2.1.1. Presenting Complex Relational Systems as Multilayer Networks

In order to jointly examine these conflict subsystems while differentiating between them so as to be able to identify different cross-type conflict processes, we can utilize the multilayer network, a generalized framework that accounts for networks characterized by differentiated nodes and differentiated ties (Boccaletti et al., Reference Boccaletti, Bianconi, Criado, Del Genio, Gómez-Gardenes, Romance, Sendina-Nadal, Wang and Zanin2014; Kivelä et al., Reference Kivelä, Arenas, Barthelemy, Gleeson, Moreno and Porter2014). The basic organizing principle of multilayer networks are network layers, which group nodes into disjoint subsets, with each layer containing one type of node. These layers and their associated node types are defined by the combination of one or more nodal attributes up to any number d. As presented in Figure 1, the Levantine conflict network has layers defined by one nodal attribute, statehood, which partitions the node set into state and nonstate actor subsets. Generally, layers can be defined as a d-tuple, with each element in the tuple varying along its corresponding attribute, meaning that the set of layers comprises every combination of possible values across all relevant nodal attributes. A multilayer network with d relevant attributes each with k levels has the following layer set where the superscript indexes the attribute and the subscript indexes the attribute level, $\{(attr_1^1,attr_1^2,\ldots ,attr_1^d),(attr_1^1,attr_1^2,\ldots ,attr_2^d),\dots ,(attr_k^1,attr_k^2,\ldots ,attr_k^d)\}$.

These network layers are used to define different types of ties in the multilayer network. Specifically, tie types are identified by the combination of types of nodes they are incident to, which can be alternatively referred to by their incident layers. The two layers in the Levantine conflict network yield three types of ties, the militarized interstate dispute incident to two state layers; the nonstate conflict incident to two nonstate layers; and the civil conflict incident to one of each layer. By convention, ties incident to the same layer are referred to as intralayer ties and those incident to different layers are referred to as interlayer ties (Kivelä et al., Reference Kivelä, Arenas, Barthelemy, Gleeson, Moreno and Porter2014).

Represented as a multilayer network, complex interdependent systems such as conflict in post-Cold War Levant can be approached using the variety of methods developed for network analysis. However, as most of these methods were initially developed for monoplex networks, extensions are required so they can account for the complex features of multilayer networks. As my purpose is to examine the generative process that underlies these systems, I focus on extensions to the ERGM.

2.2.1. Data Structure

Networks are commonly represented using an adjacency matrix Y = {Y _ij} where Y _ij indicates the presence of a tie between nodes i and j. For example, the MID network with three nodes {Syria, Lebanon, Israel} indexed from 1 to 3, and two undirected ties {{Syria, Israel}, {Lebanon, Israel}} can be represented as

(2)$$ {Y} = \left[ \matrix{ Y_{1,1}=0 & Y_{1,2}=0 & Y_{1,3}=1 \cr Y_{2,1}=0 & Y_{2,2}=0 & Y_{2,3}=1 \cr Y_{3,1}=1 & Y_{3,2}=1 & Y_{3,3}=0 } \right]. $$

For the purposes of statistical inference of multilayer networks via the ERGM, the set of ties on a multilayer network with ℓ layers can be grouped by type by partitioning the matrix into ℓ × ℓ blocks:

(3)$$ \bi {Y} = \left[\matrix{ \bi {Y}^{1,1} & \bi {Y}^{1,2} & \cdots & \bi {Y}^{1,\ell} \cr \bi {Y}^{2,1} & \bi {Y}^{2,2} & \cdots & \bi {Y}^{2,\ell} \cr \vdots & \vdots & \ddots & \vdots \cr \bi {Y}^{\ell,1} & \bi {Y}^{\ell,2} & \cdots & \bi {Y}^{\ell,\ell} } \right]. $$

This procedure places each network layer, with its intralayer ties, into one of the ℓ main diagonal blocks, with the off-diagonal blocks in turn containing interlayer ties. As noted, depending on the specific multilayer network structure, these off-diagonal blocks may be constrained in different ways, discussed in more detail later. The matrix from Equation (2) would therefore occupy the (1, 1) block of the full adjacency matrix of the Levantine conflict network, which is graphically illustrated in Figure 1. In addition to the MID ties on the (1, 1) block, the (2, 2) block represents the nonstate conflict ties, and the off-diagonal blocks (1, 2) and (2, 1), which contain the same information because this network is undirected, represent the interlayer, civil conflict ties.

An additional step in constructing the multilayer adjacency matrix is to specify the constraints appropriate for the overall structure of the network. Applying sampling constraints to the adjacency matrix, usually in the form of disallowing ties to be formed between certain nodes, is a relevant consideration to all ERGM examinations of networks. In any network, contextual characteristics of the system can require that ties never be formed between certain nodes. For example, in a temporally-pooled examination of interstate relations, state entry and exit from the interstate system mean that there are states whose existence do not overlap, such as for Czechoslovakia (exit in 1993) and Palau (entry in 1994). Aside from these contextually idiosyncratic requirements, certain types of multilayer networks require systematic constraints on their off-diagonal blocks. I discuss these in more detail in the application section.

2.2.2. Local Network Structures in Multilayer Networks

The main contribution of the multilayer extension to ERGMs is in the network dependence structures that can be used to account for phenomenon across different types of ties. These cross-layer dependence structures should be specified by the researcher based on appropriate theoretical expectations. In the application section I demonstrate the use of various cross-layer dependence structures, but a full discussion of possible multilayer terms is beyond the scope of this paper. For a detailed description of a broad set of these structures, readers are referred to Wang et al. (Reference Wang, Robins, Pattison and Lazega2013, Reference Wang, Robins, Pattison and Lazega2016). Here, to illustrate the basic construction of a cross-layer network structure, I discuss how clustering structures on a monoplex version of the Levantine conflict network differ from that on multilayer version of the same network.

On a monoplex network, the lowest form of tie-clustering is represented by a two-star, defined as two ties that share a node (Robins et al., Reference Robins, Snijders, Wang, Handcock and Pattison2007). The corresponding network statistic is a count of two-stars on the network. On the multilayer network with three types of conflict ties, there are six different types of two-star clusters, as shown in Figure 3. In order to obtain the size of the desired subset, ties are only counted when they belong to the appropriate blocks. For example, to obtain a count of two-star comprised of a MID and a civil war, instead of computing the statistic over the entire network, only the Y^1,1 and Y^1,2 blocks from Figure 2 are used. This logic of identifying and limiting counted ties by matrix block remains the same for all cross-layer network structures regardless of their complexity.

Figure 2. Adjacency matrix for post-Cold War levantine conflict network.

Figure 3. Six different types of two-star conflict clustering.

After a complex system has been modeled as a multilayer network and the network represented as an adjacency matrix, statistical inference on the generative process of the system is an exercise in specifying the theoretically appropriate model terms and sampling constraints for the ERGM.

3.1.1. Multiplex Networks

Multiplex networks are those where different types of ties exist between the same set of real-world actors. Each of these tie types exists on a different network layer. This is a commonly observed network structure, as actors within a system are rarely restricted to only a single type of relation. A policy network where actors share different types of information using different communication channels, or an interstate network of both conflict and trade ties are both examples of multiplex networks. The multiplex network requires constraints placed on the off-diagonal blocks of the adjacency matrix. By definition there is only one set of real-world actors represented by multiple nodes on the network, and interlayer ties are used to couple nodes representing the same actors across layers. These self-coupling interlayer ties are almost always treated as fixed. Specifically, when the relational data is represented as an adjacency matrix, the off-diagonal blocks of the matrix are identity matrices. This data structure is presented in Figure 4 using the German ChemG policy network data from Leifeld and Schneider (Reference Leifeld and Schneider2012). The identity matrices are considered when calculating network statistics but are not modeled as outcomes. Existing approaches to statistical inference on multiplex networks (e.g., Leifeld and Schneider, Reference Leifeld and Schneider2012; Heaney, Reference Heaney2014; Song, Reference Song2014) often treat one layer of the network as the “outcome” network that is allowed to vary, while fixing the other layers as exogenous predictors. Under certain conditions this approach can be reasonable, but elsewhere the underlying assumptions are violated. In these cases, where the two networks are mutually reinforcing, the inferential result will be subject to simultaneity bias on the estimated parameters.

3.1.2. Extensions to Leifeld and Schneider (Reference Leifeld and Schneider2012)

The question Leifeld and Schneider posed is what drives tie formation on these two policy communication networks. They base their argument on a transaction costs approach. As they argue, policy actors send information to each other as attempts at influencing their policy positions. Since establishing contact is costly, actors are more likely to utilize existing channels of communication as opportunity structures to attempt influence at a relatively low cost. Based on this logic, they derive a number of expectations. Here, I limit my multiplex extension to two of these hypotheses, as this is intended as an illustrative exercise.

First, Leifeld and Schneider argue that information exchange is likely to be reciprocated. To test this, they include a term for reciprocal tie formation in both of their political and scientific communication models. For both models, the term was statistically significant with a positive sign, indicating that reciprocal tie formation is as hypothesized a part of the underlying generative process for the observed networks. These monoplex, within-layer examinations, however, cannot account for the potential for more complex forms of reciprocity to occur across network layers. Consider that within a given actor-dyad, where two types of ties are possible for both directions, there are ten different observable combinations of information flow. Figure 5 outlines these combinations with the exception of no information exchange. Of these ten possible information-sharing scenarios between two actors, four include some form of reciprocal tie formation that cannot be modeled without considering cross-layer dependence. For example, when structure F is included as a model term, it captures the tendency for one kind of information exchange to be reciprocated through other means, which is expected in policy networks with actor specialization. The other three structures (G, H, and I) capture complex forms of conditional reciprocity. In the multiplex model I include these cross-layer dependence terms.

Figure 5. Census of dyad configurations in a directed duplex network.

Second, the authors argue that actors who exchange political information are more likely to exchange scientific information and vice versa. For this kind of mutual reinforcement between political and scientific communication, the authors modeled the two networks separately while including the other network as an exogenous predictor. However, specifying edge or node covariates that are simultaneous with the outcome variable in an ERGM as exogenous will result in the same type of simultaneity bias as when endogenous covariates are specified as exogenous in classical regression models. The multiplex approach allows for the inclusion of a dependence term that directly measures cross-layer reinforced tie formation. This term is visualized as structure E in Figure 5, which I include in my multiplex extension.

In this extension, I fit two models of the ChemG multiplex network. The first is a layer independence model that includes both network layers but no cross-layer dependence terms. The model specification here follows exactly those found in Leifeld and Schneider (Reference Leifeld and Schneider2012). The second is a cross-layer dependence model that, in addition to the within-layer terms from the layer independence model, adds cross-layer dependence terms described above. The full reciprocity term captured by structure I is omitted to avoid multicollinearity.

3.2.1. Node-colored Networks

A node-colored network comprises different types of actors, represented by “colors” that function as categorical labels for actor type (Kivelä et al., Reference Kivelä, Arenas, Barthelemy, Gleeson, Moreno and Porter2014). Nodes are grouped by their colors into different layers, which means that intralayer ties are those that exist between nodes of the same color, and interlayer ties between nodes with different colors. Because actors in node-colored networks never exist on more than one layer, interlayer ties are not used for coupling nodes across layers as they are in multiplex networks. Instead, they represent ties between two different real-world actors and are therefore substantively-meaningful relational outcomes that can be modeled. As such, whereas a multiplex network has as many types of ties as it does layers, a node-colored network with k colors has up to $((_2^k))$ types of ties. By this definition, bipartite networks are a specific case of node-colored networks where intralayer ties are assumed away. For example, political contributions modeled as a bipartite network assumes independence between different receivers and between different donors aside from shared cross-type partners (e.g., Heaney and Leifeld, Reference Heaney and Leifeld2018). Unless a node-colored network is truly bipartite, it is desirable to model it as a multilayer network by treating intralayer ties as outcomes.

Because the interlayer ties of node-colored networks are usually relational outcomes, systematic constraints are generally not placed on the off-diagonal blocks of node-colored networks. The exception to this is when the interlayer ties represent hierarchical affiliation ties (e.g., employment, citizenship) between nodes in different layers (Wang et al., Reference Wang, Robins, Pattison and Lazega2013). For example, a congressional member network and a congressional staffer network (Montgomery and Nyhan, Reference Montgomery and Nyhan2017) can be represented as a node-colored network with congressional members on one layer and their staffers on another. In these networks, if affiliations are generally stable and not of theoretical interest, the off-diagonal blocks may reasonably be fixed at their observed values.

3.2.2. Modeling Conflict Clustering across conflict Types

In this application I focus on the tendency for conflicts to cluster. I base my model on a study by Gleditsch et al. (Reference Gleditsch, Salehyan and Schultz2008), which examines the relationship between interstate conflict and the presence of civil wars in a country. More specifically, the authors examine MID onset for all politically-relevant dyads during the period of 1948–2000 and find that the presence of civil conflict in at least one state in the dyad predicts MID onset. This relationship should manifest as clustering of interstate and civil conflicts in the system. In their model specification, Gleditsch, Salehyan, and Schultz include the presence of civil conflict in the dyad as an exogenous predictor to MID onset. Specifying one type of conflict as the outcome and the others as exogenous predictors is the current standard in modeling the relationship between different types of conflicts, using the ERGM or otherwise. As illustrated using the Levantine conflict network earlier, what this approach misses is that the presence of different types of intrastate conflict is highly likely to be endogenous to interstate conflict through both simultaneity and confounding.

My multilayer extension departs from Gleditsch et al.'s (Reference Gleditsch, Salehyan and Schultz2008) study in a number of ways. First, using the multilayer approach, I model all potential conflicts in the global system (i.e., interstate, civil, and nonstate conflicts) as outcomes instead of focusing on politically-relevant interstate dyads. Second, to limit the methodological scope of my examination to something that is tractable for this paper, I focus on a single temporal cross-section of the global conflict system instead of a time-series. Specifically, I examine the period immediately following the end of the Cold War, from 1989 to 1995. This period was marked by a drastic change in the balance of power between states and nonstate actors in the periphery in favor of the latter (Kalyvas and Balcells, Reference Kalyvas and Balcells2010). The overall strategic environment at the time, ripe for potential conflict clustering across types, presents an apt opportunity for illustrating the importance of modeling interdependence across different conflict processes.^{Footnote 3}

Similar to the ChemG application, I fit a layer independence and a cross-layer dependence model. The two model specifications are identical except for how dependence across different types of conflicts is modeled. Specifically, both models include within-layer star terms to capture the tendency for within-type conflict clustering, but where the layer independence model approaches cross-type clustering by specifying one type of conflict as an exogenous predictor of another, the cross-layer dependence model uses cross-layer dependence star terms. Further, the cross-layer dependence model also includes a three-way clustering term that captures the tendency for all three types of conflict to cluster at the same time, which is illustrated in Figure 7. This term cannot be modeled in a straightforward manner using conventional approaches. A discussion of the data used, network creation procedure, and modeling steps is included in the online appendix.

Figure 7. Local network structure for three-way clustering: cross-layer three path.