Test On Stochastic Block Model:local Smoothing And Extreme Value Theory

The stochastic block model is widely used in modeling the community structures in network data.In this paper,to obtain a consistent estimator of the number of com-munities,we present a new sequential testing procedure,based on the locally smoothed adjacency matrix and the extreme value theory.Under the null hypothesis that the com-munity number is equal to a predetermined number,the test statistics converges to the Type Ⅰ extreme value distribution,and otherwise,it explodes fast and the divergence rate could even reach n in the strong signal case where n is the size of the network,guaran-teeing high detection power.This method is simple to use and serves as an alternative approach to the novel one in Lei(2016)using random matrix theory.To detect the change of the community structure,we also propose a two-sample test for the stochastic block model with two observed adjacency matrices.Simulation studies justify the theory.We apply our method to the political blog data set and find reasonable group structures.