DERGMs: Degeneracy-restricted exponential random graph models

Exponential random graph models, or ERGMs, are a flexible class of models for networks. Recent work highlights difficulties related to the models' ill behavior, dubbed `degeneracy', such as most of the probability mass being concentrated on a very small subset of the parameter space. This behavior limits both the applicability of an ERGM as a model for real data and parameter estimation via the usual MCMC algorithms. To address this problem, we propose a new exponential family of models for random graphs that build on the standard ERGM framework. We resolve the degenerate model behavior by an interpretable support restriction. Namely, we introduce a new parameter based on the graph-theoretic notion of degeneracy, a measure of sparsity whose value is low in real-worlds networks. We prove this support restriction does not eliminate too many graphs from the support of an ERGM, and we also prove that degeneracy of a model is captured precisely by stability of its sufficient statistics. We show examples of ERGMs that are degenerate whose counterpart DERGMs are not, both theoretically and by simulations, and we test our model class on a set of real world networks.