Whether a particular model specification is appropriate in a given context is in part an empirical question, regardless of how coherent the underpinning theoretical framework. Almost certainly, therefore, we will come to entertain different dependence assumptions and different models within the broad ERGM class as we work with new forms of network data, and as we gain experience in understanding the strengths and limitations of particular model specifications.

Butts (2006) proposed a form of dependence termed “reciprocal path dependence,” the idea that two tie-variables are conditionally dependent if their presence creates a directed cycle of any length in the graph. Statistics for the resulting homogeneous model are the counts of cycles of each length in the graph, the so-called cycle census.

Although closure is testified as being a central mechanism in tie formation, we saw that too much closure is responsible for the instabilities of the pure Markov model (Figures 6.5 and 6.6). Robins, Pattison, Snid- jers, and Wang (2009) also argued that as there are costs associated with establishing and maintaining relationships with people, there may be additional costs associated with maintaining big groups. Consequently, both from the perspective of creating models that generate realistic graphs and based on theoretical considerations, tendencies for nonclosure might be considered. Robins, Pattison, Snijders, and Wang (2009) proposed configurations such as edge-triangles and bow ties, representing particular mechanisms that counteracted closure (Figure 6.13). The edge-triangle configuration is interpreted as a form of brokerage from the central node (the one with three ties). Similarly, the bow tie may be seen to represent partially overlapping group membership, where there is one node that belongs to two triangles. Here, the triangle is interpreted as the simplest form of a group. There is ongoing work examining the complexities of these additional configurations.

In these cases, a means of building model specifications from a particular construal about dependencies has provided a valuable path to model building. Equally important, though, in an empirical setting, is the means of evaluating how adequately a model has been specified. This is an issue that is taken up in Chapter 12.

Conclusion

The central premise of the ERGM modeling endeavor is its dual interpretation as a model for ties and for the graph. This duality mirrors both theoretical considerations - the way individuals form ties but at the same time are constrained and affected by structure - and technical considerations - we provide a model for tie-variables but conditional on the rest of the graph. In this chapter, we define different model specifications in terms of localized structures and attempt to interpret these in terms of substantively interesting effects. In Chapter 7, we show how the dual consideration for ties and overall structure give rise to and motivate the inclusion of the subgraph counts presented in this chapter. We explain how the dependence assumptions relate to subgraphs and how the probability of a tie is informed by neighboring structures.