  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.
