Exponential Random Graph Models (ERGM/p*)

Exponential random graph models (ERGM/p*) are a class of statistical models principally used in the analysis of social networks but are more generally applicable to a variety of relational data sets. The models have analogues in spatial statistics and quantum mechanics. There are various classes of the models, and once a particular model specification is chosen and parameter values selected, they can be used to simulate probability distributions of graphs. More pertinently, for a particular model specification applied to empirical network data, parameters can be estimated to draw conclusions about the prevalence of small network patterns in the data and to permit inference about the type of processes that may have produced the network structure.

For a given number of nodes, an exponential random graph model establishes a probability distribution across the entire set of possible graphs. At the same time, a model provides an estimated probability of a network tie between two nodes, conditional on the presence of other ties in a neighborhood of possible ties. The neighborhood is determined by hypothesized dependencies among network ties, as discussed below, and sets up small patterns (subgraphs) of network ties that affect the presence or absence of a given tie. These patterns are usually termed configurations. Broadly, the model can be thought of as representing a network built up by cumulation of these particular configurations. For instance, in directed networks, reciprocated ties are often included as a configuration in exponential random graph models because reciprocation is an important structural feature of human social networks. In such cases, contingent on other effects in the model, networks with more reciprocated ties are more likely (have higher probability in a graph distribution), and a particular network tie is more likely if its reciprocated tie is in place.

The models are parameterized in terms of configurations, with a positive parameter indicating that a configuration occurs in the network more commonly than expected by chance. Configurations in a model are usually chosen with a view to important theoretical and empirical themes in the social network literature, including activity and popularity effects (reflected in degree distributions) and network closure effects (reflected in the amount of clustering or triangulation in the network). So for undirected networks, configurations often include stars and triangles, and for directed networks, reciprocated ties, in- and out-stars, and transitive and cyclic triads. Star-based parameters are used to model the degree distribution, and triangulation-based parameters to model network closure.

Often, all nodes in the network are treated as equivalent (homogeneous models), so that the labeling of the node does not matter. In that case, configurations of isomorphic subgraphs are accorded the same parameter (for example, all reciprocated ties are assumed to have an equal effect on graph probability.) However, it is also possible to incorporate effects related to nodal attributes, including activity and popularity for nodes with particular attributes, and homophily processes based on shared attributes. Therefore, attributes introduce nodal heterogeneity into the model. If a node is uniquely distinctive, however, then it can be treated as exogenous, with ties associated with it used as predictors of other network ties but not themselves modeled.

...