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 based on 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; conversely, we prove its divergence when the model is underfitted. This dual theoretical foundations enable two computationally efficient sequential testing algorithms to consistently determine the number of communities without prior knowledge. Numerical experiments on simulated and real-world multi-layer networks demonstrate the accuracy and efficiency of our approaches in estimating the number of communities.