Typical structure of sparse exponential random graph models

TL;DR

This paper analyzes the typical structure of sparse exponential random graph models, showing how fractional powers of sufficient statistics can prevent degeneracy and revealing their resemblance to Erdős–Rényi graphs with planted structures.

Contribution

It introduces a method to avoid degeneracy in sparse ERGMs using fractional powers and establishes the typical graph structure through rigorous mean-field and large deviation analyses.

Findings

Fractional powers of sufficient statistics prevent degeneracy in sparse ERGMs.

Typical samples resemble Erdős–Rényi graphs with planted cliques or bipartite structures.

Structural behavior is characterized in large deviations regime for homomorphism counts.

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 $F_k$. 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 na\"ive 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 Erd\H{o}s--R\'enyi 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 Erd\H{o}s--R\'enyi graph in the large deviations regime for excess $F_k$-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 [5] on the asymptotic solution for the associated entropic variational problem. A technical ingredient of independent interest is a stability form of Finner's generalized H\"older inequality.