A New Spectral‐Based Goodness‐of‐Fit for Stochastic Block Models With Strong Power

ABSTRACT The stochastic block model stands as one of the most extensively utilized network models for the purpose of community detection. In recent years, testing the number of communities and testing the latent community label are two important problems for stochastic block models, which help us recover the precise community structure so that the network can be better utilized to analyse real problems. Based on the linear spectral statistic of the normalized adjacency matrix, we propose a new test statistic with strong power. We use random matrix theory techniques to prove that the null distribution of our test statistic is the standard normal distribution. Furthermore, we extend our test statistic to degree‐corrected stochastic block models. Simulation experiments demonstrate the effectiveness of our test statistic, while real‐world data studies validate its practical applicability.