Sub-critical Exponential random graphs: concentration of measure and some applications

TL;DR

This paper establishes new concentration of measure results for sub-critical exponential random graph models, including inequalities and a CLT, enhancing understanding of their probabilistic behavior.

Contribution

It provides the first comprehensive concentration results for ERGMs in the sub-critical phase, including Poincaré inequalities, Gaussian concentration, and a CLT, with new proof techniques.

Findings

Proved Poincaré inequality for ERGMs in the sub-critical phase.

Established Gaussian concentration for Lipschitz functions of ERGMs.

Derived a CLT for ERGM observables.

Abstract

The exponential random graph model (ERGM) is a central object in the study of clustering properties in social networks as well as canonical ensembles in statistical physics. Despite some breakthrough works in the mathematical understanding of ERGM, most notably in (Bhamidi, Bresler, Sly, 2011) through the analysis of a natural Heat-bath Glauber dynamics, and in (Chatterjee, Diaconis, 2013), (Eldan, Gross, 2018) via a large deviation theoretic perspective, several basic questions have remained unanswered owing to the lack of exact solvability unlike the much studied Curie-Weiss model (Ising model on the complete graph). In this paper, we establish a series of new concentration of measure results for the ERGM throughout the entire sub-critical phase, including a Poincar\'e inequality, Gaussian concentration for Lipschitz functions, and a central limit theorem. In addition, a new proof of a quantitative bound on the $W_1-$Wasserstein distance to Erd\H{o}s-R\'enyi graphs, previously obtained in (Reinert, Ross, 2017), is also presented. The arguments rely on translating temporal mixing properties of Glauber dynamics to static spatial mixing properties of the equilibrium measure and have the potential of being useful in proving similar functional inequalities for other Gibbsian systems beyond the perturbative regime.