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

Abstract

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, β-models, p 1models, 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, neardegeneracy can affect simplistic ERGMs lacking structure, but wellposed 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

Link predictions for incomplete network data with outcome misclassification

“…Our method complements existing parametric network models by allowing misclassified outcome data and unreported/latent links. As many parametric network models (eg, ERGM) are considered as an incomplete-data generating process, 44 our jointly modeling of misclassification and network is as well. We extend the basis of existing network models from likelihood to expected likelihood to account for the misclassified outcome variables.…”

Testing biological network motif significance with exponential random graph models

“…They take into account that networks are finite. Indeed, far from requiring very large networks to fit the requirements of mean-field theories, they are dependent on network size and do not scale consistently to infinity (Rolls et al, 2013;Shalizi and Rinaldo, 2013;Schweinberger et al, 2019) -a property that can be used to estimate population size from network samples (Rolls and Robins, 2017). They can handle modular organization or community structure (Fronczak et al, 2013), samples from larger networks (Handcock and Gile, 2010;, and missing data (Robins et al, 2004;.…”

Selection of Exponential-Family Random Graph Models via Held-Out Predictive Evaluation (HOPE)

“…However, these criteria rely on a number of theoretical assumptions that are frequently problematic in a network modeling context. First, edge variables in typical network models are nonindependent, making it difficult to determine the effective sample size needed for size-corrected AIC and BIC calculations (Hunter et al, 2008a); indeed, at this time the theoretical justification for these criteria is unclear in the case of models for single networks with dyadic dependence (though see Kolaczyk & Krivitsky, 2015;Schweinberger et al, 2017, for some possible directions). Second, likelihood calculations for complex ERGMs rely on stochastic approximations (e.g., bridge sampling) that become expensive for large networks and where high precision is needed.…”

“…This is related to the inconsistency of dependence models under naive subsampling, when the presence of unmeasured vertices is not accounted for(Shalizi & Rinaldo, 2013); procedures that do allow consistent estimation are discussed bySchweinberger et al (2017).…”