Hypothesis testing in sparse weighted stochastic block model

Community detection is a fundamental task in network data mining. Various algorithms have been proposed to detect the communities of a network. However, the output of these algorithms are meaningful only if community structure exists in the network. It is necessary to statistically test the presence of community structure before applying any community detection algorithms. The existing algorithms or testing procedures mainly focus on unweighted graph, that is, the edge presence or absence is coded as a binary variable. However, most real-world networks have weights. Recently, several algorithms have been devised to detect communities in weighted networks. In this paper, we consider the fundamental problem whether community structure exists in a weighted network. Specifically, we propose a test statistic based on the number of weighted triangles and edges, derive its limiting distribution under the null hypothesis and analyze its power. The simulation results and real data application show that the proposed test can achieve high power.