasticity is strictly concave

convexity in this case being nonexistent (even at low *C* levels along the front edge). Comparing the two elasticities (equations 16 and 17) yields the following dominance principle: epochal probability of war *W* is more sensitive to change in escalation probability *!* than to change By contrast, arc elasticity of war probability *W* is strictly concave, as seen in Figures 4c and d, convexity in this case being nonexistent (even at low *C* levels along the front edge). (*@⌘ <* 0). This makes minimal epochs with only two crises rather special, which is not immediately intuitive. This particular property vanishes in all epochs with a multiplicity of crises beyond just two (*C ≥* 3). Concavity in point elasticity of *W* with respect to *!* accelerates as the number of crises surpasses the first single digits. (*@⌘ <* 0). This makes minimal epochs with only two crises rather special, which is not immediately intuitive. This particular property vanishes in all epochs with a multiplicity of crises beyond just two (*C ≥* 3). Concavity in point elasticity of *W* with respect to *!* accelerates as the number of crises surpasses the first single digits.

in number of crises *C*, because point elasticity *⌘!* is greater than arc elasticity *⌘c*. The different but joint effects of escalation probability *!* and number of crises *C* on epochal war probability *W* are analyzed and understood by calculating the gradient of *W* with respect to both variables using the nabladot operator, as follows: Comparing the two elasticities (equations 16 and 17) yields the following dominance principle: epochal probability of war *W* is more sensitive to change in escalation probability *ω* than to change in number of crises *C*, because point elasticity *ηω* is greater than arc elasticity *ηc* . The different but joint effects of escalation probability *ω* and number of crises *C* on epochal war probability *W* are analyzed and understood by calculating the gradient of *W* with respect to both variables using the nabladot operator, as follows: By contrast, arc elasticity of war probability *W* is strictly concave, as seen in Figures 4c and d, convexity in this case being nonexistent (even at low *C* levels along the front edge). Comparing the two elasticities (equations 16 and 17) yields the following dominance principle: epochal probability of war *W* is more sensitive to change in escalation probability *!* than to change in number of crises *C*, because point elasticity *⌘!* is greater than arc elasticity *⌘c*. The different but joint effects of escalation probability *!* and number of crises *C* on epochal war probability *W* are analyzed and understood by calculating the gradient of *W* with respect to both variables using the nabladot operator, as follows: By contrast, arc elasticity of war probability *W* is strictly concave, as seen in Figures 4c and d, convexity in this case being nonexistent (even at low *C* levels along the front edge). Comparing the two elasticities (equations 16 and 17) yields the following dominance principle: epochal probability of war *W* is more sensitive to change in escalation probability *!* than to change in number of crises *C*, because point elasticity *⌘!* is greater than arc elasticity *⌘c*. The different but joint effects of escalation probability *!* and number of crises *C* on epochal war probability *W* are analyzed and understood by calculating the gradient of *W* with respect to both variables using the nabladot operator, as follows:

$$\nabla W = \partial\_{\omega} \left[ 1 - (1 - \omega)^{C} \right] \mathbf{i} + \Delta\_{C} \left[ 1 - (1 - \omega)^{C} \right] \mathbf{j} \tag{18}$$

$$\mathbf{j} = C(1 - \omega)^{C - 1}\mathbf{i} + \omega(1 - \omega)^{C}\mathbf{j},\tag{19}$$

based on perfect bilateral vertical symmetries around the *P* = *!* = 0*.*5 axis. This is another political property not apparent from simple inspection of the basic models but consistent with formal fundamental symmetry and equivalence between causal logic conjunction and disjunction (De Morgan's laws) associated with the probability of international events (AND-based conjunctive) which is a two-dimensional vector function W = (*!, C*). The resulting vector field of this hybrid gradient is in Figure 4e and corresponding vector magnitude or norm *|*W*|*(*!, C*), now a scalar function, is in Figure 4f. We see from these results that both probability vector fields and norms *|*W*|* and *|*E*|*, in Figures which is a two-dimensional vector function W = (*!, C*). The resulting vector field of this hybrid gradient is in Figure 4e and corresponding vector magnitude or norm *|*W*|*(*!, C*), now a scalar function, is in Figure 4f. We see from these results that both probability vector fields and norms *|*W*|* and *|*E*|*, in Figures which is a two-dimensional vector function **W** = **Ψ**(*ω*, *C*). The resulting vector field of this hybrid gradient is in Figure 4e and corresponding vector magnitude or norm |**W**|(*ω*, *C*), now a scalar function, is in Figure 4f.

and epochal war probability (OR-based disjunctive), respectively. 18 2e and f and Figures 4e and f, resemble each other—a surprising result from comparative analysis based on perfect bilateral vertical symmetries around the *P* = *!* = 0*.*5 axis. This is another political property not apparent from simple inspection of the basic models but consistent with formal fundamental symmetry and equivalence between causal logic conjunction and disjunction (De Morgan's laws) associated with the probability of international events (AND-based conjunctive) and epochal war probability (OR-based disjunctive), respectively. 2e and f and Figures 4e and f, resemble each other—a surprising result from comparative analysis based on perfect bilateral vertical symmetries around the *P* = *!* = 0*.*5 axis. This is another political property not apparent from simple inspection of the basic models but consistent with formal fundamental symmetry and equivalence between causal logic conjunction and disjunction (De Morgan's laws) associated with the probability of international events (AND-based conjunctive) and epochal war probability (OR-based disjunctive), respectively. We see from these results that both probability vector fields and norms |**W**| and |**E**|, in Figures 2e and f and Figures 4e and f, resemble each other—a surprising result from comparative analysis—based on perfect bilateral vertical symmetries around the *P = ω* = 0.5 axis. This is another political property not apparent from simple inspection of the basic models but consistent with formal fundamental symmetry and equivalence between causal logic conjunction and disjunction (De Morgan's laws) associated with the probability of international events (AND-based conjunctive) and epochal war probability (OR-based disjunctive), respectively.

#### 4.3 Case 3: Frequency of war and systemic polarity (Midlarsky's law)

Having just examined the probability of war at the relational level, in this last case study we again change our frame of reference, this time turning to the systemic level of analysis. The formal theoretical (and empirically supported) explanation for the annual frequency of warfare as a result of fundamental po-

NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS

Figura 4. Elasticities and gradient of the epochal probability of war *W*. (a) 3D surface of point elasticity *⌘!*(*W*) with respect to crisis escalation probability *!*; (b) contour plot of (a); (c) 3D surface of arc elasticity *⌘c*(*W*) with respect to epochal number of crises *C*; (d) contour plot of (c) ; (e) vector field of the dot-gradient vector function *r· W*; (f) contour plot of (e). 19 Figure 4. Elasticities and gradient of the epochal probability of war *W*. (a) 3D surface of point elasticity *ηω*(*W*) with respect to crisis escalation probability *ω*; (b) contour plot of (a); (c) 3D surface of arc elasticity *ηc* (*W*) with respect to epochal number of crises *C*; (d) contour plot of (c) ; (e) vector fi eld of the dot-gradient vector function simple inspection of the hybrid function under investigation. The analytical process thus far has focused on scalar properties of the IR hybrid function under investigation. The first nabladot operation is to calculate the hybrid gradient of *Z* to discover the magnitude *and* direction of changes in *Z* as a function of changes in *X* and *Y* . The result of applying the nabladot operator (a vector operator) to scalar hybrid function *'* is a hybrid vector function Φ = *r· '* with *x*- and *y*-components. The hybrid gradient in two dimensions is the scalar *W*; (f) contour plot of (e).

*r· ' ⌘ @x'* i + ∆*y'* j*,* (1)

vector product calculated using the new *nabladot vector operator r·* (note the dot within the nabla

2014; 2017; 2019; 2020; 2020).<sup>3</sup> Note that the resulting nabladot gradient of hybrid function *'* is a striated vector field with a first-order partial derivative component along the *x*-axis (continuous) and a first-order partial difference component along the *y*-axis (discrete), hence the striation of

The absolute and standardized norms of the hybrid gradient *r· '* are calculated next, along with corresponding graphs for investigating the resulting vector field. Each pair of plots for a vector field and corresponding norm should use identical domains to facilitate understanding through comparative analysis. Cardinal directions (N, E, S, W) are used for simple orientation in graphs. Other hybrid operations of nabladot calculus equivalent to the divergence, curl, Laplacian, Hessian, and Jacobian are subsequently calculated to shed additional (and usually new) light on the original function *Z* = *'*(*X, Y* ) through the medium of nabladot operators, each supported by additional

The main results of nabladot analysis shed new light on fundamental, real-world, substantive properties and features of the original hybrid function under investigation, features that remain hidden or inaccessible through other forms of analysis. Each main formal expression is accompanied by an interpretation in plain English, although this is not always possible without some loss of precision or clarity. Some results can be somewhat complicated nonlinear functions that do not further simplify; we prefer them that way rather than introducing artificial approximations which may be simpler but unrealistic or unnatural objects, unlike real IR phenomena. In most cases an ensemble of images and visual analytics (Thomas and Cook 2005; Wellin 2013) of complicated

Among the most important substantive (and testable) results from nabladot analysis are the dominance principles mentioned above—they explain which independent variable has dominant effect on the dependent variable, a major theoretical (and arguably policy) question impossible to answer *ex ante*—as well as other characteristic phenomena of interest (e.g., discrete striations, inflection or "tipping" points, asymptotes and other singularities, constant or invariant subfields, and others) revealed by geometric and topological information. In addition, interesting scalar and vector fields of *'* become accessible to direct investigation through formal tools of nabladot calculus and analysis. A novel and valuable feature of this approach is that nabladot calculus provides exact results in analytical investigations where the classical infinitesimal calculus of hybrid IR functions would provide approximations with errors over the discrete domain of independent

<sup>3</sup> IR scholars rarely consider the presence of vectors in international relations, other than metaphorically. This analysis demonstrates the rigorous analysis of vectors and vector fields in IR using formal methods from nabladot calculus, as in the next section. To contain notation, we shall use i and j to denote unit vectors along continuous and discrete dimensions, respectively, rather than create new unit vectors for each variable. <sup>4</sup> Use of the partial derivative with respect to *<sup>Y</sup>* (a discrete variable) instead of the partial difference—which is often used in approximations—produces a measurable error that varies in magnitude depending on the structure of *'* and values of *Y* . Measurable discrepancies between the two operators (nabladot and classical

nabla) are demonstrable but beyond the present scope due to space limitations (Cioffi 2021).

the vector field's topology.<sup>4</sup>

graphic analyses.

variable(s).

10

symbol), which is defined as follows:

functions can add significant clarity.

litical uncertainty among great powers was pioneered by Manus I. Midlarsky in 1974. This causal theory is based on aspects of strategic uncertainty rooted in Shannon's information entropy and related concepts. The specific *explanandum* is the annual amount of warfare experienced in the international system, which is an emergent property generated not just by crises (as in the previous case) but also outright conquest, revenge, colonization, and all other types of wars. In any given year, Midlarsky's theory predicted that annual global war frequency *φ* is determined by systemic polarity Θ among the great powers.7 Formally, 4.3 Case 3: Frequency of war and systemic polarity (Midlarsky's law) Having just examined the probability of war at the relational level, in this last case study we again change our frame of reference, this time turning to the systemic level of analysis. The formal theoretical (and empirically supported) explanation for the annual frequency of warfare as a result of fundamental political uncertainty among great powers was pioneered by Manus I. Midlarsky in 1974. This causal theory is based on aspects of strategic uncertainty rooted in Shannon's information entropy and related concepts. The specific *explanandum* is the annual amount of 4.3 Case 3: Frequency of war and systemic polarity (Midlarsky's law) Having just examined the probability of war at the relational level, in this last case study we again change our frame of reference, this time turning to the systemic level of analysis. The formal theoretical (and empirically supported) explanation for the annual frequency of warfare as a result of fundamental political uncertainty among great powers was pioneered by Manus I. Midlarsky in 1974. This causal theory is based on aspects of strategic uncertainty rooted in Shannon's information entropy and related concepts. The specific *explanandum* is the annual amount of 4.3 Case 3: Frequency of war and systemic polarity (Midlarsky's law) Having just examined the probability of war at the relational level, in this last case study we again change our frame of reference, this time turning to the systemic level of analysis. The formal theoretical (and empirically supported) explanation for the annual frequency of warfare as a result of fundamental political uncertainty among great powers was pioneered by Manus I. Midlarsky in 1974. This causal theory is based on aspects of strategic uncertainty rooted in Shannon's information entropy and related concepts. The specific *explanandum* is the annual amount of

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CLAUDIO CIOFFI-REVILLA

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CLAUDIO CIOFFI-REVILLA

$$
\phi = K \log \Theta,\tag{20}
$$

which is a hybrid bivariate nonlinear function, where *K* is a continuous proportionality parameter over the closed interval [1, 10], historically, and systemic polarity Θ ≥ 1 is discrete. The case when Θ = 1 (a single hegemonic power) is the most peaceful, since *φ*(Θ = 1) = 0, and the function's (*k*, *Θ*)-domain once again lies within Cartesian quadrant I. types of wars. In any given year, Midlarsky's theory predicted that annual global war frequency *φ* is determined by systemic polarity ⇥ among the great powers.<sup>7</sup> Formally, *φ* = *K* log ⇥*,* (20) which is a hybrid bivariate nonlinear function, where *K* is a continuous proportionality parameter types of wars. In any given year, Midlarsky's theory predicted that annual global war frequency *φ* is determined by systemic polarity ⇥ among the great powers.<sup>7</sup> Formally, *φ* = *K* log ⇥*,* (20) which is a hybrid bivariate nonlinear function, where *K* is a continuous proportionality parameter types of wars. In any given year, Midlarsky's theory predicted that annual global war frequency *φ* is determined by systemic polarity ⇥ among the great powers.<sup>7</sup> Formally, *φ* = *K* log ⇥*,* (20) which is a hybrid bivariate nonlinear function, where *K* is a continuous proportionality parameter 4.3 Case 3: Frequency of war and systemic polarity (Midlarsky's law) Having just examined the probability of war at the relational level, in this last case study we again change our frame of reference, this time turning to the systemic level of analysis. The formal theoretical (and empirically supported) explanation for the annual frequency of warfare as a result 4.3 Case 3: Frequency of war and systemic polarity (Midlarsky's law) Having just examined the probability of war at the relational level, in this last case study we again change our frame of reference, this time turning to the systemic level of analysis. The formal

CLAUDIO CIOFFI-REVILLA

The 3D surface graph of equation 20 is illustrated in Figure 5a, which shows war frequency rising on a hilly slope as polarity Θ and *K* increase. The associated contour plot in Figure 5b looks "straight down hill," highlighting the relatively mild gradient on the skirting slopes of the hybrid function, which is a surface bound by a linear slope along *K* but a logarithmically convex slope along Θ, a radial pattern observable in the plot's isocontours. over the closed interval [1*,* 10], historically, and systemic polarity ⇥ *≥* 1 is discrete. The case when ⇥ = 1 (a single hegemonic power) is the most peaceful, since *φ*(⇥ = 1) = 0, and the function's (*k, ✓*)-domain once again lies within Cartesian quadrant I. The 3D surface graph of equation 20 is illustrated in Figure 5a, which shows war frequency rising on a hilly slope as polarity ⇥ and *K* increase. The associated contour plot in Figure 5b looks "straight down hill," highlighting the relatively mild gradient on the skirting slopes of the hybrid function, which is a surface bound by a linear slope along *K* but a logarithmically convex over the closed interval [1*,* 10], historically, and systemic polarity ⇥ *≥* 1 is discrete. The case when ⇥ = 1 (a single hegemonic power) is the most peaceful, since *φ*(⇥ = 1) = 0, and the function's (*k, ✓*)-domain once again lies within Cartesian quadrant I. The 3D surface graph of equation 20 is illustrated in Figure 5a, which shows war frequency rising on a hilly slope as polarity ⇥ and *K*increase. The associated contour plot in Figure 5b looks "straight down hill," highlighting the relatively mild gradient on the skirting slopes of the hybrid function, which is a surface bound by a linear slope along *K* but a logarithmically convex over the closed interval [1*,* 10], historically, and systemic polarity ⇥ *≥* 1 is discrete. The case when ⇥ = 1(a single hegemonic power) is the most peaceful, since *φ*(⇥ = 1) = 0, and the function's (*k, ✓*)-domain once again lies within Cartesian quadrant I. The 3D surface graph of equation 20 is illustrated in Figure 5a, which shows war frequency rising on a hilly slope as polarity ⇥ and*K* increase. The associated contour plot in Figure 5b looks "straight down hill," highlighting the relatively mild gradient on the skirting slopes of the hybrid function, which is a surface bound by a linear slope along *K* but a logarithmically convex of fundamental political uncertainty among great powers was pioneered by Manus I. Midlarsky in 1974. This causal theory is based on aspects of strategic uncertainty rooted in Shannon's information entropy and related concepts. The specific *explanandum* is the annual amount of warfare experienced in the international system, which is an emergent property generated not just by crises (as in the previous case) but also outright conquest, revenge, colonization, and all other types of wars. In any given year, Midlarsky's theory predicted that annual global war frequency *φ* is determined by systemic polarity ⇥ among the great powers.<sup>7</sup> Formally, theoretical (and empirically supported) explanation for the annual frequency of warfare as a result of fundamental political uncertainty among great powers was pioneered by Manus I. Midlarsky in 1974. This causal theory is based on aspects of strategic uncertainty rooted in Shannon's information entropy and related concepts. The specific *explanandum* is the annual amount of warfare experienced in the international system, which is an emergent property generated not just by crises (as in the previous case) but also outright conquest, revenge, colonization, and all other types of wars. In any given year, Midlarsky's theory predicted that annual global war frequency *φ*

Each independent variable increases *φ* in a different way, as shown in Figures 5a and b. Calculating the partial derivative and partial difference of *φ* with respect to *K* and Θ, respectively: slope along ⇥, a radial pattern observable in the plot's isocontours. Each independent variable increases *φ* in a different way, as shown in Figures 5a and b. Calculating the partial derivative and partial difference of *φ* with respect to *K* and ⇥, respectively: slope along ⇥, a radial pattern observable in the plot's isocontours. Each independent variable increases *φ* in a different way, as shown in Figures 5a and b. Calculating the partial derivative and partial difference of *φ* with respect to *K* and ⇥, respectively: slope along ⇥, a radial pattern observable in the plot's isocontours. Each independent variable increases *φ* in a different way, as shown in Figures 5a and b. Calculating the partial derivative and partial difference of *φ* with respect to *K* and⇥, respectively: *φ* = *K* log ⇥*,* (20) is determined by systemic polarity ⇥ among the great powers.<sup>7</sup> Formally, *φ* = *K* log ⇥*,* (20)

$$\begin{aligned} \partial\_k \phi &= \ln \Theta & \text{(for } K \text{'s effect on } W \text{, in Figure 5c)} & \text{(21)}\\ \Delta\_\theta \phi &= K \left[ \ln(\Theta + 1) - \ln \Theta \right] & \text{(for } \Theta \text{'s effect on } \phi \text{, in Figure 5e)} & \text{(22)} \end{aligned}$$

which is a hybrid bivariate nonlinear function, where *K* is a continuous proportionality parameter

Here, too, we see that changes in parameter *K* and polarity ⇥ have clearly different effects on war frequency *φ*, in this case through a simple univariate discrete function that is strictly discrete in ⇥, and through a more complicated bivariate hybrid function (Figures 5c and d) with two terms in both dimensions, respectively. Congruent political effects. Changes in parameter *K* or polarity ⇥ have proportional effects on *φ*, as shown by strictly positive values of both graphs of derivatives and differences in Figures 5c (2D) and e (3D), respectively. The graph of *@kφ* lacks a contour plot, since it has univariate domain in ⇥. Here, too, we see that changes in parameter *K* and polarity ⇥ have clearly different effects on war frequency *φ*, in this case through a simple univariate discrete function that is strictly discrete in ⇥, and through a more complicated bivariate hybrid function (Figures 5c and d) with two terms in both dimensions, respectively. Congruent political effects. Changes in parameter *K* or polarity ⇥ have proportional effects on *φ*, as shown by strictly positive values of both graphs of derivatives and differences in Figures 5c (2D) and e (3D), respectively. The graph of *@kφ* lacks a contour plot, since it has univariate domain in ⇥. Here, too, we see that changes in parameter *K* and polarity ⇥ have clearly different effects on war frequency *φ*, in this case through a simple univariate discrete function that is strictly discrete in ⇥, and through a more complicated bivariate hybrid function (Figures 5c and d) with two terms in both dimensions, respectively. Congruent political effects. Changes in parameter *K* or polarity ⇥have proportional effects on *φ*, as shown by strictly positive values of both graphs of derivatives and differences in Figures 5c (2D) and e (3D), respectively. The graph of *@kφ* lacks a contour plot, since it has univariate domain in ⇥. Here, too, we see that changes in parameter *K* and polarity Θ have clearly different effects on war frequency *φ*, in this case through a simple univariate discrete function that is strictly discrete in Θ, and through a more complicated bivariate hybrid function (Figures 5c and d) with two terms in both dimensions, respectively. **Congruent political effects***.* Changes in parameter *K* or polarity Θ have proportional effects on *φ*, as shown by strictly positive values of both graphs of derivatives and differences in Figures 5c (2D) and e (3D), respectively. The graph of The 3D surface graph of equation 20 is illustrated in Figure 5a, which shows war frequency rising on a hilly slope as polarity ⇥ and *K* increase. The associated contour plot in Figure 5b looks "straight down hill," highlighting the relatively mild gradient on the skirting slopes of the hybrid function, which is a surface bound by a linear slope along *K* but a logarithmically convex slope along ⇥, a radial pattern observable in the plot's isocontours. Each independent variable increases *φ* in a different way, as shown in Figures 5a and b. Calculating the partial derivative and partial difference of *φ* with respect to *K* and ⇥, respectively: *@kφ* = ln ⇥ (for *K*'s effect on *W*, in Figure 5c) (21) lacks a contour plot, since it has univariate domain in Θ. (*k, ✓*)-domain once again lies within Cartesian quadrant I. The 3D surface graph of equation 20 is illustrated in Figure 5a, which shows war frequency rising on a hilly slope as polarity ⇥ and *K* increase. The associated contour plot in Figure 5b looks "straight down hill," highlighting the relatively mild gradient on the skirting slopes of the hybrid function, which is a surface bound by a linear slope along *K* but a logarithmically convex slope along ⇥, a radial pattern observable in the plot's isocontours. Each independent variable increases *φ* in a different way, as shown in Figures 5a and b. Calculating the partial derivative and partial difference of *φ* with respect to *K* and ⇥, respectively:

Isomorphism of *φ* and *@kφ*. War frequency *φ* and its rate of change with respect to *K* are isomorphic, as shown by Figure 5c and the estimated hybrid function *φ*ˆ(*K,* ⇥) fitted to Isomorphism of *φ* and *@kφ*. War frequency *φ* and its rate of change with respect to *K* are isomorphic, as shown by Figure 5c and the estimated hybrid function *φ*ˆ(*K,* ⇥)fitted to Isomorphism of *φ* and*@kφ*. War frequency *φ* and its rate of change with respect to *K* are isomorphic, as shown by Figure 5c and the estimated hybrid function *φ*ˆ(*K,* ⇥) fitted to ∆*✓φ* = *K* [ln(⇥ + 1) *−* ln ⇥] (for ⇥'s effect on *φ*, in Figure 5e)*.* (22) **Isomorphism of** *φ* **and**  *@kφ* = ln ⇥ (for *K*'s effect on *W*, in Figure 5c) (21) ∆*✓φ* = *K* [ln(⇥ + 1) *−* ln ⇥] (for ⇥'s effect on *φ*, in Figure 5e)*.* (22) *.* War frequency *φ* and its rate of change with respect to *K* are isomorphic, as shown by Figure 5c and the estimated hybrid fun-

Here, too, we see that changes in parameter *K* and polarity ⇥ have clearly different effects on war frequency *φ*, in this case through a simple univariate discrete function that is strictly discrete in ⇥, and through a more complicated bivariate hybrid function (Figures 5c and d) with two terms

original data in Figure 5d (redrawn from Midlarsky 1974, 420, fig. 3).

original data in Figure 5d (redrawn from Midlarsky 1974, 420, fig. 3).

original data in Figure 5d (redrawn from Midlarsky 1974, 420, fig. 3).

on 1815-1945 war onset data should not be retested on earlier and later datasets.

on 1815-1945 war onset data should not be retested on earlier and later datasets.

20

univariate domain in ⇥.

on 1815-1945 war onset data should not be retested on earlier and later datasets.

on 1815-1945 war onset data should not be retested on earlier and later datasets.

<sup>7</sup> Midlarsky's theory remains one of the most complete and empirically validated formal theories in international relations—and, surprisingly, one of the least known among conflict researchers. The initial validation based

original data in Figure 5d (redrawn from Midlarsky 1974, 420, fig. 3).

univariate domain in ⇥.

<sup>7</sup> Midlarsky's theory remains one of the most complete and empirically validated formal theories in international relations—and, surprisingly, one of the least known among conflict researchers. The initial validation based

20

20

20

20

<sup>7</sup> Midlarsky's theory remains one of the most complete and empirically validated formal theories in international relations—and, surprisingly, one of the least known among conflict researchers. The initial validation based

Isomorphism of *φ* and *@kφ*. War frequency *φ* and its rate of change with respect to *K* are isomorphic, as shown by Figure 5c and the estimated hybrid function *φ*ˆ(*K,* ⇥) fitted to

Deceptive simplicity. Aside from the relatively simple convexity of war frequency *φ* induced by systemic polarity ⇥, this theoretical model seems rather uncomplicated, as would appear from diagnostic images in Figure 5. However, such an impression may be superficial, as

<sup>7</sup> Midlarsky's theory remains one of the most complete and empirically validated formal theories in international relations—and, surprisingly, one of the least known among conflict researchers. The initial validation based

shown next by the second part of the analysis on elasticities and vector fields.

Here, too, we see that changes in parameter *K* and polarity ⇥ have clearly different effects on war frequency *φ*, in this case through a simple univariate discrete function that is strictly discrete in

Isomorphism of *φ* and *@kφ*. War frequency *φ* and its rate of change with respect to *K* are isomorphic, as shown by Figure 5c and the estimated hybrid function *φ*ˆ(*K,* ⇥) fitted to

Deceptive simplicity. Aside from the relatively simple convexity of war frequency *φ* induced by systemic polarity ⇥, this theoretical model seems rather uncomplicated, as would appear from diagnostic images in Figure 5. However, such an impression may be superficial, as

<sup>7</sup> Midlarsky's theory remains one of the most complete and empirically validated formal theories in international relations—and, surprisingly, one of the least known among conflict researchers. The initial validation based

shown next by the second part of the analysis on elasticities and vector fields.

original data in Figure 5d (redrawn from Midlarsky 1974, 420, fig. 3).

on 1815-1945 war onset data should not be retested on earlier and later datasets.

Deceptive simplicity. Aside from the relatively simple convexity of war frequency *φ* induced by systemic polarity ⇥, this theoretical model seems rather uncomplicated, as would appear from diagnostic images in Figure 5. However, such an impression may be superficial, as shown next by the second part of the analysis on elasticities and vector fields. Deceptive simplicity. Aside from the relatively simple convexity of war frequency *φ* induced by systemic polarity ⇥, this theoretical model seems rather uncomplicated, as would appear from diagnostic images in Figure 5. However, such an impression may be superficial, as shown next by the second part of the analysis on elasticities and vector fields. Deceptive simplicity. Aside from the relatively simple convexity of war frequency *φ* induced by systemic polarity ⇥, this theoretical model seems rather uncomplicated, as would appear from diagnostic images in Figure 5. However, such an impression may be superficial, as shown next by the second part of the analysis on elasticities and vector fields. in both dimensions, respectively. Congruent political effects. Changes in parameter *K* or polarity ⇥ have proportional effects on *φ*, as shown by strictly positive values of both graphs of derivatives and differences in ⇥, and through a more complicated bivariate hybrid function (Figures 5c and d) with two terms in both dimensions, respectively. Congruent political effects. Changes in parameter *K* or polarity ⇥ have proportional effects <sup>7</sup> Midlarsky's theory remains one of the most complete and empirically validated formal theories in international relations—and, surprisingly, one of the least known among conflict researchers. The initial validation based on 1815-1945 war onset data should not be retested on earlier and later datasets.

NABLADOT ANALYSIS OF HYBRID THEORIES IN INTERNATIONAL RELATIONS

Figura 5. Annual frequency of war *φ*(*K,* ⇥) as a hybrid logarithmic function of systemic polarity ⇥ (Midlarsky's law): (a) 3D surface of the hybrid function *φ* = *K* ln ⇥; (b) contour plot of (a); (c) 2D graph of *@kφ*, the first-order partial derivative of *φ* with respect to *K*; (d) plot of the original function fitted on historical data constrained by the logarithmic condition *φ*(1) = 0; (e) 3D surface of ∆*✓φ*, the first-order partial difference of *φ* with respect to ⇥; (f) contour plot of (e). 21 Figure 5. Annual frequency of war *φ*(*K*, Θ) as a hybrid logarithmic function of systemic polarity Θ (Midlarsky's law): (a) 3D surface of the hybrid function *φ* = *K* ln Θ; (b) contour plot of (a); (c) 2D graph of Each independent variable increases *φ* in a different way, as shown in Figures 5a and b. Calculating the partial derivative and partial difference of *φ* with respect to *K* and ⇥, respectively: *@kφ* = ln ⇥ (for *K*'s effect on *W*, in Figure 5c) (21) ∆*✓φ* = *K* [ln(⇥ + 1) *−* ln ⇥] (for ⇥'s effect on *φ*, in Figure 5e)*.* (22) Here, too, we see that changes in parameter *K* and polarity ⇥ have clearly different effects on war , the fi rst-order partial derivative of *φ* with respect to *K*; (d) plot of the original function fi tt ed on historical data constrained by the logarithmic condition *φ*(1) = 0; (e) 3D surface of Δ*Θφ* the fi rst-order partial diff erence of *φ* with respect to Θ; (f) contour plot of (e).

in both dimensions, respectively.

univariate domain in ⇥.

20

frequency *φ*, in this case through a simple univariate discrete function that is strictly discrete in ⇥, and through a more complicated bivariate hybrid function (Figures 5c and d) with two terms

Congruent political effects. Changes in parameter *K* or polarity ⇥ have proportional effects on *φ*, as shown by strictly positive values of both graphs of derivatives and differences in Figures 5c (2D) and e (3D), respectively. The graph of *@kφ* lacks a contour plot, since it has

Isomorphism of *φ* and *@kφ*. War frequency *φ* and its rate of change with respect to *K* are isomorphic, as shown by Figure 5c and the estimated hybrid function *φ*ˆ(*K,* ⇥) fitted to

Deceptive simplicity. Aside from the relatively simple convexity of war frequency *φ* induced by systemic polarity ⇥, this theoretical model seems rather uncomplicated, as would appear from diagnostic images in Figure 5. However, such an impression may be superficial, as

<sup>7</sup> Midlarsky's theory remains one of the most complete and empirically validated formal theories in international relations—and, surprisingly, one of the least known among conflict researchers. The initial validation based

shown next by the second part of the analysis on elasticities and vector fields.

original data in Figure 5d (redrawn from Midlarsky 1974, 420, fig. 3).

on 1815-1945 war onset data should not be retested on earlier and later datasets.

ction isomorphic, as shown by Figure 5c and the estimated hybrid function *φ*ˆ(*K,* ⇥) fitted to original data in Figure 5d (redrawn from Midlarsky 1974, 420, fig. 3). fitted to original data in Figure 5d (redrawn from Midlarsky 1974, 420, fig. 3).

CLAUDIO CIOFFI-REVILLA

4.3 Case 3: Frequency of war and systemic polarity (Midlarsky's law)

is determined by systemic polarity ⇥ among the great powers.<sup>7</sup> Formally,

(*k, ✓*)-domain once again lies within Cartesian quadrant I.

in both dimensions, respectively.

univariate domain in ⇥.

20

slope along ⇥, a radial pattern observable in the plot's isocontours.

Having just examined the probability of war at the relational level, in this last case study we again change our frame of reference, this time turning to the systemic level of analysis. The formal theoretical (and empirically supported) explanation for the annual frequency of warfare as a result of fundamental political uncertainty among great powers was pioneered by Manus I. Midlarsky in 1974. This causal theory is based on aspects of strategic uncertainty rooted in Shannon's information entropy and related concepts. The specific *explanandum* is the annual amount of warfare experienced in the international system, which is an emergent property generated not just by crises (as in the previous case) but also outright conquest, revenge, colonization, and all other types of wars. In any given year, Midlarsky's theory predicted that annual global war frequency *φ*

which is a hybrid bivariate nonlinear function, where *K* is a continuous proportionality parameter over the closed interval [1*,* 10], historically, and systemic polarity ⇥ *≥* 1 is discrete. The case when ⇥ = 1 (a single hegemonic power) is the most peaceful, since *φ*(⇥ = 1) = 0, and the function's

The 3D surface graph of equation 20 is illustrated in Figure 5a, which shows war frequency rising on a hilly slope as polarity ⇥ and *K* increase. The associated contour plot in Figure 5b looks "straight down hill," highlighting the relatively mild gradient on the skirting slopes of the hybrid function, which is a surface bound by a linear slope along *K* but a logarithmically convex

Each independent variable increases *φ* in a different way, as shown in Figures 5a and b. Calculating the partial derivative and partial difference of *φ* with respect to *K* and ⇥, respectively:

Here, too, we see that changes in parameter *K* and polarity ⇥ have clearly different effects on war frequency *φ*, in this case through a simple univariate discrete function that is strictly discrete in ⇥, and through a more complicated bivariate hybrid function (Figures 5c and d) with two terms

Congruent political effects. Changes in parameter *K* or polarity ⇥ have proportional effects on *φ*, as shown by strictly positive values of both graphs of derivatives and differences in

Isomorphism of *φ* and *@kφ*. War frequency *φ* and its rate of change with respect to *K* are

on 1815-1945 war onset data should not be retested on earlier and later datasets.

*@kφ* = ln ⇥ (for *K*'s effect on *W*, in Figure 5c) (21) ∆*✓φ* = *K* [ln(⇥ + 1) *−* ln ⇥] (for ⇥'s effect on *φ*, in Figure 5e)*.* (22)

*φ* = *K* log ⇥*,* (20)

follows:

*⌘✓φ* = ⇥

*⌘✓φ* = ⇥

curling associated with divergence.

curling associated with divergence.

follows:

standardized gradient, as follows:

standardized gradient, as follows:

standardized gradient, as follows:

curling associated with divergence.

follows:

follows:

follows:

22

22

22

22

22

Deceptive simplicity. Aside from the relatively simple convexity of war frequency *φ* induced by systemic polarity ⇥, this theoretical model seems rather uncomplicated, as would appear from diagnostic images in Figure 5. However, such an impression may be superficial, as shown next by the second part of the analysis on elasticities and vector fields. **Deceptive simplicity.** Aside from the relatively simple convexity of war frequency *φ* induced by systemic polarity Θ, this theoretical model seems rather uncomplicated, as would appear from diagnostic images in Figure 5. However, such an impression may be superficial, as shown next by the second part of the analysis on elasticities and vector fields.

<sup>7</sup> Midlarsky's theory remains one of the most complete and empirically validated formal theories in international relations—and, surprisingly, one of the least known among conflict researchers. The initial validation based Finally, we can now investigate Midlarsky's law using standardized variables, as in the previous cases. First, we calculate and analyze percentage change in *φ* with respect to *K* and Θ, as shown by the elasticities in Figures 6a and b. Calculation of point elasticity and arc elasticity of *φ* with respect to parameter *K* and polarity Θ yields the following set of hybrid equations: CLAUDIO CIOFFI-REVILLA Finally, we can now investigate Midlarsky's law using standardized variables, as in the previous cases. First, we calculate and analyze percentage change in *φ* with respect to *K* and ⇥, as shown by the elasticities in Figures 6a and b. Calculation of point elasticity and arc elasticity of *φ* with respect to parameter *K* and polarity ⇥ yields the following set of hybrid equations: CLAUDIO CIOFFI-REVILLA Finally, we can now investigate Midlarsky's law using standardized variables, as in the previous cases. First, we calculate and analyze percentage change in *φ* with respect to *K* and ⇥, as shown by the elasticities in Figures 6a and b. Calculation of point elasticity and arc elasticity of *φ* with respect to parameter *K* and polarity ⇥ yields the following set of hybrid equations: CLAUDIO CIOFFI-REVILLA Finally, we can now investigate Midlarsky's law using standardized variables, as in the previous cases. First, we calculate and analyze percentage change in *φ* with respect to *K* and ⇥, as shown by the elasticities in Figures 6a and b. Calculation of point elasticity and arc elasticity of *φ* with respect to parameter *K* and polarity ⇥ yields the following set of hybrid equations:

$$\eta\_k \phi = 1 \qquad \text{(for } K \text{'s percentage effect on } \phi \text{, in Figure 6a)} \quad \text{(23)}$$

$$\eta\_\theta \phi = \Theta \left[ \frac{\ln(\Theta + 1)}{\ln \Theta} - 1 \right] \text{ (for } \Theta \text{'s percentage effect on } \phi \text{, in Figure 6b)}. \quad \text{(24)}$$

We immediately see that standardized effects on *φ* are again quite different from earlier unitbased results (cf. equations 21 and 22 and associated figures). Here, point elasticity represents percentage change in annual war frequency with respect to percentage change in parameter *K*, whereas arc elasticity measures percentage change in *φ* with respect to percentage annual change in polarity ⇥. The former has a constant value of 1 while the latter is a rational hybrid function with logarithms of polarity, as shown in Figures 6a and b, respectively. Interestingly, arc elasticity (Figure 6b) exhibits a singularity under unipolarity, where *φ*(*k,* 1) = *1*, indicating a major transition from unipolarity to bipolarity. In this case arc elasticity continues We immediately see that standardized effects on *φ* are again quite different from earlier unitbased results (cf. equations 21 and 22and associated figures). Here, point elasticity represents percentage change in annual war frequency with respect to percentage change in parameter *K*, whereas arc elasticity measures percentage change in *φ* with respect to percentage annual change in polarity ⇥. The former has a constant value of 1 while the latter is a rational hybrid function with logarithms of polarity, as shown in Figures 6a and b, respectively. Interestingly, arc elasticity (Figure 6b) exhibits a singularity under unipolarity, where *φ*(*k,* 1) = *1*, indicating a major transition from unipolarity to bipolarity. In this case arc elasticity continues We immediately see that standardized effects on *φ* are again quite different from earlier unitbased results (cf. equations 21 and 22and associated figures). Here, point elasticity represents percentage change in annual war frequency with respect to percentage change in parameter *K*, whereas arc elasticity measures percentage change in *φ* with respect to percentage annual change in polarity ⇥. The former has a constant value of 1 while the latter is a rational hybrid function with logarithms of polarity, as shown in Figures 6a and b, respectively. Interestingly, arc elasticity (Figure 6b) exhibits a singularity under unipolarity, where *φ*(*k,* 1) = *1*, indicating a major transition from unipolarity to bipolarity. In this case arc elasticity continues We immediately see that standardized effects on *φ* are again quite different from earlier unit-based results (cf. equations 21 and 22 and associated figures). Here, point elasticity represents percentage change in annual war frequency with respect to percentage change in parameter *K*, whereas arc elasticity measures percentage change in *φ* with respect to percentage annual change in polarity Θ. The former has a constant value of 1 while the latter is a rational hybrid function with logarithms of polarity, as shown in Figures 6a and b, respectively. by the elasticities in Figures 6a and b. Calculation of point elasticity and arc elasticity of *φ* with respect to parameter *K* and polarity ⇥ yields the following set of hybrid equations: *⌘kφ* = 1 (for *K*'s percentage effect on *φ*, in Figure 6a) (23) *⌘✓φ* = ⇥ ln(⇥ + 1) ln ⇥ *<sup>−</sup>* <sup>1</sup> � (for ⇥'s percentage effect on *φ*, in Figure 6b)*.* (24) We immediately see that standardized effects on *φ* are again quite different from earlier unitby the elasticities in Figures 6a and b. Calculation of point elasticity and arc elasticity of *φ* with respect to parameter *K* and polarity ⇥ yields the following set of hybrid equations: *⌘kφ* = 1 (for *K*'s percentage effect on *φ*, in Figure 6a) (23) ln(⇥ + 1) ln ⇥ *<sup>−</sup>* <sup>1</sup> � (for ⇥'s percentage effect on *φ*, in Figure 6b)*.* (24) We immediately see that standardized effects on *φ* are again quite different from earlier unit-

a rapid drop with increasing polarity, which is the systemic trend experienced in contemporary history of major powers after the Soviet-American Cold War around 1989. Comparing the two elasticities (equations 23 and 24; cf. also their respective graphs in Figures 6a and b) yields the following dominance principle: annual frequency of war *φ* is more sensitive to percentage change in parameter *K* than to change in systemic polarity ⇥ under unipolar and bipolar systemic structures, but the reverse is true under tripolarity and higher-order structures—because a rapid drop with increasing polarity, which is the systemic trend experienced in contemporary history of major powers after the Soviet-American Cold War around 1989. Comparing the two elasticities (equations 23 and 24; cf. also their respective graphs in Figures 6a and b) yields the following dominance principle: annual frequency of war *φ* is more sensitive to percentage change in parameter *K* than to change in systemic polarity ⇥ under unipolar and bipolar systemic structures, but the reverse is true under tripolarity and higher-order structures—because a rapid drop with increasing polarity, which is the systemic trend experienced in contemporary history of major powers after the Soviet-American Cold War around 1989. Comparing the two elasticities (equations 23 and 24; cf. also their respective graphs in Figures 6a and b) yields the following dominance principle: annual frequency of war *φ* is more sensitive to percentage change in parameter *K* than to change in systemic polarity ⇥ under unipolar and bipolar systemic structures, but the reverse is true under tripolarity and higher-order structures—because Interestingly, arc elasticity (Figure 6b) exhibits a singularity under unipolarity, where *φ*(*k*, 1) = ∞, indicating a major transition from unipolarity to bipolarity. In this case arc elasticity continues a rapid drop with increasing polarity, which is the systemic trend experienced in contemporary history of major powers after the Soviet-American Cold War around 1989. based results (cf. equations 21 and 22 and associated figures). Here, point elasticity represents percentage change in annual war frequency with respect to percentage change in parameter *K*, whereas arc elasticity measures percentage change in *φ* with respect to percentage annual change in polarity ⇥. The former has a constant value of 1 while the latter is a rational hybrid function with logarithms of polarity, as shown in Figures 6a and b, respectively. Interestingly, arc elasticity (Figure 6b) exhibits a singularity under unipolarity, where *φ*(*k,* 1) = based results (cf. equations 21 and 22 and associated figures). Here, point elasticity represents percentage change in annual war frequency with respect to percentage change in parameter *K*, whereas arc elasticity measures percentage change in *φ* with respect to percentage annual change in polarity ⇥. The former has a constant value of 1 while the latter is a rational hybrid function with logarithms of polarity, as shown in Figures 6a and b, respectively. Interestingly, arc elasticity (Figure 6b) exhibits a singularity under unipolarity, where *φ*(*k,* 1) =

the relationship reverses between ⇥ = 2 and 3. This is a surprising qualitative transition that is invisible in the original model but is clear once the dimensions are standardized by elasticities. The different and joint effects of parameter *K* and polarity ⇥ on war frequency *φ* can be seen by calculating the gradient of *φ* with respect to both variables using the nabladot operator, as *r· φ* = *@<sup>k</sup>* (*K* ln ⇥)i + ∆*✓* (*K* ln ⇥)j (25) = log ⇥ i + *K* [(ln ⇥ + 1) *−* ln ⇥] j*,* (26) the relationship reverses between ⇥ = 2 and 3. This is a surprising qualitative transition that is invisible in the original model but is clear once the dimensions are standardized by elasticities. The different and joint effects of parameter *K* and polarity ⇥ on war frequency *φ* can be seen by calculating the gradient of *φ* with respect to both variables using the nabladot operator, as *r· φ* = *@<sup>k</sup>* (*K* ln ⇥)i + ∆*✓* (*K*ln ⇥)j(25) = log ⇥ i + *K* [(ln ⇥ + 1) *−* ln ⇥] j*,* (26) the relationship reverses between ⇥ = 2 and 3. This is a surprising qualitative transition that is invisible in the original model but is clear once the dimensions are standardized by elasticities. The different and joint effects of parameter *K* and polarity ⇥ on war frequency *φ* can be seen by calculating the gradient of *φ* with respect to both variables using the nabladot operator, as *r· φ* = *@<sup>k</sup>* (*K* ln⇥)i + ∆*✓* (*K* ln ⇥)j (25) = log ⇥ i + *K* [(ln ⇥ + 1) *−* ln ⇥]j*,* (26) Comparing the two elasticities (equations 23 and 24; cf. also their respective graphs in Figures 6a and b) yields the following dominance principle: annual frequency of war *φ* is more sensitive to percentage change in parameter *K* than to change in systemic polarity Θ under unipolar and bipolar systemic structures, but the reverse is true under tripolarity and higher-order structures—because the relationship reverses between Θ = 2 and 3. This is a surprising qualitative transition that is invisible in the original model but is clear once the dimensions are standardized by elasticities. *1*, indicating a major transition from unipolarity to bipolarity. In this case arc elasticity continues a rapid drop with increasing polarity, which is the systemic trend experienced in contemporary history of major powers after the Soviet-American Cold War around 1989. Comparing the two elasticities (equations 23 and 24; cf. also their respective graphs in Figures 6a and b) yields the following dominance principle: annual frequency of war *φ* is more sensitive to percentage change in parameter *K* than to change in systemic polarity ⇥ under unipolar and bipolar systemic structures, but the reverse is true under tripolarity and higher-order structures—because the relationship reverses between ⇥ = 2 and 3. This is a surprising qualitative transition that is *1*, indicating a major transition from unipolarity to bipolarity. In this case arc elasticity continues a rapid drop with increasing polarity, which is the systemic trend experienced in contemporary history of major powers after the Soviet-American Cold War around 1989. Comparing the two elasticities (equations 23 and 24; cf. also their respective graphs in Figures 6a and b) yields the following dominance principle: annual frequency of war *φ* is more sensitive to percentage change in parameter *K* than to change in systemic polarity ⇥ under unipolar and bipolar systemic structures, but the reverse is true under tripolarity and higher-order structures—because the relationship reverses between ⇥ = 2 and 3. This is a surprising qualitative transition that is

which is a two-dimensional vector function Φ = (*K,* ⇥). The resulting vector field of this hybrid gradient is in Figure 6c, which shows: (i) heterogeneity along the two dimensions; (ii) general southwest-northeast orientation; (iii) increasing intensity as *K* increases for low values of polarity with a hot spot in the NW corner; (iv) divergence from a line at approximately 60 degrees; and (5) which is a two-dimensional vector function Φ = (*K,* ⇥). The resulting vector field of this hybrid gradient is in Figure 6c, which shows: (i) heterogeneity along the two dimensions; (ii) general southwest-northeast orientation; (iii) increasing intensity as *K* increases for low values of polarity with a hot spot in the NW corner; (iv) divergence from a line at approximately 60 degrees; and (5) which is a two-dimensional vector function Φ = (*K,* ⇥). The resulting vector field of this hybrid gradient is in Figure 6c, which shows: (i) heterogeneity along the two dimensions; (ii) general southwest-northeast orientation; (iii) increasing intensity as *K* increases for low values of polarity with a hot spot in the NW corner; (iv) divergence from a line at approximately 60 degrees; and (5) The different and joint effects of parameter *K* and polarity Θ on war frequency *φ* can be seen by calculating the gradient of *φ* with respect to both variables using the nabladot operator, as follows: invisible in the original model but is clear once the dimensions are standardized by elasticities. The different and joint effects of parameter *K* and polarity ⇥ on war frequency *φ* can be seen by calculating the gradient of *φ* with respect to both variables using the nabladot operator, as invisible in the original model but is clear once the dimensions are standardized by elasticities. The different and joint effects of parameter *K* and polarity ⇥ on war frequency *φ* can be seen by calculating the gradient of *φ* with respect to both variables using the nabladot operator, as

better view of (i) the hybrid gradient field Φ that drops from the NW region into the uni- and

which is a two-dimensional vector function Φ = (*K,* ⇥). The resulting vector field of this hybrid

with a hot spot in the NW corner; (iv) divergence from a line at approximately 60 degrees; and (5)

which is a two-dimensional vector function Φ*⇤* = *⇤*(*K,* ⇥). The resulting vector field of this hybrid standardized gradient is seen in Figure 6e, which shows differences that are best highlighted by the corresponding 3D plot in Figure 6e. Here we see that the standardized gradient with respect to polarity undergoes a precipitous decline from unipolarity to bipolarity, after which it tapers off much more gradually and this feature is independent of *K*, consistent with elasticity results.

which is a two-dimensional vector function Φ*⇤* = *⇤*(*K,* ⇥). The resulting vector field of this hybrid standardized gradient is seen in Figure 6e, which shows differences that are best highlighted by the corresponding 3D plot in Figure 6e. Here we see that the standardized gradient with respect to polarity undergoes a precipitous decline from unipolarity to bipolarity, after which it tapers off much more gradually and this feature is independent of *K*, consistent with elasticity results.

Given such marked differences between elasticity functions, it is best to investigate the

*<sup>K</sup>* ln ⇥*@<sup>k</sup>* (*<sup>K</sup>* ln ⇥)<sup>i</sup> <sup>+</sup>

�

which is a two-dimensional vector function Φ*⇤* = *⇤*(*K,* ⇥). The resulting vector field of this hybrid standardized gradient is seen in Figure 6e, which shows differences that are best highlighted by the corresponding 3D plot in Figure 6e. Here we see that the standardized gradient with respect to polarity undergoes a precipitous decline from unipolarity to bipolarity, after which it tapers off much more gradually and this feature is independent of *K*, consistent with elasticity results.

*<sup>K</sup>* ln ⇥*@<sup>k</sup>* (*<sup>K</sup>* ln ⇥)<sup>i</sup> <sup>+</sup>

ln(⇥ + 1) ln ⇥ *<sup>−</sup>* <sup>1</sup>

�

Given such marked differences between elasticity functions, it is best to investigate the

better view of (i) the hybrid gradient field Φ that drops from the NW region into the uni- and

better view of (i) the hybrid gradient field Φ that drops from the NW region into the uni- and

which is a two-dimensional vector function Φ = (*K,* ⇥). The resulting vector field of this hybrid

with a hot spot in the NW corner; (iv) divergence from a line at approximately 60 degrees; and (5)

ln(⇥ + 1) ln ⇥ *<sup>−</sup>* <sup>1</sup>

�

⇥

⇥

Given such marked differences between elasticity functions, it is best to investigate the

The corresponding vector magnitude or norm *|*Φ*|*, a scalar function, is shown by the contour plot in Figure 6d, on the same domain as the field. This shows other clear patterns, including a better view of (i) the hybrid gradient field Φ that drops from the NW region into the uni- and bi-polar basin at minimal values of *K*—clearly where the most peaceful worlds are found—and (ii)

Given such marked differences between elasticity functions, it is best to investigate the

ln(⇥ + 1) ln ⇥ *<sup>−</sup>* <sup>1</sup>

which is a two-dimensional vector function Φ*⇤* = *⇤*(*K,* ⇥). The resulting vector field of this hybrid standardized gradient is seen in Figure 6e, which shows differences that are best highlighted by the corresponding 3D plot in Figure 6e. Here we see that the standardized gradient with respect to polarity undergoes a precipitous decline from unipolarity to bipolarity, after which it tapers off much more gradually and this feature is independent of *K*, consistent with elasticity results.

which is a two-dimensional vector function Φ*⇤* = *⇤*(*K,* ⇥). The resulting vector field of this hybrid standardized gradient is seen in Figure 6e, which shows differences that are best highlighted by the corresponding 3D plot in Figure 6e. Here we see that the standardized gradient with respect to polarity undergoes a precipitous decline from unipolarity to bipolarity, after which it tapers off much more gradually and this feature is independent of *K*, consistent with elasticity results.

Given such marked differences between elasticity functions, it is best to investigate the

⇥

The corresponding vector magnitude or norm *|*Φ*|*, a scalar function, is shown by the contour plot in Figure 6d, on the same domain as the field. This shows other clear patterns, including a better view of (i) the hybrid gradient field Φ that drops from the NW region into the uni- and bi-polar basin at minimal values of *K*—clearly where the most peaceful worlds are found—and (ii)

a distinct view of the high ridge beyond *K ≥* 5 and *✓ ≥* 2.

*<sup>r</sup>· ⇤<sup>φ</sup>* <sup>=</sup> *<sup>K</sup>*

*<sup>K</sup>* ln ⇥*@<sup>k</sup>* (*<sup>K</sup>* ln ⇥)<sup>i</sup> <sup>+</sup>

= 1i + ⇥

ln(⇥ + 1) ln ⇥ *<sup>−</sup>* <sup>1</sup>

*<sup>K</sup>* ln ⇥*@<sup>k</sup>* (*<sup>K</sup>* ln ⇥)i<sup>+</sup>

standardized gradient, as follows:

curling associated with divergence.

standardized gradient, as follows:

a distinct view of the high ridge beyond *K ≥* 5 and *✓ ≥* 2.

*<sup>r</sup>· ⇤<sup>φ</sup>* <sup>=</sup> *<sup>K</sup>*

a distinct view of the high ridge beyond *K ≥* 5 and *✓ ≥* 2.

*<sup>r</sup>· ⇤<sup>φ</sup>* <sup>=</sup> *<sup>K</sup>*

a distinct view of the high ridge beyond *K ≥* 5 and *✓ ≥* 2.

*<sup>r</sup>· ⇤<sup>φ</sup>* <sup>=</sup> *<sup>K</sup>*

= 1i + ⇥

= 1i + ⇥

= 1i + ⇥

a distinct view of the high ridge beyond *K ≥* 5and *✓ ≥* 2.

*<sup>r</sup>· ⇤<sup>φ</sup>* <sup>=</sup> *<sup>K</sup>*

= 1i + ⇥

ln(⇥ + 1) ln ⇥ *<sup>−</sup>* <sup>1</sup>

*<sup>K</sup>* ln ⇥*@<sup>k</sup>* (*<sup>K</sup>* ln ⇥)<sup>i</sup> <sup>+</sup>

$$\nabla \phi = \partial\_k \left( K \ln \Theta \right) \mathbf{i} + \Delta\_\theta \left( K \ln \Theta \right) \mathbf{j} \tag{25}$$

⇥

�

⇥

�

$$\dot{\mathbf{i}} = \dot{\log} \dot{\Theta} \,\mathbf{i} + \dot{K} \left[ (\ln \Theta + 1) - \dot{\ln} \Theta \right] \,\mathbf{j},\tag{26}$$

*<sup>K</sup>* ln ⇥∆*✓* (*<sup>K</sup>* ln ⇥)<sup>j</sup> (27)

*<sup>K</sup>* ln ⇥∆*✓* (*<sup>K</sup>* ln ⇥)j(27)

*<sup>K</sup>* ln ⇥∆*✓* (*<sup>K</sup>* ln ⇥)<sup>j</sup> (27)

j*,* (28)

*<sup>K</sup>* ln ⇥∆*✓* (*<sup>K</sup>* ln ⇥)<sup>j</sup> (27)

j*,* (28)

j*,* (28)

j*,* (28)

*<sup>K</sup>* ln ⇥∆*✓* (*<sup>K</sup>* ln ⇥)<sup>j</sup> (27)

j*,* (28)

follows:

follows:

CLAUDIO CIOFFI-REVILLA

CLAUDIO CIOFFI-REVILLA

*⌘✓φ* = ⇥

*⌘✓φ* = ⇥

ln(⇥ + 1) ln ⇥ *<sup>−</sup>* <sup>1</sup>

�

ln(⇥ + 1) ln ⇥ *<sup>−</sup>* <sup>1</sup>

with logarithms of polarity, as shown in Figures 6a and b, respectively.

history of major powers after the Soviet-American Cold War around 1989.

�

respect to parameter *K* and polarity ⇥ yields the following set of hybrid equations:

Finally, we can now investigate Midlarsky's law using standardized variables, as in the previous cases. First, we calculate and analyze percentage change in *φ* with respect to *K* and ⇥, as shown by the elasticities in Figures 6a and b. Calculation of point elasticity and arc elasticity of *φ* with

with logarithms of polarity, as shown in Figures 6a and b, respectively.

Interestingly, arc elasticity (Figure 6b) exhibits a singularity under unipolarity, where *φ*(*k,* 1) = *1*, indicating a major transition from unipolarity to bipolarity. In this case arc elasticity continues a rapid drop with increasing polarity, which is the systemic trend experienced in contemporary

We immediately see that standardized effects on *φ* are again quite different from earlier unitbased results (cf. equations 21 and 22 and associated figures). Here, point elasticity represents percentage change in annual war frequency with respect to percentage change in parameter *K*, whereas arc elasticity measures percentage change in *φ* with respect to percentage annual change in polarity ⇥. The former has a constant value of 1 while the latter is a rational hybrid function

history of major powers after the Soviet-American Cold War around 1989.

Comparing the two elasticities (equations 23 and 24; cf. also their respective graphs in Figures 6a and b) yields the following dominance principle: annual frequency of war *φ* is more sensitive to percentage change in parameter *K* than to change in systemic polarity ⇥ under unipolar and bipolar systemic structures, but the reverse is true under tripolarity and higher-order structures—because the relationship reverses between ⇥ = 2 and 3. This is a surprising qualitative transition that is invisible in the original model but is clear once the dimensions are standardized by elasticities. The different and joint effects of parameter *K* and polarity ⇥ on war frequency *φ* can be seen

which is a two-dimensional vector function **Φ** = **Ψ** (*K*, Θ). The resulting vector field of this hybrid gradient is in Figure 6c, which shows: (i) heterogeneity along the two dimensions; (ii) general southwest-northeast orientation; (iii) increasing intensity as *K* increases for low values of polarity with a hot spot in the NW corner; (iv) divergence from a line at approximately 60 degrees; and (5) curling associated with divergence. *r· φ* = *@<sup>k</sup>* (*K* ln ⇥)i + ∆*✓* (*K* ln ⇥)j (25) = log ⇥ i + *K* [(ln ⇥ + 1) *−* ln ⇥] j*,* (26) which is a two-dimensional vector function Φ = (*K,* ⇥). The resulting vector field of this hybrid gradient is in Figure 6c, which shows: (i) heterogeneity along the two dimensions; (ii) general southwest-northeast orientation; (iii) increasing intensity as *K* increases for low values of polarity *r· φ* = *@<sup>k</sup>* (*K* ln ⇥)i + ∆*✓* (*K* ln ⇥)j (25) = log ⇥ i + *K* [(ln ⇥ + 1) *−*ln ⇥] j*,* (26) which is a two-dimensional vector function Φ = (*K,* ⇥). The resulting vector field of this hybrid gradient is in Figure 6c, which shows: (i) heterogeneity along the two dimensions; (ii) general southwest-northeast orientation; (iii) increasing intensity as *K* increases for low values of polarity

Finally, we can now investigate Midlarsky's law using standardized variables, as in the previous cases. First, we calculate and analyze percentage change in *φ* with respect to *K* and ⇥, as shown by the elasticities in Figures 6a and b. Calculation of point elasticity and arc elasticity of *φ* with

*⌘kφ* = 1 (for *K*'s percentage effect on *φ*, in Figure 6a) (23)

(for ⇥'s percentage effect on *φ*, in Figure 6b)*.* (24)

We immediately see that standardized effects on *φ* are again quite different from earlier unitbased results (cf. equations 21 and 22 and associated figures). Here, point elasticity represents percentage change in annual war frequency with respect to percentage change in parameter *K*, whereas arc elasticity measures percentage change in *φ* with respect to percentage annual change in polarity ⇥. The former has a constant value of 1 while the latter is a rational hybrid function

Interestingly, arc elasticity (Figure 6b) exhibits a singularity under unipolarity, where *φ*(*k,* 1) = *1*, indicating a major transition from unipolarity to bipolarity. In this case arc elasticity continues a rapid drop with increasing polarity, which is the systemic trend experienced in contemporary

Comparing the two elasticities (equations 23 and 24; cf. also their respective graphs in Figures 6a and b) yields the following dominance principle: annual frequency of war *φ* is more sensitive to percentage change in parameter *K* than to change in systemic polarity ⇥ under unipolar and bipolar systemic structures, but the reverse is true under tripolarity and higher-order structures—because the relationship reverses between ⇥ = 2 and 3. This is a surprising qualitative transition that is invisible in the original model but is clear once the dimensions are standardized by elasticities. The different and joint effects of parameter *K* and polarity ⇥ on war frequency *φ* can be seen

(for ⇥'s percentage effect on *φ*, in Figure 6b)*.* (24)

respect to parameter *K* and polarity ⇥ yields the following set of hybrid equations:

*⌘kφ* = 1 (for *K*'s percentage effect on *φ*, in Figure 6a) (23)

The corresponding vector magnitude or norm |**Φ**|, a scalar function, is shown by the contour plot in Figure 6d, on the same domain as the field. This shows other clear patterns, including a better view of (i) the hybrid gradient field **Φ** that drops from the NW region into the uni- and bi-polar basin at minimal values of *K*—clearly where the most peaceful worlds are found—and (ii) a distinct view of the high ridge beyond *K* ≥ *5* and *Θ* ≥ 2. with a hot spot in the NW corner; (iv) divergence from a line at approximately 60 degrees; and (5) curling associated with divergence. The corresponding vector magnitude or norm *|*Φ*|*, a scalar function, is shown by the contour plot in Figure 6d, on the same domain as the field. This shows other clear patterns, including a better view of (i) the hybrid gradient field Φ that drops from the NW region into the uni- and bi-polar basin at minimal values of *K*—clearly where the most peaceful worlds are found—and (ii) with a hot spot in the NW corner; (iv) divergence from a line at approximately 60 degrees; and (5) curling associated with divergence. The corresponding vector magnitude or norm *|*Φ*|*, a scalar function, is shown by the contour plot in Figure 6d, on the same domain as the field. This shows other clear patterns, including a better view of (i) the hybrid gradient field Φ that drops from the NW region into the uni- and bi-polar basin at minimal values of *K*—clearly where the most peaceful worlds are found—and (ii)

Given such marked differences between elasticity functions, it is best to investigate the standardized gradient, as follows: a distinct view of the high ridge beyond *K ≥* 5 and *✓ ≥* 2. Given such marked differences between elasticity functions, it is best to investigate the standardized gradient, as follows: a distinct view of the high ridge beyond *K ≥* 5 and *✓ ≥*2. Given such marked differences between elasticity functions, it is best to investigate the standardized gradient, as follows:

$$\nabla^\* \phi = \frac{K}{K \ln \Theta} \partial\_k \left( K \ln \Theta \right) \mathbf{i} + \frac{\Theta}{K \ln \Theta} \Delta\_\theta \left( K \ln \Theta \right) \mathbf{j} \qquad (27)$$
 
$$\dots \quad \int \ln(\Theta + 1) \quad \dots \quad \int \dots \tag{27}$$

$$= \mathbf{1i} + \Theta \left[ \frac{\ln(\Theta + 1)}{\ln \Theta} - 1 \right] \mathbf{j},\tag{28}$$

which is a two-dimensional vector function Φ*⇤* = *⇤*(*K,* ⇥). The resulting vector field of this hybrid standardized gradient is seen in Figure 6e, which shows differences that are best highlighted by the corresponding 3D plot in Figure 6e. Here we see that the standardized gradient with respect to polarity undergoes a precipitous decline from unipolarity to bipolarity, after which it tapers off much more gradually and this feature is independent of *K*, consistent with elasticity results. which is a two-dimensional vector function Φ*⇤* = *⇤*(*K,* ⇥). The resulting vector field of this hybrid standardized gradient is seen in Figure 6e, which shows differences that are best highlighted by the corresponding 3D plot in Figure 6e. Here we see that the standardized gradient with respect to polarity undergoes a precipitous decline from unipolarity to bipolarity, after which it tapers off much more gradually and this feature is independent of *K*, consistent with elasticity results. which is a two-dimensional vector function **Φ**\* = **Ψ**\* (*K*, Θ). The resulting vector field of this hybrid standardized gradient is seen in Figure 6e, which shows differences that are best highlighted by the corresponding 3D plot in Figure 6e. Here we see that the standardized gradient with respect to polarity undergoes a precipitous decline from unipolarity to bipolarity, after which it tapers off much more gradually and this feature is independent of *K*, consistent with elasticity results.

#### 5. Conclusions

*<sup>r</sup>· ⇤<sup>φ</sup>* <sup>=</sup> *<sup>K</sup>*

= 1i + ⇥

22

22

We began this chapter by observing that hybrid functions—formal models containing a combination of continuous and discrete variables—have been present in international relations theory since antiquity, *"hiding in plain sight,"* but their rigorous analysis has been impeded by the generally disjoint nature and established practices of infinitesimal calculus and discrete calculus. This problem has now been resolved by a unified approach that is feasible and fruitful, as provided by nabladot calculus.

Analysis of three separate cases in this chapter showed how and why, far from being intractable or only amenable to approximations, hybrid functions in international relations theory contain numerous interesting features and insightful properties that shed new light on our understanding of international phenomena. Nabladot analysis of each hybrid function—and subsequent comparative analysis across them—in each case revealed previously unknown and often scientifically surprising theoretical landscapes of international phenomena ranging from generic international events, to conditions of peace, crisis dynam-

Figura 6. Elasticities and gradients of the annual frequency of war *φ*. (a) point elasticity *⌘k*(*φ*) with respect to scale parameter *K*; (b) arc elasticity *⌘✓*(*φ*) with respect to systemic polarity ⇥; (c) vector field of the dot-gradient vector function *r· φ*; (d) contour plot of (c); (e) vector field of the standardized dot-gradient vector function *r· ⇤φ*; (f) 3D plot of (e). Figure 6. Elasticities and gradients of the annual frequency of war φ. (a) point elasticity *ηk* (*φ*) with respect to scale parameter *K*; (b) arc elasticity *ηΘ*(*φ*) with respect to systemic polarity Θ; (c) vector fi eld of the dot-gradient vector function investigation. The first nabladot operation is to calculate the hybrid gradient of *Z* to discover the magnitude *and* direction of changes in *Z* as a function of changes in *X* and *Y* . The result of applying the nabladot operator (a vector operator) to scalar hybrid function *'* is a hybrid vector function Φ = *r· '* with *x*- and *y*-components. The hybrid gradient in two dimensions is the scalar vector product calculated using the new *nabladot vector operator r·* (note the dot within the nabla *φ*; (d) contour plot of (c). (e) vector fi eld of the standardized dot-gradient vector function The analytical process thus far has focused on scalar properties of the IR hybrid function under investigation. The first nabladot operation is to calculate the hybrid gradient of *Z* to discover the magnitude *and* direction of changes in *Z* as a function of changes in *X* and *Y* . The result of applying the nabladot operator (a vector operator) to scalar hybrid function *'* is a hybrid vector function Φ = *r· '* with *x*- and *y*-components. The hybrid gradient in two dimensions is the scalar *φ*; (f) 3D plot of (e).

where, by convention, i and j denote unit vectors along *x*- and *y*-dimensions, respectively, and *@<sup>x</sup>* and ∆*<sup>y</sup>* denote the first-order derivative and first-order difference with respect to *X* and *Y* (Cioffi symbol), which is defined as follows: where, by convention, i and j denote unit vectors along *x*- and *y*-dimensions, respectively, and *@<sup>x</sup>* ics, and warfare; all within the unifi ed methods enabled by the hybrid nabladot operator and associated concepts, rather than through disjoint calculi or errorprone approximations.

2014; 2017; 2019; 2020; 2020).<sup>3</sup> Note that the resulting nabladot gradient of hybrid function *'* is a striated vector field with a first-order partial derivative component along the *x*-axis (continuous) and a first-order partial difference component along the *y*-axis (discrete), hence the striation of the vector field's topology.<sup>4</sup> The absolute and standardized norms of the hybrid gradient *r· '* are calculated next, along with and ∆*<sup>y</sup>* denote the first-order derivative and first-order difference with respect to *X* and *Y* (Cioffi 2014; 2017; 2019; 2020; 2020).<sup>3</sup> Note that the resulting nabladot gradient of hybrid function *'* is a striated vector field with a first-order partial derivative component along the *x*-axis (continuous) and a first-order partial difference component along the *y*-axis (discrete), hence the striation of the vector field's topology.<sup>4</sup> Th ese new theoretical landscapes and research frontiers are exciting and their application is still in a preliminary but already a demonstrably promising stage. Dominance principles that rank the infl uence of causal independent variables, singularities, previously undetected phase transitions, the deep nature of

graphic analyses.

graphic analyses.

variable(s).

variable(s).

10

10

corresponding graphs for investigating the resulting vector field. Each pair of plots for a vector

vector product calculated using the new *nabladot vector operator r·* (note the dot within the nabla

comparative analysis. Cardinal directions (N, E, S, W) are used for simple orientation in graphs. Other hybrid operations of nabladot calculus equivalent to the divergence, curl, Laplacian, Hessian, and Jacobian are subsequently calculated to shed additional (and usually new) light on the original function *Z* = *'*(*X, Y* ) through the medium of nabladot operators, each supported by additional

field and corresponding norm should use identical domains to facilitate understanding through comparative analysis. Cardinal directions (N, E, S, W) are used for simple orientation in graphs. Other hybrid operations of nabladot calculus equivalent to the divergence, curl, Laplacian, Hessian, and Jacobian are subsequently calculated to shed additional (and usually new) light on the original function *Z* = *'*(*X, Y* ) through the medium of nabladot operators, each supported by additional

The main results of nabladot analysis shed new light on fundamental, real-world, substantive properties and features of the original hybrid function under investigation, features that remain hidden or inaccessible through other forms of analysis. Each main formal expression is accompanied by an interpretation in plain English, although this is not always possible without some loss of precision or clarity. Some results can be somewhat complicated nonlinear functions that do not further simplify; we prefer them that way rather than introducing artificial approximations which may be simpler but unrealistic or unnatural objects, unlike real IR phenomena. In most cases an ensemble of images and visual analytics (Thomas and Cook 2005; Wellin 2013) of complicated

The main results of nabladot analysis shed new light on fundamental, real-world, substantive properties and features of the original hybrid function under investigation, features that remain hidden or inaccessible through other forms of analysis. Each main formal expression is accompanied by an interpretation in plain English, although this is not always possible without some loss of precision or clarity. Some results can be somewhat complicated nonlinear functions that do not further simplify; we prefer them that way rather than introducing artificial approximations which may be simpler but unrealistic or unnatural objects, unlike real IR phenomena. In most cases an ensemble of images and visual analytics (Thomas and Cook 2005; Wellin 2013) of complicated

Among the most important substantive (and testable) results from nabladot analysis are the dominance principles mentioned above—they explain which independent variable has dominant effect on the dependent variable, a major theoretical (and arguably policy) question impossible to answer *ex ante*—as well as other characteristic phenomena of interest (e.g., discrete striations, inflection or "tipping" points, asymptotes and other singularities, constant or invariant subfields, and others) revealed by geometric and topological information. In addition, interesting scalar and vector fields of *'* become accessible to direct investigation through formal tools of nabladot calculus and analysis. A novel and valuable feature of this approach is that nabladot calculus provides exact results in analytical investigations where the classical infinitesimal calculus of hybrid IR functions would provide approximations with errors over the discrete domain of independent

Among the most important substantive (and testable) results from nabladot analysis are the dominance principles mentioned above—they explain which independent variable has dominant effect on the dependent variable, a major theoretical (and arguably policy) question impossible to answer *ex ante*—as well as other characteristic phenomena of interest (e.g., discrete striations, inflection or "tipping" points, asymptotes and other singularities, constant or invariant subfields, and others) revealed by geometric and topological information. In addition, interesting scalar and vector fields of *'* become accessible to direct investigation through formal tools of nabladot calculus and analysis. A novel and valuable feature of this approach is that nabladot calculus provides exact results in analytical investigations where the classical infinitesimal calculus of hybrid IR functions would provide approximations with errors over the discrete domain of independent

<sup>3</sup> IR scholars rarely consider the presence of vectors in international relations, other than metaphorically. This analysis demonstrates the rigorous analysis of vectors and vector fields in IR using formal methods from nabladot calculus, as in the next section. To contain notation, we shall use i and j to denote unit vectors along continuous and discrete dimensions, respectively, rather than create new unit vectors for each variable. <sup>4</sup> Use of the partial derivative with respect to *<sup>Y</sup>* (a discrete variable) instead of the partial difference—which is often used in approximations—produces a measurable error that varies in magnitude depending on the structure of *'* and values of *Y* . Measurable discrepancies between the two operators (nabladot and classical

<sup>3</sup> IR scholars rarely consider the presence of vectors in international relations, other than metaphorically. This analysis demonstrates the rigorous analysis of vectors and vector fields in IR using formal methods from nabladot calculus, as in the next section. To contain notation, we shall use i and j to denote unit vectors along continuous and discrete dimensions, respectively, rather than create new unit vectors for each variable. <sup>4</sup> Use of the partial derivative with respect to *<sup>Y</sup>* (a discrete variable) instead of the partial difference—which is often used in approximations—produces a measurable error that varies in magnitude depending on the structure of *'* and values of *Y* . Measurable discrepancies between the two operators (nabladot and classical

nabla) are demonstrable but beyond the present scope due to space limitations (Cioffi 2021).

nabla) are demonstrable but beyond the present scope due to space limitations (Cioffi 2021).

23

*r· ' ⌘ @x'* i + ∆*y'* j*,* (1)

*r· ' ⌘ @x'* i + ∆*y'* j*,* (1)

symbol), which is defined as follows:

functions can add significant clarity.

functions can add significant clarity.

probabilistic causality, and other scientifically intriguing results have intrinsic value for our understanding of international relations. As "progressive problemshifts", in the sense of Lakatos (1973; cf. also Gillespie 1976 and Moore 2001), this novel and emergent corpus of scientific knowledge also provides rich and creative foundations for more advanced analyses that enhance our theoretical as well as practical understanding.

### Acknowledgements

*Grazie mille* to professors Marco Cesa and Sonia Lucarelli for the kind invitation to contribute to this volume of papers in honor of my esteemed teacher and lifelong friend, Professor Umberto Gori. In retrospect, the earliest seeds of the nabladot calculus presented in this chapter were nucleated from my 1974 doctoral thesis, directed by Professor Gori at the University of Florence's *Scuola di Scienze Politiche e Sociali Cesare Alfieri*. Thanks to the Fenwick Library and the College of Science of George Mason University for use of the Mathematica system to create the figures and verify calculations. The editors and typographic staff of Florence University Press provided the excellent LaTeX template used to produce this chapter.

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