Goodness-of-fit test for multi-layer stochastic block models

Community detection in multi-layer networks is a fundamental task in complex network analysis across various areas like social, biological, and computer sciences. However, most existing algorithms assume that the number of communities is known in advance, which is usually impractical for real-world multi-layer networks. To address this limitation, we develop a novel goodness-of-fit test for the popular multi-layer stochastic block model. The test statistic is derived from a normalized aggregation of layer-wise adjacency matrices. Under the null hypothesis that a candidate community count is correct, we establish the asymptotic normality of the test statistic using recent advances in random matrix theory. This theoretical foundation enables a computationally efficient sequential testing algorithm to determine the number of communities. Numerical experiments on simulated and real-world multi-layer networks demonstrate the accuracy and efficiency of our approach in estimating the number of communities.