Typical structure of sparse exponential random graph models

Abstract

We consider general exponential random graph models (ergms) where the sufficient statistics are functions of homomorphism counts for a fixed collection of simple graphs ${\mathit{F}}_{\mathit{k}}$. Whereas previous work has shown a degeneracy phenomenon in dense ergms, we show this can be cured by raising the sufficient statistics to a fractional power. We rigorously establish the naïve mean-field approximation for the partition function of the corresponding Gibbs measures, and in case of “ferromagnetic” models with vanishing edge density show that typical samples resemble a typical Erdős–Rényi graph with a planted clique and/or a planted complete bipartite graph of appropriate sizes. We establish such behavior also for the conditional structure of the Erdős–Rényi graph in the large deviations regime for excess ${\mathit{F}}_{\mathit{k}}$-homomorphism counts. These structural results are obtained by combining quantitative large deviation principles, established in previous works, with a novel stability form of a result of (Adv. Math. 319 (2017) 313–347) on the asymptotic solution for the associated entropic variational problem. A technical ingredient of independent interest is a stability form of Finner’s generalized Hölder inequality.