Exponential-Family Models of Random Graphs: Inference in Finite-, Super-, and Infinite Population Scenarios

Exponential-family Random Graph Models (ERGMs) constitute a large statistical framework for modeling sparse and dense random graphs, short- and long-tailed degree distributions, covariates, and a wide range of complex dependencies. Special cases of ERGMs are generalized linear models (GLMs), Bernoulli random graphs, $\beta$-models, $p_1$-models, and models related to Markov random fields in spatial statistics and other areas of statistics. While widely used in practice, questions have been raised about the theoretical properties of ERGMs. These include concerns that some ERGMs are near-degenerate and that many ERGMs are non-projective. To address them, careful attention must be paid to model specifications and their underlying assumptions, and in which inferential settings models are employed. As we discuss, near-degeneracy can affect simplistic ERGMs lacking structure, but well-posed ERGMs with additional structure can be well-behaved. Likewise, lack of projectivity can affect non-likelihood-based inference, but likelihood-based inference does not require projectivity. Here, we review well-posed ERGMs along with likelihood-based inference. We first clarify the core statistical notions of "sample" and "population" in the ERGM framework, and separate the process that generates the population graph from the observation process. We then review likelihood-based inference in finite-, super-, and infinite-population scenarios. We conclude with consistency results, and an application to human brain networks