Motif estimation via subgraph sampling: The fourth-moment phenomenon

Network sampling is an indispensable tool for understanding features of large complex networks where it practically impossible to search over the entire graph. In this paper, we develop a framework statistical inference counting network motifs, such as edges, triangles and wedges, in widely used subgraph model, each vertex sampled independently, induced by vertices observed. We derive necessary sufficient conditions consistency asymptotic normality natural Horvitz–Thompson (HT) estimator, which can be constructing confidence intervals hypothesis testing motif counts based on particular, show that HT estimator exhibits interesting fourth-moment phenomenon, asserts (appropriately centered rescaled) converges distribution standard normal whenever its 3 (the distribution). As consequence, exact thresholds various graph ensembles, sparse graphs with bounded degree, Erd?s–Rényi random graphs, regular dense graphons.