2019/09/24 by Shirshendu Ganguly, Kyeongsik Nam, Ganguly, Shirshendu +1 · 1 citation
Mathematics · Physics and Astronomy · #Complex Network Analysis Techniques #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1909.11080
openalex publication_date 2019/09/24 · openalex created_date 2022/11/10 · openalex updated_date 2026/07/28
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 Poincaré inequality, Gaussian concentration for Lipschitz functions, and a central limit theorem. In addition, a new proof of a quantitative bound on the W1-Wasserstein distance to Erdős-Rényi 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.