Metastable wells in exponential random graph models

Exponential random graph models (ERGMs) are exponential tilts of Erdos–Renyi models where higher-order interactions promote the presence of small subgraphs like triangles. These models exhibit metastable behavior at low temperatures (strong interaction) and decompose as mixtures of phase measures or metastable wells. We present a variety of novel results for these metastable wells, including quantitative CLTs and sharp bounds on the Wasserstein distance between an ERGM in a metastable well and a corresponding Erdos–Renyi model. A common thread in these results is the careful analysis of relevant quantities under a natural Markov chain sampling algorithm.