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

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