Dependence Graphs Involving Attributes
Attribute variables can also be introduced into dependence graphs, either as exogenous predictors of network ties (social selection effects) or outcome variables from network structure (social influence effects). These types of models are presented in Chapters 8 and 9. There are some additional complexities in introducing exogenous predictors and different variable types into a dependence graph formulation that are beyond the
Figure 7.8. Sufficient subgraphs for directed Markov graph on four vertices.
scope of this chapter. Interested readers may consult Robins, Pattison, and Elliott (2001); Robins, Elliott, and Pattison (2001); and Robins and Pattison (2005).
The ERGM is a stochastic model in which the observed network is regarded as one realization from a probability distribution on the set of possible networks or graphs X on a fixed set of nodes. The probability of any particular realization x depends on parameters and statistics associated with certain configurations of x. These configurations are not arbitrarily chosen; on the contrary, they are seen as the outcomes of particular social processes that give rise to the network.
The process of formulating ERGMs for networks may be seen as comprising the following steps. First, a dependence structure is postulated in the form of dependence graph. Second, cliques are derived from the dependence graphs. Third, the Hammersley-Clifford theorem is applied to provide a factorization with parameters based on cliques. Fourth, some homogeneity constraints are proposed to identify the model. As a result, the general form of the model has parameters and statistics relating to the presence of various types of local network configurations.
Elaborations of this basic form of the model permit different forms of dependence assumptions among network variables to be incorporated in a hierarchy of model forms. In addition, exogenous variables (e.g., node attributes or geographic location) can be built in. This general approach to model building also permits the construction of models for patterns of social influence within a network (i.e., models that allow individual-level attributes to be predicted from network ties).
Work on other possible forms of dependence is ongoing. The advantages of a systematic approach of understanding dependence are more than theoretic: the type of data that need to be collected in a snowball sample to estimate a large-scale ERGM is informed by the dependence structure. Moreover, it is possible to identify a hierarchy of dependence assumptions and so investigate further model elaborations systematically. These issues are taken up in the final chapter of this book.