{"title":"Power enhancement and phase transitions for global testing of the mixed membership stochastic block model","authors":"Louis V. Cammarata, Z. Ke","doi":"10.3150/22-bej1519","DOIUrl":null,"url":null,"abstract":"The mixed-membership stochastic block model (MMSBM) is a common model for social networks. Given an $n$-node symmetric network generated from a $K$-community MMSBM, we would like to test $K=1$ versus $K>1$. We first study the degree-based $\\chi^2$ test and the orthodox Signed Quadrilateral (oSQ) test. These two statistics estimate an order-2 polynomial and an order-4 polynomial of a\"signal\"matrix, respectively. We derive the asymptotic null distribution and power for both tests. However, for each test, there exists a parameter regime where its power is unsatisfactory. It motivates us to propose a power enhancement (PE) test to combine the strengths of both tests. We show that the PE test has a tractable null distribution and improves the power of both tests. To assess the optimality of PE, we consider a randomized setting, where the $n$ membership vectors are independently drawn from a distribution on the standard simplex. We show that the success of global testing is governed by a quantity $\\beta_n(K,P,h)$, which depends on the community structure matrix $P$ and the mean vector $h$ of memberships. For each given $(K, P, h)$, a test is called $\\textit{ optimal}$ if it distinguishes two hypotheses when $\\beta_n(K, P,h)\\to\\infty$. A test is called $\\textit{optimally adaptive}$ if it is optimal for all $(K, P, h)$. We show that the PE test is optimally adaptive, while many existing tests are only optimal for some particular $(K, P, h)$, hence, not optimally adaptive.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5000,"publicationDate":"2022-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bernoulli","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3150/22-bej1519","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 3
Abstract
The mixed-membership stochastic block model (MMSBM) is a common model for social networks. Given an $n$-node symmetric network generated from a $K$-community MMSBM, we would like to test $K=1$ versus $K>1$. We first study the degree-based $\chi^2$ test and the orthodox Signed Quadrilateral (oSQ) test. These two statistics estimate an order-2 polynomial and an order-4 polynomial of a"signal"matrix, respectively. We derive the asymptotic null distribution and power for both tests. However, for each test, there exists a parameter regime where its power is unsatisfactory. It motivates us to propose a power enhancement (PE) test to combine the strengths of both tests. We show that the PE test has a tractable null distribution and improves the power of both tests. To assess the optimality of PE, we consider a randomized setting, where the $n$ membership vectors are independently drawn from a distribution on the standard simplex. We show that the success of global testing is governed by a quantity $\beta_n(K,P,h)$, which depends on the community structure matrix $P$ and the mean vector $h$ of memberships. For each given $(K, P, h)$, a test is called $\textit{ optimal}$ if it distinguishes two hypotheses when $\beta_n(K, P,h)\to\infty$. A test is called $\textit{optimally adaptive}$ if it is optimal for all $(K, P, h)$. We show that the PE test is optimally adaptive, while many existing tests are only optimal for some particular $(K, P, h)$, hence, not optimally adaptive.
期刊介绍:
BERNOULLI is the journal of the Bernoulli Society for Mathematical Statistics and Probability, issued four times per year. The journal provides a comprehensive account of important developments in the fields of statistics and probability, offering an international forum for both theoretical and applied work.
BERNOULLI will publish:
Papers containing original and significant research contributions: with background, mathematical derivation and discussion of the results in suitable detail and, where appropriate, with discussion of interesting applications in relation to the methodology proposed.
Papers of the following two types will also be considered for publication, provided they are judged to enhance the dissemination of research:
Review papers which provide an integrated critical survey of some area of probability and statistics and discuss important recent developments.
Scholarly written papers on some historical significant aspect of statistics and probability.