A fourth moment theorem for estimating subgraph counts

 Given a large graph, one is often interested in efficiently estimating various local statistics. In this talk, we'll discuss the distribution of one possible estimator arising from counting monochromatic subgraphs in random vertex colorings. We focus on the asymptotic normality of these counts, particularly for monochromatic triangles, and provide new, local influence-based necessary and sufficient conditions.  The conditions we obtain combine ideas from Boolean analysis as well as classical fourth-moment theorems originating from normal approximation results in the Wiener space.