### Test on Stochastic Block Model: Local Smoothing and Extreme Value Theory

WU Fan1, KONG Xinbing2, XU Chao3

1. 1. School of Mathematics, Southeast University, Nanjing 211819, China;
2. School of Statistics and Mathematics, Nanjing Audit University, Nanjing 211815, China;
3. School of Information Engineering, Nanjing Audit University, Nanjing 211815, China
• Received:2020-07-14 Revised:2021-03-22 Online:2022-08-25 Published:2022-08-02
• Contact: WU Fan,Email:wufan_wf@yeah.net
• Supported by:
Wu's work was supported by the National Natural Science Foundation of China under Grant No. 71971118; Kong's work was supported by the National Natural Science Foundation of China under Grant No. 71971118; Xu's work was supported by Major Natural Science Projects of Universities in Jiangsu Province under Grant No. 20KJA520002.

WU Fan, KONG Xinbing, XU Chao. Test on Stochastic Block Model: Local Smoothing and Extreme Value Theory[J]. Journal of Systems Science and Complexity, 2022, 35(4): 1535-1556.

In this paper, to obtain a consistent estimator of the number of communities, the authors present a new sequential testing procedure, based on the locally smoothed adjacency matrix and the extreme value theory. Under the null hypothesis, the test statistic 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, guaranteeing 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, the authors also propose a two-sample test for the stochastic block model with two observed adjacency matrices. Simulation studies justify the theory. The authors apply the proposed method to the political blog data set and find reasonable group structures.
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