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

Published in *Statistical Science*, 2019

Exponential-family Random Graph Models (ERGMs) constitute a broad statistical framework for modeling both sparse and dense random graphs with features such as short- or long-tailed degree distributions, covariate eﬀects, and a wide range of complex dependencies. Special cases of ERGMs include network equivalents of generalized linear models (GLMs), Bernoulli random graphs, β-models, p 1 -models, and models related to the Markov random ﬁelds employed in spatial statistics and image analysis. While ERGMs are widely used in practice, questions have been raised about their theoretical properties. These include concerns that some ERGMs are near-degenerate and that many ERGMs are non-projective. To address such questions, careful attention must be paid to model speciﬁcations and their underlying assumptions, and to the inferential settings in which 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 for typical ERGM settings. We ﬁrst clarify the core statistical notions of “sample” and “population” in the ERGM framework, separating 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 illustrative application to human brain networks.

**Keywords:**social network, exponential-family random graph model, ERGM, model degeneracy, projectivity.

Recommended citation:

Schweinberger, M., Krivitsky, P. N., Butts, C. T., and **Stewart, J.** " Exponential-Family Models of Random Graphs: Inference in Finite-, Super-, and Infinite-Population Scenarios " *Statistical Science*, to appear (2019).