Modeling Network Evolution Using Graph Motifs

Network structures are extremely important to the study of political science. Much of the data in its subfields are naturally represented as networks. This includes trade, diplomatic and conflict relationships. The social structure of several organization is also of interest to many researchers, such as the affiliations of legislators or the relationships among terrorist. A key aspect of studying social networks is understanding the evolutionary dynamics and the mechanism by which these structures grow and change over time. While current methods are well suited to describe static features of networks, they are less capable of specifying models of change and simulating network evolution. In the following paper I present a new method for modeling network growth and evolution. This method relies on graph motifs to generate simulated network data with particular structural characteristic. This technique departs notably from current methods both in form and function. Rather than a closed-form model, or stochastic implementation from a single class of graphs, the proposed “graph motif model” provides a framework for building flexible and complex models of network evolution. The method is computationally based, relying on graph theoretic and machine learning techniques to grow networks. The paper proceeds as follows: first a brief review of the current literature on network modeling is provided to place the graph motif model in context. Next, the graph motif model is introduced, and a simple example is provided. As a proof of concept, three classic random graph models are recovered using the graph motif modeling method: the Erdős-Rènyi binomial random graph, the Watts-Strogatz “small world” model, and the Barabási-Albert preferential attachment model. In the final section I discuss the results of these simulations and subsequent advantage and disadvantages presented by using this technique to model social networks. 1 ar X iv :1 10 5. 09 02 v1 [ st at .M E ] 4 M ay 2 01 1 The study of networks is one of the most interdisciplinary fields in contemporary scholarship. With its origins in graph theory, and migration to the social science largely via sociology (Freeman, 2004), many political scientists have discovered the value of these methods. The primary reason for this is that much of the data relevant to political science can be represented as a network. In network science the primary unit of analysis is the edge, or link between two actors. Likewise, many subfields in political science study interactions and organizations that are naturally modeled as a network. At the macro-level in international relations this includes trade, diplomatic and conflict relationships, while at micro-level networks can be used to study the structure of terrorist organizations. For comparative politics this may include government coalitions networks, or party affiliations. Finally, in American politics this can include campaign finance contributions or legislative co-sponsorship networks. Given the breadth of possible applications, network analysis is a growing methodological subfield within political science. The work within this niche can be crudely divided into two applications of network analysis: structurally descriptive, or networks as dependent variables. Research in the former category has a relatively long and rich history, with well-established methods for describing the structure of networks. 1 These methods include measures of actor centrality, whereby the relative position or role of actors is based on their number and type of edges within the network. In political science these methods have most frequently been applied to international relations studies. For example, structurally descriptive methods have been used to illustrate how social capital transferred through inter-governmental organizations’ membership networks can create conflicts between states (Hafner-Burton and Montgomery, 2006). Centrality-based methods have also been used to identify key actors in the conflict in Chechnya, and and these central actors vary in type, i.e., civilian, military, etc (Hämmerli et al., 2006). Additionally, structural similarities within a network of conflict dyads among countries have even been used to describe international conflict patterns (Maoz et al., 2006). In American politics, structurally descriptive methods have been used to identify the most influential members of the U.S. Congress based on their co-sponsorship networks (Fowler, 2006). In many applications of network analysis in political science authors attempt to explain a behavior or observed outcome from the structural features of the network being studied. In the example of the co-sponsorship network study, the position of the legislators inferred by their centrality is used to predict the number of legislative amendments proposed by members. This type of inference is common in descriptive network analysis. It is difficult, however, to know the direction of causality. 1The most complete reference on statistically descriptive methods is Social Network Analysis: Methods and Applications, by Wasserman and Foust (Wasserman and Faust, 1994).