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# The Spread of Conflict in International Relations

## Summary and Keywords

International relations as a subfield in political science has always been fundamentally concerned about the relations between actors and how they lead to conflictual or cooperative outcomes. However, despite this inherent interest in relations between actors, the gap between theoretical conceptualization and empirical estimation considerably widened until spatial econometric and network analytic estimation approaches allowed researchers to address interdependencies in multiactor settings empirically. However, the discipline needs to strengthen the link between theoretical and empirical network analysis by integrating formal theoretical advances and fully embracing inferential statistical network approaches that are available to researchers. Formal theories of network behavior need to be further developed to establish systematic insights into the conditions under which complex network structures arise and how they affect actor behavior. Rigorous theoretically guided research will form the basis of linking network theories of conflict and cooperation and empirical testing.

# Introduction

International relations as a subfield in political science has always been fundamentally concerned about the relations between actors and how they lead to conflictual or cooperative outcomes (Waltz, 1979; Keohane, 1984; Wendt, 1992; Lake & Powell, 1999; Kydd & Walter, 2006; Walter, 2009; Hafner-Burton, Kahler, & Montgomery, 2009; Maoz, 2010; Ward, Stovel, & Sacks, 2011; Dorussen, Gartzke, & Westerwinter, 2016). However, despite this inherent interest in relations between actors, the gap between theoretical conceptualization and empirical estimation considerably widened until spatial econometric (Franzese & Hays, 2008; Gleditsch & Ward, 2008; Neumayer & Plümper, 2010) and network analytic (Hoff & Ward, 2004; Snijders, 2001) estimation approaches allowed researchers to address interdependencies in multiactor settings empirically. In 2009, Hafner-Burton, Kahler, & Montgomery proposed a three-part agenda for future application of network analysis in international relations by (a) applying the empirical network analysis to international relations, (b) testing existing network theories of international relations using these tools, and (c) test existing international relations theories with empirical network analysis tools. This call was inspired by early network approaches in international relations and conflict studies (Most & Starr, 1980; Starr & Most, 1983; Kirby & Ward, 1987; Faber, 1987; Montgomery, 2005; Brams, Mutlu, & Ramirez, 2006; Ward, Siverson, & Cao, 2007; Dorussen & Ward, 2008) that provided an initial glimpse at the opportunities that network analysis could bring to the field. Since then, empirical and theoretical network approaches to international relations and conflict studies have gained wide acceptance in the discipline. Network analytic advances have allowed researchers to address not only problems of international conflict and cooperation (Kinne, 2013; Ward et al., 2013) but also transnational (Böhmelt, 2009; Desmarais & Cranmer, 2013; Murdie, 2014), intrastate (Siegel, 2011; Metternich et al., 2013), and intraorganizational (Larson, 2016; Perliger & Pedahzur, 2011) patterns of peace and conflict.

However, the discipline needs to strengthen the link between theoretical and empirical network analysis by integrating formal theoretical advances and fully embracing inferential statistical network approaches that are available to researchers (Granato & Scioli, 2004; Clarke & Primo, 2007; Aldrich, Alt, & Lupia, 2008). An interesting aspect of current research is that empirical network advances have been applied to international relations much earlier than network theoretical insights. Theoretical insights tested on dyadic data have been quickly extended to multiactors settings without challenging whether the assumptions behind dyadic theories can travel to seamlessly multiactor contexts. This is problematic because international relations and conflict studies have been a successful discipline precisely because of its strong focus on linking theoretical and empirical insights (e.g., escalation patterns in two-player settings [Ward, 1984]). On the empirical end, the challenge exists that multiactor theories imply a violation of many assumptions made in standard empirical approaches. Nonetheless, international relations scholars have been very happy to extend the empirical implications of dyadic theories to multiactors settings, without thoroughly investigating the empirical implications of n-player situations (Poast, 2010). In many cases this may be appropriate and insightful to use dyadic theories as heuristic to guide empirical research, but the acceptance of this approach has dampened the discipline’s appetite for truly multiactor theories (e.g., multilateral bargaining).

One way to address multiactor phenomena in international relations and conflict studies is to take a network approach (Hafner-Burton, Kahler, & Montgomery, 2009; Maoz, 2010; Ward, Stovel, & Sacks, 2011; Dorussen, Gartzke, & Westerwinter, 2016). A network theoretical perspective allows researchers to identify the conditions under which conflict and cooperation spreads in multiactor settings. It focuses our attention on the relational aspects of conflict and cooperation, which are at the very core of international relations and conflict studies. However, the main reason that a network perspective can foster theoretical advances is because it already provides a conceptual framework that facilitates concentration on all components that are relevant for the spread of conflict and cooperation (Metternich, Minhas, & Ward, 2017). This may be very obvious to network analysts, but it may be helpful to highlight the main components of a network for readers less familiar with the foundational concepts. On a very basic level, networks consist of actors (nodes) and their relations (edges). Actors in international relations applications can be, e.g., states, rebel organizations, or pro-government militias. Relations between these actors can be conflictual (e.g., fighting) but also cooperative (alliances, peace agreements). Importantly, networks of cooperation (e.g., alliances) can exist alongside or even induce networks of conflict (e.g., militarized inter-state disputes). However, networks are not only a collection of actors and their relations. Network theoretical approaches argue that the overall structure of the network impacts actor behavior (e.g., certain alliance structures may be more or less prone to free-riding) and that actors seek to form networks that maximize favorable outcomes (e.g., establishing trade treaties that maximize economic prosperity).

To leverage the network perspective, both, formal theoretical work on networks and empirical estimation strategies need to be further developed in international relations with a special focus on the linkage between theory and empirics. Formal theories of network behavior are a promising approach to gain rigorous and systematic insights to the conditions under which complex network structures form and the effects these structures have on actor behavior. Rigorous theoretical guidance will allow effective linking of network theories of conflict and cooperation and empirical testing.

This article first alludes to formal network theories, primarily developed in economics, that have clear implications for international relations and conflict studies applications. A short overview of empirical network approaches is presented, most of which have already have been applied in international relations and conflict studies. Turning to existing network studies in this field, current and future ways of combining theoretical and empirical network approaches are highlighted. Finally, a brief framework of truly network theoretical contributions is introduced that can help to specify empirical implications.

# Formal Theories of Networks

Formal network models are interested in (a) the choices actors make forming networks and (b) the behavior of actors given a certain network structures. Actors are assumed to maximize their utility by forming particular linkages to other actors or through actions on existing network structures. The promise of a formal approach is that one can discover regular patterns of network formation and behavior on networks despite the complex interactions that arise in a multiactor strategic environment. It is the same promise that, for example, bargaining theory fulfilled for dyadic behavior on the international and domestic level (Fearon, 1995; Powell, 2002; Walter, 2009). This incentive is important to the study of formal network games because many insights from dyadic settings are very difficult to generalize to n-player situations. Especially, equilibrium outcomes in non-cooperative bargaining games, which have been at the forefront of formal theory within the field of international relations and conflict studies, become sensitive to specific game forms in multiplayer settings. Herrero (1985) demonstrates how in infinite repeated play (Rubinstein bargaining) with several players multiple equilibria outcomes are sustainable, which may include inefficient outcomes where large parts of the contested outcome is destroyed. This is a fundamental challenge for international relations and conflict studies, because many of our theoretical advances depend on dyadic formal games. It is therefore necessary to take a closer look at existing formal network games and to identify classes of games that can be adopted or reinterpreted in an international relations context. This article highlights some advances in the formal network literature, and while only covering very few classes of games, it demonstrates potential value for international relations and conflict studies to pay further attention to this literature.

## Network Formation

Implicitly, network formation is at the very core of international relations and conflict studies (Dorff & Ward, 2013; Gallop, 2016). For example, when states (Morrow, 1991) or rebel organizations (Cunningham, Bakke, & Seymour, 2012; Bapat & Bond, 2012; Christia, 2012) are thinking about which alliances to form or join, they are engaging in a network formation process (Warren, 2010). Similarly, decisions about joining trade agreements or with whom to form international organizations are fundamentally about building complex systems of interactions that take the shape of a network. More generally, these situations can be considered n-player games where the nodes in a network N = {1, …, n} are the respective actors in the network (e.g., states, rebel organizations, or individuals). In network formation games, actors create links, or edges in the network terminology, that maximize their utility. Hence, every actor i receives utility ui(g) from a potential network g. In most formal games, the formation process is not unilateral, and any actor j that i would like to form a link with has to agree to this relationship. While there may be important examples where actors are coerced into networks (e.g., being a former colonial state, forced to fight for a rebel organization), many important applications in international relations and conflict studies assume mutual acceptance of ties. Alliance formation and trade links are good examples where all actors in the relationship must agree to take part in the network. While actors may have slightly different incentives to join a network or abstain from it, most network formation processes in international relations and conflict studies leave actors some individual degree of choice.

Given that many potential networks (g) can be realized, the ways in which a set of networks are likely to form must be narrowed down. A good starting point is the Nash Equilibrium, but its limitations become obvious in a network setting (Jackson, 2010). Imagine a network formation game where actors announce simultaneously which ties they want to form. If actor i includes j in its announcement and j includes i, the link ij is realized. In a Nash Equilibrium of a network formation game, neither player has an incentive to unilaterally add or delete any number of actors from its announced list. But because the focus of the Nash equilibrium is on unilateral action, very unrealistic equilibria emerge. For example, if two actors would like to form a tie, there are two Nash equilibria. The obvious one is where both actors i and j announce to form link ij. Because they prefer ij over any other outcome, any unilateral deletion of actors on their linkage list would make them worse off. However, if actors i and j both announce not to form link ij, this is also a Nash Equilibrium because, again, a unilateral deviation from their announced list will not make them better off. Only if they both would choose to change their announced list would they be able to attain their preferred outcome, but this is beyond the concept of the Nash Equilibrium.

Based on this discussion, it becomes clear that equilibria in network formation games need to rely on pairwise deviation or stability rather than unilateral deviation or stability concepts. Jackson and Wolinsky (1996) introduce a fairly simple solution concept that conditions equilibrium outcomes on joint actions. They define a network as pairwise stable if

(1)

$Display mathematics$

(2)

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This conditional equation implies that a network is pairwise stable if no player wants to delete an edge and no two players want to add an additional link between them. This is an important departure from the Nash Equilibrium because the equilibrium definition now includes mutual consent. Of course, there are further refinements of pairwise stability (Jackson, 2010), but they all stress that mutual consent is an important part in forming network ties. When applying network formation games to international relations and conflict studies, it is important to consider a solution concept that reflects the theoretically implied causal mechanism. In an extreme case, e.g., empire networks, mutual consent of forming the network may not be relevant. It may be that few actors can enforce a certain structure, which needs to be reflected in the actors’ utility functions as well as the solution concept. Stability of networks will also depend on actors abilities to write binding contracts, which is especially important in the coalition formation literature (Chae & Yang, 1994; Ray, 2007; Hyndman & Ray, 2007; Tan & Wang, 2010).

## Network Behavior

Once the characteristics of the network are created, actors will condition their behavior on its structure (Bramoullé & Kranton, 2007; Galeotti et al., 2010). Two types of games have direct implications for international relations and conflict studies: (a) public good games, where individuals benefit from actors’ efforts they are connected with, and (b) bargaining games where the network conditions who actors can bargain with.

### Public Good Games

Public good games on networks focus on local public good provision, where the “local” is defined through the network structure (Jackson & Wolinsky, 1996; Bramoullé & Kranton, 2007; Galeotti et al., 2010; Bramoullé, Kranton, & D’Amours, 2014). In a sense, the nonexcludability and nonrivalry of public goods (Olson, 1965) is restricted through network connections. These models are interesting because they can explain variation in free-riding behavior among larger number of actors. The Cold War is a good example to illustrate this logic, because not everyone was able to free-ride on military efforts of the United States or the Soviet Union. In fact, only actors connected to one of the two main actors (e.g., United States) through alliances (e.g., NATO) were able to benefit from the respective military expenditures. Network-based public-good games can allow for precise equilibrium conditions in multiactor settings (Bramoullé, Kranton, & D’Amours, 2014).

Based on Bramoullé, Kranton, and D’Amours (2014), the intuition behind network games of public goods provision can be demonstrated. Consider a simple model with N states, which must make decisions about arming their respective military. Every state i invests a certain level of arming $y≥0$ to obtain their optimal level $y¯$ of security. However, some states will be able to benefit from the military build-up of other states. Whether i can benefit from j’s arming efforts will depend on whether they are connected in an N × N matrix W, where wij is the element in the ith row and jth column. Thus, if $wij=1$, i can benefit from j’s efforts directly. The extent to which i enjoys positive externalities from j’s efforts depends on a parameter $δ∈[0,1)$ which denotes the substitutability between i’s contribution and those of its connected states. The greater δ‎, the more i benefits from j’s arming efforts. The utility of an individual state can therefore be written as follows:

$Display mathematics$

Here, bi is a differentiable benefit function that is strictly increasing and concave in yi, i.e., $b(0)=0,b′=0$ and $b″<0$. Of course, arming is associated with some marginal unit costs $ci>0$, such that $b′(0)>ci>b′(+∞)$. Following Bramoullé, Kranton, and D’Amours (2014), the best response function for every state i in this game is:

$Display mathematics$

where $y¯$ denotes i’s optimal effort absent strategic interdependence.

The best response follows the simple intuition that state i will contribute the amount of effort $yi*$ that ensures its optimal level of military spending $y¯$ is provided taking the contributions of other states into account. If the military spending of other states is not sufficient to reach $y¯$, i makes up the difference $yi*$ and makes no contribution otherwise. Because states gain utility from the military spending of connected states, this can induce free-riding among alliances. The work of Bramoullé, Kranton, and D’Amours (2014) or Galeotti et al. (2010) is interesting because they can identify fairly precise network conditions under which particular equilibrium behavior should be observed. Hence, despite the complex network setting, these formal theories allow the formulation of empirical implications that can be tested.

### Bargaining Games

However, network bargaining depends very much on the game protocol that is implemented. In an article by Cai (2000), one player bargains with all other players sequentially. This means that network bargaining turns into a series of bilateral bargaining situations, but the other actors can observe and condition their behavior on previous bargaining outcomes. Abreu and Manea (2012) also turns network bargaining into a sequence of bilateral agreements, but there can be multiple sellers and buyers in the network. The network defines the potential sellers and buyer relationships and a link is drawn at each period from a probability distribution. Again, all other actors can observe the previous bargaining outcomes. Because in these kind of games actors alternate through all possible bargaining configurations sequentially, equilibrium outcomes depend on the patience of actors. This of course has direct implications for international relations and conflict studies, for example, trade negotiations may involve more patient actors than civil war negotiations and that should shape the empirically observable outcomes.

# Empirical Approaches to Interdependencies

Empirical network approaches have seen a tremendous development over the last years (Maoz et al., 2006; Hays, Kachi, & Franseze, Jr., 2010; Steinwand, 2011; Cranmer, Desmarais & Kirkland, 2012; Minhas, Hoff, & Ward, 2016b). But their success in the field of political science has not only been dependent on the methodological advances but also on their accessibility through readily available software (Hunter et al., 2008; Minhas, Hoff, & Ward, 2016a). While early work in international relations and conflict studies relied on descriptive network statistics, it has been increasingly replaced by statistical methods that either focus on the conditions under which networks form or the conditions under which actors demonstrate particular behavior on existing networks. The existing inferential statistical network models consider that individuals or dyads do not exist in a social vacuum, which challenges traditional independence assumptions between observations. Poast (2010) and Cranmer and Desmarais (2016) provide very good discussions of why it is necessary to move away from a purely dyadic analysis in contexts where strategic interdependence exists. While Poast (2010) provides a very specific k-adic solution for this problem, the following discussion focuses on two large classes of statistical estimators: those that focus on network formation processes and those that are more concerned with strategic behavior on existing networks.

## Network Formation

Network formation models have made great inroads into international relations and conflict studies (Cranmer, Desmarais, & Menninga, 2012). Most applications use either exponential random graph models (ERGMs) (Cranmer, Desmarais, & Menninga, 2012) or latent network approaches (Ward, Stovel, & Sacks, 2011). The main difference between the approaches is that ERGMs model a particular network as a realization from a distribution of potential networks. The latent network approaches on the other hand conceptualizes networks as a realization of individual choices drawn from a probability distribution. Latent network models can be seen as a type of random effect model, while ERGMs more explicitly model the network structure. Both approaches attempt to deal with interdependent observations, when standard assumptions of statistical models are violated. In addition, these approaches consider network features themselves as influencing the behavior of actors.

### Exponential Random Graph Models

Exponential random graph models (ERGMs) have been implemented in different areas of political science and represent one of the most used inferential network approaches (Cranmer, Desmarais, & Menninga, 2012; Corbetta, 2013). This approach allows researchers to test how specific network features are affecting the realization of observed networks. The idea behind ERGM is the modelling of joint probability density from which observed networks are drawn. The parameters are estimated by maximizing the probability of a particular observed network $y∈y$ over all other potential networks $z∈Y$. Hence the probability of observing a network y is conditional on parameters θ‎ and respective network measures g(.):

$Display mathematics$

This maximization is conditional on actor-specific and dyad-specific characteristics, but most importantly on endogenous network features (e.g., outdegree popularity). The advantage of the ERGM is that researchers can explicitly test the extent to which particular features of the network (e.g., outdegree popularity) are responsible for the network formation process. Thus, scholars can test different network theoretical implications against each other. Another advantage of using ERMGs is that they have been well adapted to deal with time-varying data implementing ERGMS (TERGMS; Hanneke, Fu, & Xing, 2010) or so-called stochastic-actor–oriented models (Snijders, 2001). The stochastic-actor–oriented models developed by Snijders et al. (e.g., Steglich, Snijders, & Pearson, 2010) have the advantage of modelling simultaneously evolving social networks, which are able to distinguish selection effects from social influence. This approach is also well implemented in the software RSienna (Ripley et al., 2016). The challenge when modeling ERGMs is that researchers need good knowledge about the inherent network dynamics because leaving out features that matter in the network formation process will lead to biased estimates (similar to omitted variable bias). Hence, all relevant network features need to be explicitly specified in the model. It follows that if the researcher has little knowledge about the true network generating process, misspecification will likely occur.

### Latent Network Models

A different approach is taken by latent space models (Hoff, Raftery, & Handcock, 2002; Hoff & Ward, 2004; Hoff, 2005). The latent space approaches can be seen as extensive random-effects models, where higher-dimension random effects capture network effects to establish conditional independence between observations (Minhas, Hoff, & Ward, 2016a). When a set of random effects is included in the model, dyads can be treated in as independent. However, the random effects should not only be treated as nuisance, because their structure allows for the identification of patterns within the network (Cao, 2012; Minhas, Hoff, & Ward, 2016a). Latent network approaches usually want to model a latent variable $θij$ that through a link function g is generating observed outcomes $yij(yij=g(θij))$.

The latent variable $θij$ is a function of modeled sender (Si), receiver (Rj), and dyad specific effects $Xij$. For example, in the context of conflict contagion (Buhaug & Gleditsch, 2008; Beardsley, 2011) it could be that conflicts in poor countries are more likely to spread to other countries (sender effect) or that dictatorships are more likely to observe spill-ins of conflicts (receiver effects) independent of dyadic effects (e.g., conflicts are more likely to spillover between dictatorships (structural equivalence)). In addition to these observable explanatory variables, the latent variable $θij$ is conditional on an error $eij$.

$Display mathematics$

Accounting for further network features the latent space approach decomposes the error term to account for unobserved factors that may be driving the network structure. These random effects can easily be linked to existing network concepts. Actor-specific network effects can be broken down into sender $si$ and receiver $rj$ effects. Many centrality measures, e.g., prestige, can be captured by actor specific effects, because there may be unobservable factors on the actor level that make it more or less likely that they will form relationships with other members of the network.

$Display mathematics$

However, dyadic effects, such as reciprocity, are inherently unobservable network features. These dyadic-level random effects are captured by a dyadic-specific random effect $εij$. Once actor and dyad specific random effects are accounted for, higher-order characteristics of the network $α(ui,vj)$ (also referred to as third-order effects) need to be captured. These characteristics include endogenous network features, such as balancing, stochastic equivalence, or homophily, that contribute to the formation of a network. Several approaches have been developed to capture these higher-order effects as random effects, but the most common one used in political science is the latent factor model. In this approach, each actor has an unobserved vector of characteristics. Similarity in these unobserved characteristics (e.g., state i allying with similar states as state j) positions actors closer in a k dimensional space (Minhas, Hoff, & Ward, 2016a).

## Network Behavior

Dealing empirically with the endogenous behavior of interdependent actors has been the focus of spatial econometrics (Beck, Gleditsch, & Beardsley, 2006; Franzese & Hays, 2008; Darmofal, 2009). Most spatial econometric estimation approaches take a particular network structure as given, which defines the interdependencies between actors (e.g., Bove & Böhmelt, 2016). Instead of assuming that a particular behavior y is only influenced by individual level covariates X, spatial econometric approaches take into account that the behavior of other actors directly influences their peers (Ward & Gleditsch, 2002; Gleditsch & Ward, 2008). Imagine that a standard formation would be $y=Xβ+ε$. However, spatial econometric approaches assume $y=ρWy+Xβ+ε$, where W is a n ×‎ n matrix containing information on linkages between actors. The parameter ρ‎ describes the extent to which the behavior of connected actors Wy impacts on behavior of interest.

Spatial econometric models can usually be distinguished between spatial-lag models that explicitly model the dependencies $y=ρWy+Xβ+ε$ and the spatial error model $y=Xβ+u$ where the error u get decomposed into a spatial error component pWu and a (usually) normally distributed error component ε‎. Hence, $u=pWu+ε$. In political science, there seems to be a slight preference for implementing spatial lag models, because authors are often explicitly interested in the degree of spatial interdependence rather than treating it as nuisance. Increasingly, researchers are attempting to endogenize the creation of the network (Hays, Kachi & Franceze, Jr., 2010; Steinwand, 2011), and these approaches will certainly gain greater attention in the future.

This theory provides importance guidance to spatial econometric models when constructing the W matrix (the underlying network of relations). Information on which actors behave interdependently and the degree to which they do is stored in the W matrix. Misspecification of the W matrix will have similar effects as omitted variable bias. Hence, researchers applying spatial econometric approaches must rely on theoretical arguments and extensive robustness checks with alternative specifications. Plümper and Neumayer (2010b) provide a very good discussion about the construction of the W matrix and how to integrate information about directionality and particular dependencies between actors (see also Neumayer & Plümper, 2016).

# Linking Network Theory and Empirical Implications

The main challenge for international relations and conflict studies scholars is to combine theoretical and empirical network approaches. Ideally, formal network theories should inform the appropriate empirical model to gain insights to our research questions. However, network theories and network empirics can interact in many different ways. This section highlights a few recent studies that provide examples of how network theory and network empirics can be combined to enhance our understanding of cooperation and conflict in multiactor settings.

## Gallop (2016)

A promising approach of linking theoretical and empirical network models is taken by Gallop (2016). Gallop (2016) analyzes the conditions under which networks of cooperation form in international relations. The article provides a theoretical formal model, which is solved for pairwise stability (Pairwise Nash Equilibrium) but uses real-world data to identify the equilibria algorithmically. Gallop (2016) is then able to compare the network statistics of the observed cooperation network to the theoretically derived, but empirically informed, cooperation network. Cooperative events are extracted from machine-coded data provided by the ICEWS project (Boschee et al., 2015). Gallop (2016) can demonstrate that, while there are some differences between the theoretically derived and the empirically observed network, the overall fit is relatively good. This insightful approach combines theoretical and empirical work by providing a direct comparison between the network expected and the actual realized network.

## Larson and Lewis (2017)

The work of Larson and Lewis (2017) is fascinating; they explore the causal mechanisms of information flow on networks through field experiments. They investigate whether ethnically homogenous or heterogeneous villages are more likely to spread information. They seed information in villages that are similar on a number of dimensions but differ in the extent to which they are ethnically homogenous or heterogeneous. Larson and Lewis (2017) demonstrate that ethnically homogenous networks are more prone to information spread, even if the density of the network is considerably lower. While one has to be cautious to overstate their empirical insights gathered from only two villages, their article provides ex-post explanations of these findings by simulating the spread of information on networks with varying degrees of trust. Their argument is that low levels of trust diminish the spread of information in ethnically heterogenous networks even if they show high levels of density. Ideally, such simulations should inform the research’s empirical expectations, which seems not to be the case, but the use of field experiments to gather more information on the spread of information is an important avenue of future research. As in the contribution by Gallop (2016), combining network simulations and empirical work is a very promising avenue of research because it allows the researcher to make fairly precise empirical predictions in complex network settings.

## Hays, Schilling, and Boehmke (2015)

Hays, Schilling, and Boehmke (2015) develop a spatial econometric estimation approach to WWI participation or states. They are interested in understanding the timing of entering WWI depending on the relationship to already fighting states’ in regards to (a) geographic distance, (b) alliances, and (c) rivalry. The choice of the respective networks (W), geographic distance, alliances, and rivalry are grounded in existing theoretical work. While no formal argument explicitly models the decision to join the war, the choice of a particular or a set of W matrices is nicely linked to the existing international relations literature. This is important because in a spatial econometric framework the choice of the underlying W matrix should be theoretically motivated (Plümper & Neumayer, 2010b), but it is endogenously modeled (Hays, Kachi & Franceze, Jr., 2010; Steinwand, 2011). Using a spatial duration approach, the authors find that especially alliance networks and rivalry networks decrease the time until joining WWI. What is usually helpful to show in this context is whether accounting for strategic dependence can increase our ability to predict the outcome of interest. A good example of this can be found in Franzese, Hays, and Cook (2016).

## Chyzh (2016)

A very promising approach to combining formal and empirical analyses is taken by Chyzh (2016). She highlights that domestic elites may see repression as an optimal strategy to stay in power but are hampered by direct international trade to implement human rights abuses. This occurs because trade partners that respect human rights may impose sanctions and conditionalities on trade if repression is observed. Hence, governments that view repression as a beneficial strategy have incentives to form indirect rather direct trade ties with states (or more precisely businesses from these states) that would punish human-rights abuses with lower levels of trade. Deriving her empirical implications from a formal theoretical game that relies on the solution concept of pairwise stability, she expects to find that states that respect human rights are more likely to have direct trade ties than states that do not respect human rights. Using the RSiena approach (an actor-based approach for dynamic networks), Chyzh (2016) demonstrates considerable support for her main hypothesis. This chapter is a very good attempt of providing a formal network model and testing its implications with an empirical network approach. The RSienna approach can nicely model the endogenous nature of actor level outcomes (human rights abuses) and the network level outcomes (trade), which is a fairly close fit to the formal model that is presented in the paper.

## Plümper and Neumayer (2015)

Plümper and Neumayer (2015) provide a novel empirical approach of testing alliance behaviors in existing networks. First, they are interested in changes in military spending of the United States’ NATO allies, rather than looking at levels. Second, they define free-riding not only in respect to U.S. military spending but also military spending of the Soviet Union. If allies neither respond to U.S. military spending nor to Soviet Union spending they are defined as free-riders. This article demonstrates that a network theoretical approach does not always necessitate a fully endogenous spatial or network model. Because the Plümper and Neumayer (2015) assume that NATO allies only condition on the United States and do not condition their choice to arm on any other states, they can estimate a fairly simple model that nonetheless fully captures the empirical implications of the theoretical model. Hence, network theories do not always necessitate network analytic statistical methods.

# Moving Forward

Networks are a fundamental element of international relations and conflict studies and the discipline has made great progress theoretically and empirically accounting for complex dependencies. This article highlights the need to integrate formal theoretical and empirical network models to establish a firm foundation for a network oriented research agenda in international relations and conflict studies. Of course, not everything that looks like a network is necessarily driven by strategic behavior. Hence, when theorizing about network structures, it is important to reflect on Galton’s problem of whether the clustering of particular events is the result of exogenous treatment, that effects a cluster of units, or whether clustering is actually a result of endogenous contagion where, e.g., actors condition their behavior on expected or observed behavior of their neighbors. This distinction is important because it will condition the type of estimation strategy employed to a respective empirical problem. If clustering of behavior is driven by exogenous factors, theoretical and empirical approaches that consider endogenous network approaches are not needed.

However, in many cases clustering or joint behavior is a result of interdependent behavior which constitutes the very relational aspect that is at the core of international relations thinking. But how can thinking about international relations processes be structured so that it considers the network features of a particular empirical or theoretical problem? A good starting point is to identify the potential sources of network effects and to theoretically consider their relative importance, which can later be tested empirically. The source of network effects will often determine the type of estimation strategy employed in the empirical application. Usually, four components should be considered when thinking about network dynamics: actor effects that relate to the sender and/or the receiver, dyad effects between actors, and higher-order network features (e.g., homophily) that condition the behavior of actors, in addition to actor- and dyad-level effects.

Sender effects are an important aspect of network formation and behavior on networks because they speak to the conditions under which particular actors will take an active role in forming a network or spreading certain types of behavior on a network. Initial theories of the democratic peace were strongly arguing that democratic states per se would be less likely to start conflicts with other states. This view of the democratic peace is strongly sender centric, and even though empirically this theoretical view has not gained support, many important phenomena in international relations are sender dependent. In a different context, Metternich, Minhas, and Ward (2017) show that countries with excluded ethnic groups are more likely to spread conflict, independent of other possible network effects. Sender characteristics are also important because they help explain which actors play a crucial role in spreading or mitigating behavior, which can condition policy responses that are geared towards especially (non-)contagious actors. It may also help explain why exponential spread of behavior or outcomes occurs, driven by only few actors in a network.

Receiver effects pertain to the conditions under which actors are more or less likely to receive things from the network or are more likely to be sought out as partners in network formation processes. While sender effects are more about how contagious particular actors are, receiver effects deal with the immunity of actors in a network (Braithwaite, 2010). For example, when asking why some countries were affected by the Arab Spring and others were not, this question can easily be framed in terms of the characteristics of a country that make it more or less likely to receive conflicts.

Of course, not only actor specific characteristics matter in network dynamics. In many instances, common characteristics or traits matter most for linkage formation. Among the most prominent examples for dyadic effects in international relations and conflict studies is the democratic peace (Russett, 1993; Rousseau et al., 1996; Bueno de Mesquita et al., 1999). The democratic peace is a dyadic phenomenon because the joint characteristics of two states make it more or less likely that they engage in armed conflict. Thus, it not only matters that one state in a dyad is democratic to establish the democratic peace but also that both states in a dyad are democratic to substantially decrease their propensity to fight each other.

Network processes are not only driven by actor and dyad-specific effects but also higher-order network effects. These relate to concepts such as balancing, stochastic equivalence, or homophily (see Wasserman & Faust, 1994). In international relations and conflict studies, fundamental concepts have always related to network features. For example, in neorealism (Waltz, 1979) there are particular expectations about a balanced international system. However, the tendency towards more actor-centric approaches (Mitchell, 2017) has weakened the influence of system-level theories. The presentation of theoretically derived higher-order hypotheses is less common, with some notable exceptions (e.g., Plümper & Neumayer, 2010a; Cranmer, Desmarais, & Kirkland, 2012).

The challenge for network approaches in international relations and conflict studies will be to develop a firm network theoretical basis that can systematically inform empirical network models. Similar to the role bargaining theory has played to inform dyadic level empirical studies, a network theoretical foundation is needed that can guide and structure the discipline. Formal approaches to network processes can help the discipline develop true multiactor theories (e.g., Cunningham (2006)). International relations and conflict studies have empirically outpaced our theoretical explanations of network dynamics. It is time for network theory development to close the gap.

# Acknowledgment

Nils W. Metternich acknowledges support from the Economic and Social Research Council (ES/L011506/1).

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