ERGMs

Exponential Random Graph Models (ERGMs) are a family of statistical models for social networks. They treat the observed network as a single realization from a probability distribution over all possible graphs on a fixed set of nodes. In an ERGM, the probability of a graph x is given by P(X = x) = exp{ θ^T s(x) } / Z(θ), where s(x) is a vector of network statistics (for example, the number of edges, triangles, two-stars, or other configurational features) and θ is a corresponding vector of parameters. Z(θ) is a normalizing constant that sums exp{ θ^T s(y) } over all graphs y on the same node set.

These statistics encode features of interest such as network density, transitivity, degree distribution, and mixing with covariates. Estimation of θ typically uses maximum likelihood methods. Because Z(θ) is intractable for all but very small networks, practitioners rely on Markov chain Monte Carlo maximum likelihood estimation (MCMLE) or, as a faster alternative, maximum pseudolikelihood estimation (MPLE). Model assessment often involves simulating networks from the fitted model and comparing them with the observed network on various diagnostics.

Common challenges include model degeneracy, where the fitted model concentrates probability on highly unlikely graphs (e.g., empty or complete graphs). Careful specification of s(x) and checking goodness-of-fit are important. Extensions include dynamic ERGMs (TERGMs or STERGMs) for evolving networks and curved ERGMs that allow more flexible, non-linear parameterizations.

ERGMs are widely used in sociology, political science, and criminology to study how social structure, homophily, reciprocity, and other processes shape network formation. They provide a principled framework for inference about the mechanisms that generate relational data.