Given a large network 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 a 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.