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

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 by Bhamidi, Bresler, and Sly [Ann. Appl. Probab. 21 (2011), pp. 2146–2170] through the analysis of a natural Heat-bath Glauber dynamics, and by Chatterjee and Diaconis [Ann. Statist. 41 (2013), pp. 2428–2461; Eldan [Geom. Funct. Anal. 28 (2018), pp. 1548–1596; and Eldan and Gross [Ann. Appl. Probab. 28 (2018), pp. 3698–3735] 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 Poincaré inequality, Gaussian concentration for Lipschitz functions, and a central limit theorem. In addition, a new proof of a quantitative bound on the W 1 − W_1- Wasserstein distance to Erdős-Rényi graphs, previously obtained by Reinert and Ross [Ann. Appl. Probab. 29 (2019), pp. 3201–3229], 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.