  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
