Militarized Interstate Disputes

We turn now to an analysis of one of the canonical datasets for IR, the MID data, which was collected by the Correlates of War project and widely employed in empirical studies of IR. We use the SRM to analyze the dependencies in dyadic data for international conflict annually from 1816–2001. We break down this data in three different ways. First, we create an aggregated matrix representing the years 1952–2001. This matrix mirrors much of the work done in IR, where data on conflictual interactions is treated as binary. Secondly, because the SRM can support a variety of data types, we create another version of the aggregated matrix that contains counts of the number of dispute onsets over the 50-year period (that is, a running count of whether there was an MID in a given year). Thus if country i initiates three different disputes during the 50-year period against country j (or is on the side of an initiator), then a count of 3 is shown in the aggregated matrix for X_ij. We run the SRM on each of these aggregated matrices and report the results. Thirdly, we create an individual sociomatrix for each year of the MID data from 1816–2001. The rows and columns of each matrix represent the countries that existed in the international system for that year. The matrices then contain counts for each MID that occurred between any given dyad (or set of countries) in the matrix. The country initiating the dispute, and all countries on the side of the initiator, are coded 1 (if there are two different disputes for that year between the same two countries, it is coded 2, and so on). With this set-up, we can then run the SRM on each matrix-year and assess how the different SRM effects and variances change over time.

We begin with an analysis of two aggregated matrices for the period 1952–2001, the most recent available data. The network representation of this data is shown in Figure 1. In this graph the nodes are the countries in the international system during this time period, and the links are determined by the presence of MIDs between countries.

Fig. 1

Network graph representing countries involved in MIDs, 1952–2001

Note: countries with the highest number of ties (MIDs) are the farthest away from the center.

Next we examine the variances for the network of countries involved in MIDs during the 1952–2001 period (Table 2). It is clear that there is a lot of variance within this global network of countries, and it is even more apparent that the unique (dyadic) variance is much more salient than the individual actor or partner variance. If an analyst thought the salience of one dyad was independent of other dyads, then you would expect that the actor-partner and dyadic covariances would be essentially zero, but this is clearly not the case.

Table 2

Variance Structure of Social Relations Model for 50 Years of MID Data

The sender and receiver (target) effects for both the network of total conflict onsets and an aggregated matrix of MID counts are shown in Figures 2 and 3, respectively. In each graphic, the countries are plotted from low to high, illustrating the exponential shape of the effects of individual countries on the overall pattern of militarized disputes since the end of World War II. We can see that incorporating counts of onsets in the matrix gives us a complete picture of conflict dynamics and the major players in this system. In the counts matrix, Russia and the United States top the list, followed by Iraq and China. Iran is close behind.

Fig. 2

The actor (sender) effect of every country from a binary matrix representing the number of onsets of MIDs, 1952–2001

Note: these are sorted from low to high. Top actors are labeled.

Fig. 3

The actor (sender) effect of every country from an aggregated matrix of counts representing the number of onsets of MIDs, 1952–2001

Note: these are sorted from low to high. Top actors are labeled.

In both matrices, the actors with the most impact are not the countries that would necessarily score the highest on conventional power metrics (though there is obviously some overlap). Not only can actor effects be negative (countries below the lower horizontal line), but the most salient countries in this domain have scores on actor effects that are two orders of magnitude larger than those at the median level of impact. Notably, the nodes with the most ties in Figure 10 (such as the United States, Iraq and Russia) place high in the actor effects plot in Figure 3. Similarly, we see that the United States and Russia rank highest on receiver (target) effects.

The basic point of this analysis is that by considering the dyadic interactions in the context of their interdependencies, a different picture emerges—one that reflects these dependencies as well as the independent power of countries to affect MIDs separately, both as actors and targets. If we turn to the analysis of how these actor and target effects vary over the same period, we see a more nuanced picture (shown in Figure 4). Here, we see that the sender (actor) effects and partner (target) effects vary substantially over time for the United States, Russia and China. Such figures clearly support the idea that the current approach to dyadic analysis in IR is incomplete and demonstrate how the SRM can reveal overlooked patterns in the data.

Fig. 4

Individual actor effects for USA, Russia and China as both senders and targets over time

Scholars have long debated the role of reputation and repeated interactions in conditioning the ebb and flow of the international system. While directly addressing such questions is beyond the scope of this article, the results from the SRM shed light on the influence of major players in the international system in an easily interpretable and systematic way. To further this point, it is instructive to examine the actor effects of the United States and Russia over the most recent period. We present the unique relationship effects for the United States and Russia in Figures 5 and 6, respectively.

Fig. 5

The evolution of US-Russia actor (sender) and target (receiver) effects, 1815–2001

Note: the red dots (in online version) and the corresponding arrows highlight the trajectory during the Second World War, while other years are shown in shades of gray. Darker shades indicate more recent years.

Fig. 6

Dyadic, actor and target variance in the network of MIDs, 1816–2001

Figure 5 illustrates that Russia and the United States have actor effects that are largely independent of one another, and are not extraordinary in most years. The individual years from 1816 to 1938, and subsequent to 1945, are shown as gray dots with decreasing darkness as they recede in time (that is, the earliest years are the lightest, the most recent the darkest). Median levels are shown with thin rules, which transect the mass of points in the scatter plot. In 1939 an extraordinary cycle begins as the impact of Russia (the Soviet Union at this point in time) rises substantially until 1940. By 1941 the impact of the United States also jumps to a new level. Throughout WWII, these two actors have high levels of correlated impact on world politics, in terms of the conflict domain, but each declines rather quickly as the war draws to an end in 1945. By this point in time, each of the actor effects of Russia and the United States have returned to their pre-WWII averages, where they remain for the most part throughout the Cold War.Footnote ¹⁰ A similar correlation is seen in the pair's partner (target) effects as shown in Figure 5(b). Notably, target effects for 1940, 1943, 1944 and 1945 are quite similar.

Finally, we examine the dyadic versus the actor and target components of the network. The results shown in Figure 6 are produced by running the SRM on each matrix-year from 1816–2001. Here the dyadic variance in the SRM of militarized disputes is not only substantially larger than either the actor or target effects, but the movement of the variance over time has more variance. At major crisis points in the evolution of conflictual international relations we can see spikes in all three, but the dyadic variance has enormous spikes before the global wars, as well as during the 1960s and prior to the dismantling of the Soviet empire.

International Trade

The benefit of using the SRM is multilayered. First, we have shown that it directly models the dependencies in the data. However, such an approach can also then be used to compare multiple contexts of substantive interest. As a brief example, we compare the SRM results from 2000 trade data with the results generated by the MID data used in our study. Scholars have long theorized about the intersection of military might and economic prosperity as an indicator of global power; comparing such data using the SRM sheds light on the influence of those actors. Figure 7a shows the actor effects for data on the top 10 exporters in the year 2000; Figure 7b plots them against the actor effects generated for the same countries, in the same year, using the MID data.

Fig. 7

Individual actor effects for top 10 exporters in 2000

Sources: IMF trade data and actor effects for the same states active in the MID dataset during the same year.

Next, we depart from the first half of our analysis and demonstrate how the SRM can be extended by incorporating covariates of interest. There are various approaches to incorporating covariates, such as structural equation modeling (Kenny, Kashy and Cook Reference Kenny, Kashy and Cook2006) and random effects ANOVA (Bond and Lashley Reference Bond and Lashley1996). These kinds of models can be estimated using MLwiN.Footnote ¹¹ Hoff et al. (Reference Hoff, Fosdick, Volfovsky and Stovel2012) have also recently developed a wider range of techniques for nomination networks that also estimates the covariance structures within the SRM framework.Footnote ¹² Another way this can be done is to estimate a latent space model, which integrates components of the basic SRM by incorporating the decomposition of variance into the error term of a bilinear mixed-effects model. This estimation also allows for a straightforward way to incorporate covariates specific to the actor, partner and dyadic relationships. In addition, the latent space model also builds on the SRM analysis by estimating third-order dependences (Hoff and Ward Reference Hoff and Ward2004).

Similar to second-order dependence, third-order dependence is commonly recognized in the social network analysis literature. Aspects of this higher-order dependence include transitivity, balance and clusterability (Wasserman and Faust Reference Wasserman and Faust1994).Footnote ¹³ An often-used example in the literature on third-order dependence considers the following relationships among a triad {i, j, k}. If we know that i considers j a friend and j is a friend of k, then the probability that k will also be a friend of i is likely to be higher than for a random person outside this triad, since i and k are at least indirectly connected in the friendship network by virtue of their separate linkages to j. In other words, knowing something about the relationships in the first two dyads in a triad often tells us something about the relations in the third dyad. As we think about the nature of the third-order dependence in some network datasets, a conceptualization of an ‘unobserved’ or latent ‘social space’ in which every network actor is embedded is very fruitful. Thus, for example, the observation of two links, i→j and j→k, suggests that i and k are not too far away from each other in this social space (which is often unobservable) and therefore are also likely to have a link between them.

Third-order dependence is an expression of the underlying probability of a link between two actors. We do not observe the complete set of all of these network characteristics, but we can infer them from the pattern of dyadic linkages. If we can map out the latent positions of each actor in the ‘social space,’ we can then assume that the ties in the network are conditionally independent. A series of latent models has been recently developed by Hoff, Raftery and Handcock (Reference Hoff, Raftery and Handcock2002) and Hoff (Reference Hoff2005) in which latent vectors, say u_i and v_j, for any two actors i and j are used to locate the actors in the ‘social space’ in order to account for third-order dependence. The latent model includes an estimate of the latent positions of each country in the investment network (u_i and v_j), which index the propensities for country pairs to have similar interaction patterns toward other countries. Put simply, if two countries share similar positions in the latent space, they have a higher probability of interacting with each other. This means a high level of bilateral transnational investment. The latent space model for dyadic data is based on an analysis of the actual dyadic matrices. A simple version of that model is given by the following specification:Footnote ¹⁴

$${{y}_{i,j}} = \beta _{d}{\!\!'} {{x}_{i,j}}\, + \,\beta _{s}{\!\!'} {{x}_i}\, + \,\beta _{r}{\!\!'} {{x}_j}\, + \,{{a}_i}\, + \,{{b}_j}\, + \,{{\gamma }_{i,j}}\, + \,u_{i}{\!'} {{v}_j},$$

This implies the following moments for γ_i,j, which we estimate rather than ignore:

$$E(\gamma _{{i,j}}^{{\rm{2}}} )\, = \,\sigma _{r}^{{\rm{2}}} \, + \,\sigma _{s}^{{\rm{2}}} \, + \,\sigma _{\gamma }^{{\rm{2}}} ;\; \; E({{\gamma }_{i,j}}{{{\epsilon}}_{j,i}})\, = \,\rho \sigma _{\gamma }^{{\rm{2}}} \, + \,{\rm{2}}{{\sigma }_{rs}};\; \; E({{\gamma }_{i,j}}{{{\epsilon}}_{i,k}})\, = \,\sigma _{r}^{{\rm{2}}} ;\; \; E({{\gamma }_{i,j}}{{{\epsilon}}_{k,j}})\, = \,\sigma _{s}^{{\rm{2}}} ;\; \; E({{\gamma }_{i,j}}{{{\epsilon}}_{k,i}})\, = \,{{\sigma }_{rs}},$$

where $$$\sigma _{r}^{2} $$$ represents dependence among dyadic observations with a common partner, $$$\sigma _{s}^{2} $$$ represents dependence among measurements with a common sender and ρ represents the correlation of measurements within a dyad (that is, reciprocity). These dependencies are important.

This specification adds another layer to the SRM by configuring it to take clustering into account as well.Footnote ¹⁵ This model has been applied in a few places in political science (Ward, Siverson and Cao Reference Ward, Siverson and Cao2007; Ward and Hoff, Reference Ward and Hoff2007; Hoff and Ward Reference Hoff and Ward2004; Cao Reference Cao2009; Breunig, Cao and Luedtke Reference Breunig, Cao and Luedtke2012). It has also been extended to dynamic networks (Ward, Ahlquist and Rozenas Reference Ward, Ahlquist and Rozenas2013) and to dynamic networks combined with simple games (Metternich et al. Reference Metternich, Dorff, Gallop, Weschle and Ward2013). It has also been applied in a wide variety of studies outside the political science literature.Footnote ¹⁶

Figure 8 shows that the computationally simple SRM approach provides estimates of actor and partner effects that are fairly similar to the more elaborate estimation of a similar concept using the General Bilinear Effects Model. Though the scales are different, both approaches give very similar results for the individual effects of actors and partners in a small network of the top 10 exporters in 2000. The more elaborate approach is useful and captures some nuance that is not embedded in the SRM model. However, the basic answers for most networks will be close.

Fig. 8

Actor and target effects generated by the GBME Versus SRM approach

Reciprocity and the Laws of War

In this section we apply the SRM and GBME to a long-standing question in IR: When do states follow the laws of war? We focus on the work by Morrow, who argues that institutional commitment shapes state behavior in following the laws of war (Morrow Reference Morrow2007). Namely, such commitments are established through three mechanisms: (1) treaties set the standard of expectations for wartime behavior, (2) states can reinforce agreements by responding to violations of treaties with similar actions and (3) the joint ratification of treaties strengthens their influence and therefore creates effective deterrence through reciprocity. In addition, ratification by democracies heightens compliance, whereas ratification loses its effect on non-democracies. The concept of reciprocity is key to Morrow's arguments. Indeed, he suggests that reciprocity is crucial not only for understanding behavior at the state level, but also for interpreting the behavior of individual soldiers on the battlefield. He analyzes the reciprocal nature of states in obeying the laws of war using dyadic data on compliance. To assess these claims, he employs an ordered probit modeling approach. In this section, we will discuss how the SRM and GBME provide leverage over other approaches, particularly by highlighting both the theoretical and methodological implications of reciprocity.

The data we will use is from Morrow and Jo (Reference Morrow and Jo2006), in which the authors record compliance with the laws of war for eight different ‘issue areas’ during 1900–92: the treatment of POWs, treatment of the wounded (enemy and civilian), respect for cultural property and general treatment of civilians. Compliance is conceptualized here using a 4-point ordinal scale in which the higher the value on the scale, the worse the violations against the laws of war (0 = no violations, 1 = infrequent/minor violations, 2 = frequent/major violations and 3 = the rules are completely disobeyed). The data is organized around each individual ‘warring dyad.’Footnote ¹⁷ To illustrate our point, we use one of the issue areas (treatment of POWs) and one time period within the data (1975–92), to create an n×n sociomatrix that best represents the dyadic nature of the data.Footnote ¹⁸

First, we use SRM on the sociomatrix for the selected time period. SRM allows us to analyze reciprocity at both the individual and dyadic levels. For example at the individual level, reciprocity is captured if actor i is hostile toward all others in the group, whether or not all others in the group are hostile toward actor i. In Figure 9 the sender and target effects are sorted from low to high; the correlation of these two effects expresses the general reciprocity between individual states. The generalized dyadic reciprocity has a covariance that is roughly equal to the unique variance (both about 0.33), while the actor and partner variances are minimal, as is the actor covariance (each approximately −0.01). These data are gleaned from calculating the basic quantities of the SRM, rather than asserting a priori the values of these quantities.

Fig. 9

Sender and target effects for 1975–1992

Note: the correlation between sender and target effects = 0.94. The correlation between the two represents generalized reciprocity: the measure of how much a sender's actions influences the response of all of their partners.

For the dyadic level, we can assess whether partners within one dyad see each other as similar or interact with each other similarly. In general, we can say that for any dyadic pair, X_ij, actor i's relationship effect toward actor j is correlated with actor j's relationship effect toward actor i. This measure closely gets at the dynamics Morrow discusses in his general theoretical framework for compliance. Namely, that compliance with the laws of war is a phenomenon that is largely driven by reciprocity: when one country mistreats POWs, the other is likely to do so as well. The SRM can demonstrate whether or not such dynamics are driving the patterns of the data. Figure 10 displays Morrow's original data on the left-hand side and a network of unique effects for the same set of countries. The right-hand side graph presents the 1975–92 network of states and highlights their dependent nature. Here the ties are weighted by the SRM unique effects for each dyad, with the highest unique effects shaded darkest (green in the online version). As we can see, clear patterns emerge; the dyadic pairs Iran-Iraq and Somalia-Ethiopia have the strongest unique effects, or reciprocal violations.

Fig. 10

L: network graph of dyadic compliance from Morrow's data (1975–92); R: the network's unique effects

Note: the thinner gray (red in online version) lines indicate lower values of dyadic reciprocity while thicker, black (green in online version) ties indicate the highest values.

Lastly, we conclude with a comparison of the GBME approach to Morrow's ordered probit approach. Again we focus specifically on modeling the treatment of POWs during 1975–92. For the ordered probit replication, we include the most relevant variables from Morrow's study: the democracy score of the sender country, a binary variable that represents whether the countries in the dyad were in an alliance, a binary indicator of war and a compliance indicator for the target country using Morrow's 4-point ordinal scale (which, in his set-up, is used to measure reciprocity). In Table 3 we see that Morrow finds substantive support for the influence of democracy on compliance; democracy has a positive effect on the level of compliance for POW treatment. In his own words, “ratification by democracies, but not nondemocracies, correlates with greater compliance” (Morrow Reference Morrow2007). In addition, the ordered probit results seem to support the idea that the target state's compliance influences the compliance of the violating state. The GBME results, however, reveal a different story: when the network effects are taken into account, democracy plays no role in predicting compliance.

Table 3

Ordered Probit Estimates for Treatment of POWs, 1975–92

To replicate Morrow's study using our approach, we have to make some adjustments. His empirical analysis uses a variety of interaction terms that become numerically infeasible when one collapses a large number of years into a single analysis. The SRM and GBME work slightly differently, operating on an n×n sociomatrix. As mentioned, we use a smaller subset of the data than Morrow, and focus on the period from 1975–92. With a substantially smaller dataset to analyze, the nuanced and complicated interactions of Morrow's original analysis cannot be replicated exactly. Rather, we undertake an ‘in spirit’ replication. As we proceed, we include the democracy level of a country and whether or not the dyad is at war and whether the two countries are allied. These covariates are introduced into the GBME approach. The Bayesian analysis is run for 500,000 iterations, and every 100th is saved. As suggested by our SRM analysis, we find strong effects for reciprocity—much stronger than those in Morrow's model, as shown in Table 4.

Table 4

Coefficient Estimates and Credible Intervals for Laws of War Analysis, 1975–1992

Our empirical results give some priority to network analysis over that which ignores interdependencies. As with our analysis of MIDs and trade, our results clearly demonstrate the utility of capturing the dependencies undergirding patterns of international relations. Specifically, our analysis of the laws of war suggests that democracy, as a covariate for the members of a network, has a weak effect, and this effect is imprecisely measured. Indeed, the effect for democracy is close to (and indistinguishable from) zero once the uncertainty of the estimation is taken into account. Importantly, we find that actual, measured reciprocity is very powerful and precisely estimated. In addition, alliances are shown not to be especially powerful but may be said to reduce compliance. The network measurement of this effect is much more precise than our simplistic replication of Morrow's work, but they both point in the same direction on the impacts of alliances. In general, although dependencies captured by the covariance terms are smaller than is typical of highly dependent networks, they are not small enough to ignore, and suggest that there is indeed a strong degree of second-order network clustering.

Though neither of the network approaches yields results that are overwhelming, the GBME reveals a more precise estimate of state compliance with the laws of war and highlights the dependencies discovered by the SRM analysis. Furthermore, both the GBME and SRM analyses emphasize reciprocity—measured by capturing dyadic effects—as a critical factor underlying our theoretical understanding of the laws of war. This contribution is significant. Our application of the SRM and GBME illustrates our central argument: that both theoretical and empirical research benefits from proper assessment and modeling of interdependent, dyadic data. Even in areas that have strong intuitions derived from carefully researched game theoretic approaches, it is critical to have a method of capturing the strategic interdependencies that embody the theoretical insights of such models.