Despite calls for greater emphasis on empirical studies of international conflict and cooperation—and despite the mountains of data collected—persuasive analytical models have proven elusive. To the extent that quantitative analyses of international conflict have been rigorously conducted and offered, they remain unsatisfying in their capacity to predict behavior (see Beck 2000). However, a set of techniques seeing increasing application in this regard is network analysis, a field that offers international relations scholars 1) a collection of theories describing relationships between the structural characteristics of networks and their behaviors and 2) a collection of analytical tools for investigating these structural properties.[1]
Recent applications of network analysis to international relations theory are diverse and include efforts such as explicating network and social power as challenges to more traditional, neorealist views of structure and material power (Knoke 1990, Hafner-Burton and Montgomery 2006, and Beckfield 2008);[2] characterizing the implications of network structure on the capacity to impact illicit networks (Krebs 2002, Kinsella 2006, Enders and Su 2007, and Kahler 2009); describing the relationship between networks of intergovernmental organizations, their influence on relative positions of power, and military conflict (Hafner-Burton and Montgomery 2006); raising methodological questions regarding past studies of the Kantian peace (Ward, Siverson, and Cao 2007); the formation of international norms for the regulation of certain categories of weapons (Carpenter 2011); and shifting emphasis from actor-centered to hierarchical network models in the study of international political economy (Oatley et al 2013). Each of these marks an attempt to illuminate some causal relationship between network structure and the behavior of the nodes in their respective networks. What each of these efforts has in common, however, is their level of analysis. Each focuses primarily on measures associated with individual nodes in networks or the distribution of those measures (especially centrality), on the behavior of the nodes in their dyadic interactions, or on incorporating interdependencies into the statistical methods for analyzing these interactions (progressive corrections).[3] Further, these tend to display a tendency of international relations scholarship to disaggregate the international system into dyadic structures (either country pairs or dyad years) based on methodological limitations. Unfortunately, the international system does not exhibit such simple structures, and no matter how useful such studies may be they must be constrained by the empirical and analytical approaches they adopt.[4]
Another line of inquiry focuses on the mechanics of network formation in the international system, proposing and testing hypotheses for the mechanisms and motivations of states in forming these networks.[5] This effort has been led by Zeev Maoz (2012), with recent additions by Kinne (2013), and posits that the co-evolution of networks is a critical shaping process in patterns of international conflict and cooperation. Maoz notes, however, that data collection and analysis has been generally on networks as they are found, ignoring the evolutionary process that explains
the networks came to be (Maoz, 2012). Although he acknowledges that baseline network models have been assessed since at least 1960, Maoz recommends looking to the models in the physical, biological, and social sciences to connect observed network structures to the processes by which they are formed, beginning with to two most prominent models of network formation: preferential attachment (PA) and homophily (HO). Ultimately, Maoz finds that these models allow for the identification of linkages between micro-level processes and macro-level consequences such as structures in international politics. While his presentation is formidable and extremely heuristic, Maoz remains concerned that the model may be overly complicated and wonders whether it could be simplified without significant loss theoretical and empirical value. (2012) More recently, Kinne (2013) has refined our understanding of the structural dynamics of network formation, specifically focusing on endogenous network effects as causal in the formation of bilateral agreements between states.
Both of the lines of inquiry described above tend to ignore the behavior and influence of the network at the network level. That is, it is important to not only consider the formation of the network and the behavior of its elements, but also the
: the emergent behavior of the nodes in the network, the network itself, and the interaction between these levels. Further, while recognizing the complicated and complex nature of international relations, most network studies of interaction in international relations concentrate on dyadic pairings, in which the vast majority of interactions are trivial (Beck 2000, 22).[6] By explicitly incorporating the mutually constitutive nature of agents and structures and more realistically reflecting real-world observations, a step forward is implied, but until a critical intervening link or mechanism between the two is isolated, empirical data analysis is likely to be less persuasive than theoretical model-making or metaphysical philosophizing. We agree with Maoz that this crucial intervening link is the network. We are concerned, however, that his dismissal of the potential explanatory power of random networks may be premature and that these have not been fully explored. It is possible, however, that the noted gap is simply methodological; that is, there appears a gap in the analytical tools for network analysis as they have been imported into the field of international relations.
It is toward advancing the study of networks in international relations that this effort aims, and with this in mind borrows a tool that has proven heuristically valuable in the understanding of complex relationships in the social and physical sciences—random Boolean networks (RBNs). As part of the research agenda put forward by Hafner-Burton, Kahler, and Montgomery (2009), we seek to advance a systematic technique for evaluating the
We do not make grandiose claims of certainty with this technique. Neither do we describe or explain causative factors in the emergence and evolutionary dynamics of particular structures; we address only a time series of
network structures. As such, we are acutely aware that this analysis does not propose any definitive answers to important questions in the field of international relations. We do, however, hope to place a new and potentially powerful suite of analytical tools at the international theorist’s disposal. Not finding RBNs applied to international relations anywhere in the literature, we offer the following in the expectation that it will illuminate the dynamics of the international system and suggests a path for future empirical and theoretical research.
This discussion proceeds in four parts. First, we outline the theory, history, properties, and applications of RBNs, developing the tools necessary to apply them as a technique to a topic of interest within international relations. Second, to prepare and animate an example of the technique in practice, we briefly summarize a key debate in international theory, the role of trade relationships in levels of cooperation between nation-states. In the third section, we construct a trade-based Boolean network model of the international system at a fixed moment in time, and we show that this network may exhibit chaotic dynamical behavior based on its topology. This is followed by a demonstration that the noted fundamental topological structure of the international system—with all its attendant emergent behaviors—is not isolated but rather persists over time. We close with a brief discussion of the potential theoretical implications associated with the application of RBNs to the field of international relations, focusing on the potential for this new tool to illuminate theoretical discourse in the agent-structure debate.
Random Boolean Networks[7]
A network is nothing more than a collection of interacting and interdependent nodes, actors, or agents. A Boolean network is a generalization of a system of cellular automata[8] with a set of N nodes σi (i = 1, 2, ..., N). where each node is a binary state variable capable of taking on the Boolean values zero and one that explicitly (if highly abstractly) represents the nodes, behaviors, interactions, and interdependencies of a concrete network.[9] These nodes interact with each other via a network of interconnections between the nodes and Boolean transition functions. Each node provides a primary contribution to the behavior of other nodes in the network, and we call ki the outdegree of the ith node. A randomly-generated example network comprised of N = 6 nodes is shown in Figure 1. In this example, ki = 2 for each i, and the contributing/controlling connections between nodes are shown as directed edges with influence flowing in the direction of the arrows. Illustrating these controlling connections, the value of σ1 at time t will contribute to the behavior of the nodes σ2 and σ4 at time t + 1; conversely, the value of σ1 at time t will be directly controlled by the behavior of σ2, σ4, σ5, and σ6 at time t - 1.
Figure 1: Random network with N = 6 nodes and constant outdegree ki = 2
Note that in this example, there are no self-loops associated with any node; that is, no node is connected to itself. This is not a necessary constraint, but it raises an interesting issue regarding independence.[10] In a network with no self-loops, the state of each node has no direct dependence on its previous state; it is possible, though, for a node to influence itself via intermediaries. As noted above, the state of
σ1 at time
t will depend on the value of state
σ2 at time
t - 1, but
σ2 depends (in part) on
σ3 which depends, in turn (and in part), on
σ4. From the diagram, we see that the value of
σ4 depends (in part) on the value of
σ1. So, the value of
σ1 at time
t is a function (in part) of the value of
σ1 at time
t - 4 (the length of the cycle just described). It is this self-referential behavior that gives rise to nonlinearities and potentially complex behavior in the dynamical Boolean network; it also necessarily complicates linear regression techniques since assumptions of dyadic independence are violated.
As noted above, the
ith node in the Boolean network adds a first order contribution to the behavior of of the other nodes. In a
random Boolean network (RBN), the values for
ki are chosen from a discrete probability distribution (in the example shown in Figure 1 this is a constant distribution with
ki = 2 for every
i), and the nodes controlled are usually selected with uniform probability. To describe the mechanism through which this control is exerted, suppose the
ith node is controlled by
Ki nodes (i.e., the indegree of the
ith node in the network is
Ki), and the set of controlling nodes are represented by the set {
σi1, σi2,…,
σiKi}. The value of each node is then determined by a formula of the form
where
fi is a Boolean function of the controlling elements. The function
fi associates with each of the possible input strings (one for each binary string of length ) a value of either zero or one. Thus, for each node there are possible 2
Ki transition functions
fi, and for a network of
N nodes in a given configuration there are
possible network definitions. In an RBN, these transition functions are typically defined randomly so that
fi = 0 with probability
p and
fi = 1 with probability 1 -
p. Continuing the example begun in Figure 1, a set of randomly generated transition functions are shown in Figure 2. So, for example, if
σ1 = 0 and
σ6 = 0 at time
t, then
σ4 = 1 at time
t + 1. Since
K6 = 0 in the example, this node will have a constant value depending only on its initial condition.
Figure 2: Randomly generated transition functions for an RBN, p = 0.5
Now, given an initial condition for a network of
N nodes (i.e., an initial state selection of zero or one for each node), the network updates synchronously and deterministically cycles through a series of network states defined by the transition functions
fi.[11] Since the set of potential network states is finite—there are only 2
N distinct binary strings of length
N—the network must at some point repeat a previously held network state. Thereafter, the network progresses through a fixed cycle of network states called a
limit cycle. There may be many such limit cycles for a given network, with each cycle reached by some subset of initial conditions referred to as the
basin of attraction for that limit cycle. The set of
26 = 64 states associated with the network described in Figure 1 and Figure 2 are shown in the directed graph in Figure 3. Each point represents one of the 64 states the network can assume, and the transition from one network state to another is shown as an arrow connecting those network states. This example network has four distinct limit cycles, two of which are constant.[12]
Figure 3: Network state space and limit cycles for the example RBN
Another characteristic to note in the directed graph shown in Figure 3 is the number of states with and without precursors. A network state that cannot be reached from some other network state, based on the transition functions defining the network, is said to have no precursors. In the example shown, 52 of the 64 network states have no precursors. So, in this example, the average time (measured in the number of state transitions required) to reach a limit cycle is quite short, making this network quite predictable in its trajectory.
In developing the description for Boolean networks above, it was necessary to specify the outdegree
ki for each node in the network. As mentioned, the values for
ki are typically chosen from a discrete probability distribution. The particular distribution from which the
ki are chosen, together with the value of the parameter
p and the manner in which they are associated with the other network nodes, determine the average dynamics of the resulting RBN.
Note the generality of this abstract construction. Any system composed of discrete elements that interact and are in some sense interdependent can be modeled as an RBN. In other words, any network should be susceptible at some level to representation and analysis using these tools. Thus represented, the behavior of the agents is abstractly modeled as binary responses to stimuli and their interactions are captured via transition functions. While exceedingly abstract, these networks show a rich set of dynamic properties. Of course, one must take care to avoid reifying RBNs and ensure the links between the model and the modeled are appropriate.[13]
A final note on RBNs is warranted before proceeding. Returning briefly to Figure 1, the networks on which we overlay a Boolean dynamical structure are perfectly amenable to traditional measurements of network properties.[14] The techniques and theory associated with RBNs are therefore supplementary in terms of their approach to empirical and theoretical investigation of network properties. They do not replace more established methods; they only add new approaches to investigating networks and open new possibilities for the character of the hypotheses posed in these examinations.
Applications and Dynamics of RBNs
RBNs have been used in the study of phenomena as diverse as spin glasses (Derrida and Flyvberg 1986), evolution (Stern 1999), the social sciences (Garson 1998 and Weigel 2000), the stock market (Chowdhury 1999), and more direct applications to information flow and network theory (Richardson 2008). However, the seminal study using RBNs as an analytical tool is Stuart Kauffman’s (1969) investigation of gene regulation and cellular differentiation.[15] In this work, Kaufmann examined the consequences of a hypothesis that genetic networks are themselves (or are at least closely analogous to) RBNs in which the on-off/one-zero dynamic of the network describes the expression and repression of the nodes or genes and thus the phenotype of associated organisms (or cells within those organisms). His analysis revealed that for Boolean networks in which the expression of each network is influenced by two to three other genes (with a constant distribution for
Ki and uniformly random choices for the controlling nodes), (1) the associated network is moderately (but not perfectly) stable under random perturbation, and (2) networks are capable of achieving only a limited number of other limit cycles (or phenotypes) under random perturbation. For higher degrees of connectivity, however, the RBNs become much more phenotypically unstable, and for lower degrees of connectivity, network stability prevents changes of phenotype.[16] Kauffman also observed that the average number of limit cycles for RBNs in this narrow connectivity range is proportional to the square root of the number of nodes in the network, an observation consistent with the relationship between the number of genes (nodes) and cell types (limit cycles) in organisms. This and later work led to speculation by Kauffman (and others) that the evolution of organisms with the genetic complexity and order observed in nature is possible only in genetic regulatory networks with connectivity in this narrow range of two to three,[17] leading to the phrase
life at the edge of chaos.[18] Only in these networks is phenotypical stability great enough to allow for the development of well-adapted populations of organisms while simultaneously allowing for organism sufficiently adaptive to a changing environment.
Kauffman’s characterization of the influence of network connectivity and the Boolean parameter
p was later extended using different analytic techniques to describe a more well-defined border between order and chaos in terms of the distribution of
Ki and the parameter
p. The phase transition in such networks is characterized by the relationship
where
K is the constant indegree distribution. The resulting phase diagram is shown in Figure 4, where the shaded region shows the parameter space in which the associated RBN is highly ordered, the unshaded region indicates the parameter space in which the associated networks are chaotic, and the curve marks the phase transition between them—the edge of chaos (Derrida and Pomeau 1986).[19] In other analyses, it has been shown that the negative correlation between connectivity and stability in these RBNs is due to an average loss of structure in the system’s state space and is consistent across multiple characterizations of stability and robustness.[20] Of note in Figure 4, the range of values for the connectivity constant
K leading to ordered networks is quite small; such a range does not account for the range of connectivities observed in real genetic networks (Aldana 2003).
Figure 4: Phase diagram for RBNs with constant indegree K
A discrete distribution producing Boolean networks with interesting dynamic properties and a much wider range of parameters admitting stability is the so-called power law or scale-free distribution. A power law network is one in which the probability that an arbitrary element of the network contributes to the control of exactly
k other elements is proportional to
k-γ, where
γ is referred to as the scale-free exponent or scale-free parameter.[21] Such a distribution is characterized by a heavy tail, in a manner of speaking; that is, phenomena described by power laws have a relatively high incidence of extreme behavior.[22] In this case, a network with connections described by a power law would show a large number of nodes with few connections and a few nodes with a large number of connections, or, in the parlance of traditional network analysis, a large number of nodes with low centrality and a few with high centrality. Power laws describe a wide range of network phenomenon, including the interconnectivity of citations in scientific papers (Lotka 1926 and Price 1965),[23] protein interaction networks (Maslov 2002), networks of email correspondence (Ebel 2002), and Internet topologies (Faloutsos 1999).
Figure 5: Phase diagram for RBN with power law outdegree and scale-free parameter γ
As with the RBNs studied by Kauffman, the dynamical behavior under small perturbations of an RBN will depend principally on the parameters defining the network,
γ and
p. The perturbations considered here are those induced by randomly selecting one variable in the system and changing the current value of that state, as in Kauffman’s network analysis. These perturbations might be conceived of as exogenous influences on the closed system described by the RBN. A relationship between robustness of the network and the network parameters similar to that described in Figure 4 can be derived for these scale-free RBNs, as shown in Figure 5. The curve describing the phase transition between ordered and chaotic regions in such a network is defined by the relation
where
is the Riemann Zeta function (Aldana 2003, 51). As in Figure 4, in the shaded region the RBN is stable and ordered and in the unshaded region the RBN is chaotic. Networks with connectivity characteristics placing them in this zone that are subject to perturbation from exogenous influences have a tendency to change their network-level phenotype with relatively high frequency. That is, the system is—inherently and structurally—unstable and unpredictable.[24]
Problematizing the Pax Economica[25]
That the economic interdependence of states—what we now call economic globalization—encourages pacific interaction among those states is not a new theory. In 1748, the philosopher Charles-Louis de Secondat, Baron de Montesquieu wrote in
The Spirit of Laws, “Commerce is a cure for the most destructive prejudices; for it is almost a general rule that wherever we find agreeable manners, there commerce flourishes; and that wherever there is commerce, there we meet with agreeable manners … Peace is the natural effect of trade. Two nations who traffic with each other become reciprocally dependent; for if one has an interest in buying, the other has an interest in selling: and thus their union is founded on their mutual necessities.” (Montesquieu 2001, 346-7) Writing 150 years later, this sentiment was famously echoed by Norman Angell in the years preceding the First World War.
In The Great Illusion, he argued that economic interdependence between states would render military confrontation counterproductive and therefore any doctrine that saw nations as naturally competing units and the international system as one in which “advantage, in the last resort, goes to the possessor of preponderant military force, the weaker going to the wall” (1913, ix) was a
stage of development out of which we have passed; that commerce and industry of a people no longer depend upon the expansion of its political frontiers; that a nation’s political and economic frontiers do not now necessarily coincide; that military power is socially and economically futile, and can have no relation to the prosperity of the people exercising it; that it is impossible for one nation to seize by force the wealth or trade of another—to enrich itself by subjugating, or imposing its will by force on another; that, in short, war, even when victorious, can no longer achieve those aims for which peoples strive. (1913, x)
In other words, he predicted the “diminishing role of physical force in all spheres of human activity” as a result of the economies and interests of states being intertwined (1913, xii-xiii).
The assertions of Montesquieu and Angell have been called into question repeatedly, of course, not least by the earth-shaking events of 1756-1763 (The Seven Years War), 1803-1815 (The Napoleonic Wars), 1914-1918 (The First World War), 1939-1945 (The Second World War), etc., yet belief in the pacific effects of interdependence has not waned. For example, Thomas Friedman, in his best-selling book
The World Is Flat, argues for a concept he calls the Dell Theory of Conflict Prevention (2007, 580-95). An extension of his previously-articulated Golden Arches Theory—that as countries are woven together by global trade and rising standards of living (symbolized by the spread of McDonald’s), the cost of war would become prohibitive for both the winners and the losers—Friedman’s Dell Theory postulates that “no two countries that are part of a major global supply chain, like Dell’s, will ever fight a war against each other as long as they are both part of the same global supply chain.” (2007, 585) In the face of previous claims—such as those given voice by Montesquieu and Angell—the implicit contention by Friedman is that in previous ages the world was simply not interdependent enough to realize the pacific benefits of this interdependence.
Embedded in the ideas of Montesquieu, Angell, and especially Friedman is a demand for measures of economic interdependence that can then be turned to a search for—if their arguments are correct—negative correlations between interdependence and the incidence of militarized conflict. In this vein, scholars have applied statistical methods to illuminate the relationship between economic interdependence and warfare. The results of these investigations are diverse. Oneal and Russett, for example, find that “higher levels of economically important trade, as indicated by the bilateral trade-to-GDP ratio, are associated with lower incidences of militarized interstate disputes and war, even controlling for potentially confounding, theoretically interesting influences: geographic contiguity, the balance of power, alliance bonds, and economic growth rates.” (1997, 288) On the other hand, Barbieri has argued, “Rather than inhibiting conflict, extensive economic interdependence increases the likelihood that dyads will engage in militarized interstate disputes.” (1996, 29) There are, of course, legitimate reasons one might come to varying conclusions regarding the correlation between economic interdependence and conflict—divergence in cases considered, approaches to operationalizing measures of interdependence, consideration of confounding factors, etc. We will demonstrate that these varying conclusions and the general indeterminacy of the question may perhaps derive from something more structurally fundamental than these intervening factors. We hope to show that
the topology of the network itself may have an independent, critical intervening effect on trade-based assessments of international conflict or cooperation.
As already noted, analyses in this area—like those of Oneal, Russett, and Barbieri—are generally dominated by regression methods associating dyadic interdependence to conflict. Here, we offer an alternative methodological approach. While the networks we examine are constructed via examination of dyadic relationships, the analysis and relevant questions are based on the totality of the topological structure of the interdependent system thus created. Specifically, we develop a characterization of the international system using results from the theory of complex systems and RBNs. This characterization leads to a hypothesis problematizing the normative concept of increasing incidence of cooperation (or competition) induced via economic interdependence and suggests that increasing interdependence may, in fact, lead to systems that are inherently unpredictable and, in a sense, chaotic and unstable in their behavior. Thus, it may be fundamentally impossible to claim with any certainty that interdependence leads to any particular systemic behavior—it is possible that none (or all) of Oneal, Russett, and Barbieri are correct.
Economic Interdependence and RBNs
We now construct a series of power law networks based on a characterization of the economic interdependence of countries in the international system. This construction extends and refines the work by Maoz mentioned above. The present work will depart from Maoz in one significant and a number of minor methodological points. Maoz is primarily concerned with the
formation of networks in international relations, positing two causal logics (homophily and preferential attachment) and then comparing the networks formed via those logics to empirical data (2011, 1-4). Our analysis, on the other hand, begins with the empirical data and confirms Maoz’s findings that trade networks form power law networks (2011, 19-23), but we extend this to a discussion of the consequent dynamics of those networks when viewed as RBNs and the associated impacts on cooperation and competition in the international system. Methodologically speaking, our approach to creating a real-world network of trade relationships, constructing statistical fits for the resulting degree distribution, and statistically testing those distributions differ from and provide a useful analytical complement to the techniques employed by Maoz (2011, 15-8 and 27-8).
To construct a trade network, consider directed dyads of nations.[26] Given two nations, express the economic dependence of the
ith nation on the
jth as a percentage of the
ith nation’s total trade (imports and exports) accounted for by dyadic trade between the
ith and
jth nations (imports and exports). This indicator for the dependence of the
ith nation on the
jth is expressed by the formula
where
Dij is the total trade volume between the
ith nation and the
jth,
Ti is the total trade volume for the
ith nation, and
Sij is the proportion of the
ith nation’s trade encompassed by the
jth nation. For example, if in 1965 the United States had a total trade volume (imports and exports) of $50,285.7M and $11,410.7M of this was comprised by trade with Canada, then the share of total United States trade provided by Canada was 0.23. Conversely, if Canada’s total trade volume (imports and exports) in 1965 was $17575.6M, then the share of total Canadian trade provided by the United States was 0.65.[27]
To construct a network from the trade share data described above is a simple matter. Define each nation in the international system as a node, and consider two nations
i and
j to be adjacent (connected) in the network if the trade share describing the
ith nation’s dependence on the
jth nation meets some threshold. Each such adjacency results in a directed edge as illustrated in Figure 1; that is, if the
ith nation is economically dependent on the
jth nation, then the
jth in some sense controls or at least contributes to the behavior of the
ith. It is possible, of course, for two nations to have a mutually dependent relationship, a situation resulting in a network relationship as illustrated between
σ1 and
σ4 in Figure 1. The threshold for defining dependency is problematic, of course. It has been observed that operationalizing economic interdependence as a factor in international relations is extraordinarily complicated, not least since there is no clear consensus on exactly what constitutes such interdependence (Barbieri 1996). There is a further complication observed by Maoz and shared in our analysis; the attachment model is binary and cannot readily describe the behavior of signed or valued networks (Maoz 2012).[28] In the following analysis and for illustrative purposes, the threshold is set at 0.01 (i.e., the
ith nation depends on the
jth if one percent of the
ith nation’s trade involves the
jth).[29] Note that the dependence relationship described implies neither subservience nor subordination; it simply identifies those relationships in which two nations are, at some level, economically linked and in which the behavior of one nation is in some way dependent on the behavior of another.[30]
An example network describing the economic interdependence relationships in the international system for the year 1965 and based on the criteria described above is given in Figure 6. In this case, there are 121 nodes (representing nations) in the network, each connected to one or more additional nodes by edges directed either outward (i.e., that nation exerts proximate influence over the behavior of another) or inward (i.e., that nation is directly influenced by another). The seven nations with the greatest connectivity (the United Kingdom, West Germany, the United States, Italy, France, the Netherlands, and Japan) are indicated in the diagram, as are the five nations with the least connectivity (Mongolia, Luxembourg, North Korea, Albania, and Nepal). For clarity in the diagram, no distinction is made between edges directed inward and outward.[31]
Figure 6: Economic interdependence network diagram for 1965
Note that no equivalent to Figure 3 will be developed for this network. With 121 nodes, the network can assume a total of 2
121 possible network states—greater than the number of stars in the observable universe.[32] Also of note is that the described network does not include all of the United Nations-recognized members of the international community. Based on the criteria described above, many are economically independent from the network shown; their states are strictly exogenously determined relative to the closed system of the RBN shown.[33] Since the only effect of interest here is the influence of economic interdependence, those isolated nodes are constant with respect to this factor.[34]
The adjacency relationships discussed above define a network structure. To complete the characterization of a Boolean network requires only a description of the dynamical state of the nodes and the network system. That is, a characterization of the Boolean state variables and the associated transition functions is required. For example, if the Boolean state variables describing each node in the network are assumed to represent a posture that is, as Adam Smith might have described it, cooperation-dominant or competition-dominant (or some other relevant and empirically appropriate referent), the needs of the Boolean network description are satisfied.[35]
Regarding the characterization of the transition functions
fi that define the dynamic behavior of the network (i.e., the transition of states from cooperation-dominant to competition-dominant and vice versa) based on its current network state, it is possible to carry the analytic analogy too far. One might, for example, attempt to derive carefully prescribed, empirically or theoretically-derived transition functions from the social and political character of the individual nations. Are the nations represented by nodes in the Boolean network liberal democracies, anocracies, or autocracies? Should their relative behavior be captured in transition functions derived from theory and be governed by the so-called Kantian Peace championed by Doyle (1983a and 1983b), Oneal and Russet (1997), and others? Or should the transitions be governed in some sense by balancing behavior as championed by Waltz (1979), Gilpin (1981), and others? Is it important to consider the influence of institutional factors as described by Keohane (1984)? Even this cursory enumeration demonstrates that complications abound. Fortunately, as interesting as such an analytic derivation might be in general and as informative as it might be to a robust theoretical examination of economic interdependence, it is in a sense irrelevant to the development of the Boolean network described here. In the end, the primary point of interest in terms of the functional behavior of the network is the achieved distribution of outcomes for the dependent variables in the transition functions. The behavior of the network is governed by the parameter
p. Why this parameter takes on a given value is a subject beyond the scope of this discussion, though it may constitute a fruitful area for future theoretical and empirical work.
Having derived an RBN from the international system with a structure determined by economic interdependence among the nations in the system, we now turn to characterizing the structure of that network and the influence of that structure on network behavior. In particular, it is important to determine if the distribution of outdegree across the nodes follows a power law behavior.
Recall that the outdegree
k is said to follow a power law distribution if the probability that an arbitrary element of the network contributes to the control of exactly
k other elements is proportional to
k-γ. More formally, we say that
k is drawn from a power law distribution if the probability density function is of the form
where we will use a continuous distribution to approximate the behavior of the corresponding discrete distribution for convenience. The task before us is to compute the parameter
γ from the network data developed above; Figure 7 is a histogram showing the observed frequencies of outdegree, from which data
γ will be computed.
Figure 7: Observed outdegree frequency histogram and fitted power law distribution
Several methods, each with strengths and weaknesses, are available.[36] The approach taken here first exploits an interesting property of the power law distribution: the associated cumulative distribution function is itself a power law distribution, as is easily demonstrated. Recall that the cumulative distribution function associated with a distribution defined by
p(
k) can be expressed as the probability that the outdegree of an arbitrary node is greater than or equal to
k (Newman 2005, 326):
Computationally, data in this form are somewhat easier to manage than the data associated with the original probability density function, primarily because the cumulative data set is non-zero throughout the domain of interest and because our test statistic operates on the cumulative distribution.
Figure 8: Dynamic behavior of the RBN economic interdependence network for 1965
Returning to the problem of fitting a power law distribution to the outdegree connectivity of the economic interdependence network shown in Figure 6, the scale-free parameter for the outdegree was calculated using nonlinear regression techniques applied to the cumulative distribution data, yielding
γ = 1.723 (Brown 2001). Notice, as shown in Figure 8, that this value for the scale-free parameter places the resulting RBN well within the chaotic phase. Further, this placement is essentially independent of
p, provided 0 <
p < 1 (i.e., provided the behavior of the network is neither uniformly cooperation-dominant nor uniformly competition-dominant).[37] That is, the value for
γ is such that the Boolean network describing economic interdependence in the international community for 1965 is chaotic in the sense described by Kauffman irrespective of the value assumed by
p.
The calculated distribution for outdegree connectivity (derived from the empirical data) associated with the scale-free parameter for the economic interdependence network for 1965 is also shown in Figure 7 (as a solid curve). Note that the curve appears to agree quite well with the observed data. More precisely, one can test for statistical agreement between an empirical data set and a hypothesized distribution. For non-normal data, the Kolmogorov-Smirnov statistic, given by the maximum absolute difference between the observed and hypothesized cumulative distributions, is one powerful option for assessing the agreement between the distributions (Marsaglia 2003). In this case, the hypothesized distribution and the empirical data agree statistically (requiring a level of significance
α = 0.586 to reject the hypothesis that the empirical data is drawn from the hypothesized distribution).[38] So, it is reasonable to describe the international system in 1965 as chaotic (in the sense described for RBNs) with respect to the influence of economic interdependence in the sense described by Kauffman for RBNs. As noted above, this implies a fundamental instability in the system derived entirely from the network’s structure. That is, the network described is sensitive to small perturbations in the network state of individual nations (cooperation or competition dominance). This sensitivity implies that changes in the posture of individual nations have a relatively high likelihood of inducing significant changes in network behavior (e.g., changing basins of attraction for the network) perhaps implying a change in the international system modeled (i.e., the phenotype of the international system).
The foregoing analysis characterized an RBN based on economic interdependence in the international system for the year 1965. The process described was repeated in each year from 1965 to 2006, and the results are given in Figure 9. The top curve in Figure 9 shows the calculated value (based on nonlinear least squares regression) of the scale-free parameter for the outdegree distribution in each year from 1965 to 2006. Note that this curve is (almost) monotonically decreasing. That is, the network is becoming more connected as a function of time indicating increasing economic interdependence in terms of trade share and based on the criteria established above.[39] In terms of the induced behavior in the associated RBNs, however, increasing interdependence and decreasing values for the scale-free parameter
γ indicate movement deeper into the chaotic region of the phase diagram. In other words, economic interdependence leads to network topologies associated with less—not more—stable behavior.
Figure 9: Scale-free parameters and goodness of fit, 1965-2006
The second curve in Figure 9 shows the statistical significance for the fit of a scale-free distribution to the empirical data in each year from 1965 to 2006. The shaded box indicates those values satisfying a level of significance of
α = 0.10; that is, for those years in which the test statistic falls in the shaded region one can reject the hypothesis that the empirical data are drawn from a power law distribution with a reasonable degree of confidence. With the exception of the years from 1981-1984 and the years from 1992-2006, the data agree statistically with the given scale-free distribution.[40]
This raises the question of the manner in which the empirical data deviate from the given scale-free distribution in the periods 1981-1984 and 1992-2006. In each of these nineteen years, the test statistic—the maximum absolute difference between the hypothesized and empirical cumulative distributions—is drawn from the portion of the distribution comprised of those nodes/nations with low outdegree. Further, in every one of these nineteen years, the hypothesized distribution predicts more nodes with small outdegree than are observed in the empirical distribution. That is,
the empirical data describes a network with a greater degree of connectivity than predicted by the model.
The data show that the shift in the early 1990s is a direct result of international reorganization due to the entry of new nations into the international system that accompanied the demise of the Soviet Union (e.g., Belarus, Latvia, Moldova, Ukraine, Uzbekistan, etc.), the fracturing of Yugoslavia (Bosnia and Herzegovina, Croatia, Macedonia, etc.), the Velvet Revolution in Czechoslovakia (Czech Republic and Slovakia), etc. When these new nations entered the system, they tended to appear with well-established economic connectivity to other nations in the system and to each other.[41] The transition involving the Soviet Union (as an isolated portion of the network) is illustrated in Figure 10. In the data for 1991, the Soviet Union comprised a single node (represented by the box or closed system shown). In the following year, however, that single node became fourteen distinct nodes, each with a total degree (indegree and outdegree) between two and seventeen (considering only connections with the other nodes generated from the Soviet node); half of these nodes have a total degree greater than ten in 1992. Note that Estonia is not connected to any of the other fourteen nodes shown; Estonia is, however, connected to many nations outside this group. The addition of these new nodes has a similar effect on the connections which formerly involved only the Soviet Union.[42] This phenomenon biases the data in the direction of increased connectivity.
Figure 10: Economic Interdependencies in the Former Soviet Union, 1992
This leads to two observations. First, if the low-degree elements in the cumulative distribution are ignored, the statistical agreement is significant in all but six of the years examined.[43] Further, in each of these six years, (1992, 1997, 1999, and 2001-2003), we would fail to reject the null hypothesis that the empirical data were drawn from a scale-free distribution if we were to make the rejection criteria only slightly more stringent (from
α = 0.10 to
α = 0.05). Second, in every case discussed above, the tendencies contributing to failures in the fit to a power law distribution were driven by increased connectivity. As noted earlier, in work covering both non-power law and power law networks it is increasing connectivity that causes loss of structure in the examined networks and correlates with decreasing order in the dynamic behavior of the associated RBNs.
Agent-Structure Considerations
Random Boolean Network analysis may prove quite useful in facilitating greater fidelity or predictive power in rigorous empirical studies, but it may also add deeper theoretical insight in qualitative debates as well. For instance, in the ongoing constructivist challenge to long-standing models, the actions of agents are understood as both structurally enabled and constrained by the socio-political and environmental structures within which they operate; in turn the structure is shaped and modified by complex feedback mechanisms of the agents,
ad infinitum. This has tended to move academic study toward how networks form and change, along with the parallel changes and growth in agents, as opposed to traditional emphases on the primacy or relative influence of one or the other. Nonetheless, faced with an impasse in furthering the agendas of either position, Roxanne Doty (1997) and Colin Wight (1999) called for movement beyond parochial borders and a fresh look at methodologies from other sciences for heuristic assistance. We assert the introduction to RBN analysis is in part an answer to this increasingly vocal call.
The argument between agent and structure stems from the Thomas Carlyle’s nineteenth century query “does man make history or does history make the man?” In this formula, the answer is one or the other, requiring a Manichean response. Of course, we are all constructivists now, and the correct answer has always been both. Even Aristotle recognized this, though his terminology differed; this realization is the essence of his response to the question of whether it is better to have good men or good laws.[44] But in theory we can stretch the limits of reality and abstract out confounding variables. This allows us to determine
primacy of place or effort, and for politicians this is a vital point: where does one put one’s limited resources in a world in which agents and structures interact? Where can one maximize one’s return if an investment can be made only in structures or agents?
Carlyle insisted, “The great man, with his free force direct out of God’s own hand, is the lightning … In all epochs of the world’s history, we shall find the Great Man to have been the indispensable savior of his epoch; [The] History of the World, I said already, was the Biography of Great Men.” (1840) The place of biography was thus venerated and touted as the proper role of the historian, and the preferred historical method was therefore to seek out those individuals who instituted positive sweeping change and detail their lives as a guide to proper behavior. To be an Alexander one must emulate Alexander.
In opposition to Carlyle, sociologist Herbert Spencer argued that time tends to swallow up individual achievements and that the Great Men of history were simply the inevitable outputs of their age. His emphasis on natural selection as the mechanism for adaptive change moved Spencer to pen a sweeping history of social evolution.[45] Similarly, Karl Marx, who saw history as a Hegelian clash between the controllers of economic production and those who labored to support, and, ultimately, to overthrow them, gave primacy of meaningful change to structure. In his classic “Eighteenth Brumaire,” Marx insisted that “Men make their own history, but they do not make it just as they please; [they simply] performed the task of their time.” (1852, 595)
Such views paralleled those of the emerging sciences of economics, biology, and sociology—individual agency was not instrumental, as historians insisted; rather, collective action fashioned structures that, in turn, molded the behavior of individuals. Adam Smith, for example, saw the structure of markets shaping individual preferences such that through their myriad interactions an invisible hand provided the greatest good for the greatest number in a way that no individual master or central authority could equal (Smith 1982).[46] The most critical observation at this point in our theoretical narrative is that the distinction between agent and structure had extraordinary epistemological ramifications. If Great Men were not the nexus of history, then where was the point in studying the details of the past?
Of course, such bald differentiation was not pressed to theoretical extremes. Although advocates of one side or the other remain,[47] the sophisticated position has insisted not on determining the absolute primacy of one or the other but rather on describing the relative or even changing influence of both. It took little reflection to recognize that a balance between agent and structure was far more explanatory of real world behavior.
Thus accommodation became the critical focus, finding early articulation in Emile Durkheim’s 1893 opus,
The Division of Labour. In this seminal work, Durkheim “isolates a subtle connection between individuation and individualism, a shifting polarity between ‘man in particular’ and ‘man in general.’ The individual within the small community [is] unconscious of the general attributes which, as a man, he shares with the whole of humanity.” (Giddens 1972, 9) Durkheim identified not only the essential reciprocal nature of agent and structure, but also the varying levels from animal instinct (or “irrational passions”) to rational choice, familial and social obligation and constraint, and the broader
conscience collective. He also clearly identified the properties of social interaction as emergent characteristics “created by the organization of individuals in society.” (Giddens 1972, 43; see also Giddens 1970) In a sense, it is these properties or
emergent characteristics that we extrapolate in the present essay as the influence of the topology of the social network.
Durkheim was extraordinarily influential in the development of Anthony Giddens’ theory of
structuration, to date the most sophisticated attempt to reconcile the “escape of human history from human intentions,” (Giddens 1972, 7) which he ascribes to “various forms of Marxist functionalism” and a situating of human “action in
time and space as a continuous flow of conduct.” (Giddens1979, 3) Central to his theory is “the
duality of structure” by which he means “the essential recursiveness of social life, as constituted in social practices: structure is both medium and outcome of the reproduction of practices.” (Giddens 1979, 5)[48]
While the issue of one
or the other is effectively over, the debate over
primacy continues. In the modern era, especially in the context of international relations theory, the state has been the predominant social organization and unit of analysis, a point that many current international theorists have been vigorously insisting upon or apologizing for.[49] With Kenneth Waltz’s (1959) critique of the levels of analysis problem in
Man, the State, and War, the emphasis shifted to the relative positions of a hierarchy of agents operating within a layer of structures. Each level has actors and structures that constitute the basis for analysis, and while all have interdependent effects Waltz found the heavily structural argument of the third image the most persuasive. Although the book became profoundly influential, David Singer (1961) crystallized its importance in an influential article two years later. At stake was the determination of a research model that would generate outcomes—predictions—based on the level of analysis preferred. “[T]he problem is really not one of deciding which level is most valuable to the discipline as a whole and then demanding that it be adhered to from now unto eternity. Rather, it is one of realizing that there is this preliminary conceptual issue and that it must be temporarily resolved prior to any given research undertaking.” (Singer 1961, 91)
Taking up the issue for a new generation, Alexander Wendt (1987) and David Dessler (1989) kicked off another wave of theorizing. Wendt adapted the structurationist approach to deny ontological priority to both agents and structure, but argued for more fidelity in determining the “properties and causal powers of their primary units of analysis.” (1987, 337) For agents this comprised elements of preference formation—choice, reflexivity, and learning. For structures it included more than just military and economic power calculations; the roles of cultural and social values, norms, and institutions were also included. The central theme is the place of meaningful change in the system, how it occurs, and why. Agency allows for change through action but says little about preference formation or the role of culture, society, and law on determining preferences. Structure accounts for the exogenous factors determining preference but is theoretically immune to meaningful transformation.
When causal factors flow from both agent and structure and are mutually constitutive the essential problem becomes determining the proper theoretical emphasis. Wendt saw realization of one or the other as giving weight to neorealist or neoliberal theoretical constructs, and resolution would elevate one paradigm to Kuhnian domination while relegating the other to a merely interesting explanatory enhancement. (1987) Dessler pushed past the ontological arguments and looked directly at the epistemological importance of establishing a scientific research program to empirically derive successful explanations of observable phenomena. His Neopositivist method, which he called scientific realism, sought “a transformational approach [that] can draw explicit links between structural and unit-level theories” and would cut across levels of analysis to determine “manifestations of rule-following behavior … aspects of a common underlying ontology based on the concept of social rule … [and] a promising basis for constructing explanations of peaceful change.” (1989, 471-2) In this manner, structural determinism need not lead to dismal fatalism or agent-centered theories to reliance upon acts of heroic universalism. But isolating the influences that predominate in national relationships is critical to sound theory and even more important to extrapolating from theory real-world policies.
Conclusion
Our intent in this essay is not necessarily to refute or affirm various claims of increasing cooperation or conflict via trade relationships in the international system—the example was selected for its utility in explicating the method. Neither is it to solve or prioritize the roles of agent or structure in an ongoing debate. Our primary purpose is simply to introduce into the toolkit of international relations scholarship a method to illuminate
an aspect of network dynamics and structure heretofore not seen in the literature. We submit that the role of Boolean networks, extremely heuristic in parallel analyses in the social and physical sciences, can and should be more fully explored as a model to illuminate the international system. Indeed, following the logic of complexity theory and consistent with studies of emergent behavior, a better understanding of the topology of these networks in international relations may uncover a class of significant and independent causal effects that fundamentally alter our understanding of empirical observations of the international system. The most substantial contribution in this regard is to advance the state of the art as desired by Dressler, Doty, Wight, Maoz, and others. For the sample case presented in this analysis, the topology of interdependent international systems with sufficiently dense interconnectivities may lead to situations in which small perturbations of the system exert disproportionate—and unpredictable—effects. As the sociologist and complex system theorist John Urry writes, “Up to a point, tightening the connections between elements in the system will increase efficiency when everything works smoothly. But, if one small item goes wrong, then that can have a catastrophic knock-on effect throughout the system. The system literally switches over, from smooth functioning to interactively complex disaster.” (2003, 35) Critically, these effects are induced
at the network level and not at the traditional analytic level of the individual dyads, and the new methods presented here offer an entrée into entirely new approaches in international relations. Through the lens of random Boolean networks, such behaviors become clearer and lose the stigma of outliers and pathologies. This presents us with a new path for analytic study and new possibilities for the formulation of hypotheses regarding the international system. Further, the injection of an intermediary into the agent-structure debate, one that explicitly mediates the reciprocal influence described so eloquently by Giddens, may allow for stronger empirical analysis of the relationship than heretofore has occurred and may offer analogies rich enough to provide leverage for theoretical exploration as well.
The part of this year that raises the greatest distress: I've read some...I've thought a bit...and I've written nothing. (To be fair, some of the compensations in traveling Europe have been lovely.) Perhaps I had nothing to say...but in reading the things some think they have to say, this may be doing myself a disservice. Perhaps I focused more on others than on me...and that may be a good thing. Perhaps I lack the time...but this is the excuse too easy to offer. Perhaps I lack the discipline...and this is my new fear. The latter issues are to one degree or another within my control, however.
