###### In This Symposium

- Dyadic Research Designs: Progress or Postmortem? Read More
- The Promises and Pitfalls of Dyadic Data Read More
- Reconsidering Dyads. Again. Read More
- Dyads and Conflict – Beyond Interstate War Read More
- Make it so: Interdependence and the Next Generation Read More
- Straw Men and Red Herrings in the Dyads Debate Read More
- Systemic effects, statehood, and dyadic research designs Read More
- Embracing the limitations of modeling choices: dyadic design, theory, and simplifying assumptions Read More
- Against Dyadic Design Read More

Dyads are resilient. Despite numerous challenges to their reign, dyadic models continue to dominate empirical international relations (IR) scholarship. Cranmer and Desmarais (2016)** **take aim at dyads yet again, carefully summarizing the key reasons why dyadic models are inadequate for empirical IR scholarship while also offering network-analytic methodologies as a viable alternative. In response, both Diehl and Wright (2016)** **and Poast (2016)** **provide thoughtful, guarded defenses of the dyadic approach. Both sets of defenders articulate concerns that should not be taken lightly; and, given the ubiquity of dyadic models, I suspect that such concerns are held by a large segment of the discipline. Constructing and estimating an inferential network model, such as an exponential random graph model (ERGM) or one of its many derivations, is not a trivial task. Is it worth it? Are we really better off abandoning dyads in favor of more complex models? Yes, we are. I shall focus my comments on three issues: (1) model selection, (2) the substantive importance of interdependence, and (3) multilateralism versus bilateralism.

(1) Cranmer and Desmarais rightly point out that IR data systematically violate the i.i.d. assumption that is foundational to regression models. Diehl and Wright (2016) counter that “the dyadic approach is appropriate if one is concerned primarily with an outcome based on two actors,” offering such examples as interstate conflict, rebel-government violence in civil wars, and enduring rivalries. With regard to rivalries, Diehl and Wright (2016) state that “assuming that linked rivalries are always connected is almost as misleading as starting with the expectation that they are fully independent.” Referencing Arab-Israeli rivalries in particular, the authors conclude that “[a]ssuming full case independence seems misguided, but then again so too does assuming complete interdependence.” These observations reflect important questions about methodological assumptions. Isn’t the assumption that everything is connected just as faulty as the assumption that everything is independent? Yet, this framing of the issue is a false dichotomy, and it risks propping up a straw man caricature of the network approach.

First, we must clarify that network analysis is not divorced from dyads. Rather, dyads are the building blocks of networks. Substantive bilateral relationships—the probability of a conflict, an alliance, a trade agreement, an investment treaty, or whatever—can still be the primary focus. We can still estimate the probability that a particular bilateral tie will emerge. Network models, in part, allow us to make such probabilistic estimates more accurately, by accounting for the overall structure of the network rather than limiting the analysis strictly to attributes of and relations between some *ij* pair of states.

Second, and most importantly, network models do not assume complete *inter*dependence. Rather, they simply make no assumptions about *in*dependence. An inferential network model thus allows the analyst to explicitly model the various interdependencies that might exist within a network. Consider the example of a preferential trade agreement (PTA) network (e.g., Manger, Pickup, and Snijders 2012). Perhaps, when forming new PTAs, a given *i* prefers *j* partners that already have many PTAs in place (a so-called “preferential attachment” or “rich get richer” effect). Or, perhaps *i* prefers *j* partners that have already signed PTAs with *i*’s own PTA partners (a “transitivity” or “friend of a friend” process). Such effects, despite their endogeneity, are easily incorporated into an ERGM or similar network model, which then allows us to *explicitly test whether those interdependencies matter or not.*

Assuming the model is otherwise well specified, if ties in the PTA network do not exhibit transitivity, the corresponding parameter estimates for transitivity effects will be indistinguishable from zero. Similarly, if there is no preferential attachment process in the PTA network, this fact will be reflected in the estimates and standard errors. In fact, if the network effects are null, then the ERGM estimates of specified exogenous covariates (e.g., distance, GDP, democracy, and all the other things that might matter for PTAs) will be identical to those in a standard logit regression model (Wasserman and Pattison 1996). Put differently, inferential network models allow us to model the various exogenous influences that IR scholars typically care about, while also allowing us to determine whether interdependencies—i.e., network effects—are present. If network effects matter, we can estimate their magnitude while also reducing bias in estimates of relevant covariates. If network effects don’t matter, we are left with parameter estimates comparable to those produced by regression. In contrast, pooled dyadic regression requires us to assume ex ante that no dependencies exist in the data, while providing zero information about whether that assumption is true. Which approach seems more sensible?

(2) There is an unfortunate tendency, even among proponents of network analysis in IR, to view endogenous network influences as statistical nuisances that must be “controlled.” Certainly, as Cranmer and Desmarais (2016) make clear, if a given *ij* dyadic tie is not independent of, say, an *ik* tie (or a *ji* tie or a *jk* tie or any other tie), then we have an estimation problem. Yet, we must keep in mind that interdependencies in IR data are *substantively interesting phenomena themselves.* Why does the *ik* tie influence the *ij* tie? Such endogenous effects do not simply appear sui generis. Indeed, there is a long, rich theoretical tradition in IR of exploring the various ways in which states strategically respond to one another’s actions (e.g., Deutsch et al. 1957; Keohane 1986; Schelling 1960). Interdependence is the natural state of affairs; it is not a statistical nuisance, but something that happens for palpable political, economic, and social reasons. Cranmer and Desmarais (2016) suggest this possibility in their discussion of economic sanctions. Numerous other network examples abound. Warren (2010) finds strong evidence that states use knowledge of their antagonists’ alliances to inform their own alliance making. Manger, Pickup, and Snijders (2012) find that states pursue transitivity in their PTA ties in order to avoid the negative effects of trade diversion. In work on diplomacy, I have shown that states prefer to establish embassies in countries that already host large numbers of embassies (i.e., a preferential attachment effect), as this strategy allows a resource-constrained diplomatic corps to establish indirect contacts with diplomats from across the globe (Kinne 2014).

In a separate project, Jonas Bünte and I explore network effects in bilateral government-to-government lending—i.e., loans made by one sovereign government to another (Bünte and Kinne 2015). Employing temporal ERGMs (Desmarais and Cranmer 2012), we find that, as part of an intense competition to exercise influence in valuable markets, states condition their lending on the lending patterns of their competitors. Lenders use the existing ties within the network to determine which borrowers offer the greatest potential returns (e.g., political influence) and the lowest potential losses (e.g., credit defaults). In short, the network provides lender governments with strategically valuable—and otherwise unavailable—information.

Figure 1: Predicting Chinese Loans, 2010

To make substantive sense of these influences, we examine China’s controversial and highly publicized loans to African governments. Figure 1 shows “true positives,” i.e., African countries to whom China extended bilateral loans in 2010. We employ out-of-sample prediction—using data up to but not including the year 2010 as a training set—to compare a network model to a standard regression model. The goal is to successfully predict the newly created 2010 China-Africa loans. The regression model fares poorly, correctly predicting a loan in only one case (Mali). The network model, on the other hand, accurately predicts all nine of these new loans. In other words, if we wish to understand why, when, and where China makes bilateral loans, traditional monadic and dyadic covariates tell us very little. We must look to the network to understand China’s lending practices. More generally, when we ignore network influences, we do not merely risk biased parameter estimates. We risk ignoring the most substantively interesting and fundamental aspects of international relations.

(3) As both Poast (2016) and Cranmer and Desmarais (2016) note, multilateral ties pose a particularly difficult problem for dyadic models. Cranmer and Desmarais (2016) favor network models for such data, while Poast (2016) discusses the relative merits of both network models and *k*-adic approaches (Poast 2010). While I strongly agree that multilateral phenomena clearly do not fit within a dyadic framework, I worry that over-reliance on this particular limitation may constitute a red herring, distracting our attention from the difficulties in modeling multilateral data while also minimizing the importance of network methodologies for bilateral data. Network models of multilateral data are not easy to estimate. A standard ERGM assumes that network ties, though they may be interdependent, are separable. An *ij* tie may influence, say, an *ik* tie, but we nonetheless assume that these are distinct ties, where, in principle, one can exist without the other. With multilateral data, however, ties are often created as a package, via summitry or multilateral bargaining processes, and no one tie can exist in isolation. The North Atlantic Treaty, for example, cannot plausibly be interpreted as simply an aggregation of separate bilateral ties. Estimating an inferential network model on such data may yield misleading results. For example, multilateral ties are likely to yield networks with high levels of transitivity, but it’s not clear that a traditional transitive closure process is actually responsible for these levels of transitivity.

Just as importantly, multilateral relations—the bulk of which involve formal treaties, agreements, organizations, or other international legal phenomena—are vastly outnumbered by bilateral relations. Indeed, the number of bilateral instruments annually deposited with the UN Treaty Series is orders of magnitude larger than the number of multilateral instruments, despite the fact that, as a rule, bilateral treaties are far more underreported than their multilateral counterparts. And trends toward bilateralism have only increased in recent years. These bilateral ties hold special relevance for network analysis, as they are often strategically formed in response to—or in anticipation of—the ties of others. And while Diehl and Wright (2016) are certainly correct that, in principle, good theory should have some sense of when dyads are independent and when they are not, this expectation, in practice, places a heavy, almost omniscient burden on theorizing. Ultimately, despite good theory, the question of whether dyadic relations are independent is empirical. If interdependencies exist, then our models must account for them, even if our theory cannot explain them. Network-analytical models are appearing with greater frequency in the discipline’s journals, and I have yet to see a network model of IR data, in any issue area, that shows the assumption of independent dyadic observations to be empirically correct. Network influences are the norm, not the exception.

**References**

Bünte, Jonas, and Brandon J Kinne. 2015. “How sovereign creditors maximize political benefits bilateral loans.” Presented at Annual Conference of the American Political Science Association, San Francisco, September 4.

Cranmer, Skyler J., and Bruce A. Desmarais. 2016. “A Critique of Dyadic Design.” *International Studies Quarterly* Forthcoming.

Desmarais, Bruce A, and Skyler J Cranmer. 2012. “Statistical mechanics of networks: Estimation and uncertainty.” *Physica A: Statistical Mechanics and Its Applications* 391 (4): 1865–1876.

Deutsch, Karl W., Sidney A. Burrell, Robert A. Kann, Maurice Lee, Jr., Martin Lichterman, Raymond E. Lindgren, Francis L. Loewenheim, and Richard W. Van Wagenen. 1957. *Political Community and the North Atlantic Area: International Organization in the Light of Historical Experience*. Princeton: Princeton University Press.

Diehl, Paul F, and Thorin M Wright. 2016. “A Conditional Defense of the Dyadic Approach.” *International Studies Quarterly* Forthcoming.

Hollway, James, and Johan Koskinen. 2016. “Multilevel embeddedness: The case of the global fisheries governance complex.” *Social Networks* 44: 281–294.

Keohane, Robert O. 1986. “Reciprocity in International Relations.” *International organization* 40 (1): 1–27.

Kinne, Brandon J. 2014. “Dependent diplomacy: Signaling, strategy, and prestige in the diplomatic network.” *International Studies Quarterly* 58 (2): 247–259.

Manger, Mark S., Mark A. Pickup, and Tom A. B. Snijders. 2012. “A Hierarchy of Preferences: A Longitudinal Network Analysis Approach to PTA Formation.” *Journal of Conflict Resolution* 56 (5): 853–878.

Poast, Paul. 2010. “(Mis)Using Dyadic Data to Analyze Multilateral Events.” * Political Analysis* 18 (4): 403–425.

Poast, Paul. 2016. “Dyads Are Dead, Long Live Dyads! The Limits of Dyadic Designs in International Relations Research.” *International Studies Quarterly* Forthcoming.

Schelling, Thomas C. 1960. *The strategy of conflict*. Cambridge, Mass.: Harvard University Press.

Warren, Camber. 2010. “The Geometry of Security: Modeling Interstate Alliances as Evolving Networks.” *Journal of Peace Research* 47 (6): 697–709.

Wasserman, Stanley, and Pip Pattison. 1996. “Logit models and logistic regressions for social networks: I. An introduction to Markov graphs and p.” *Psychometrika* 61 (3): 401–425.