Power enhancement and phase transitions for global testing of the mixed membership stochastic block model

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 ${\mathit{\chi }}^2$ 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 ${\mathit{\beta }}_n(K,P,h)$, 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 optimal if it distinguishes two hypotheses when ${\mathit{\beta }}_n(K,P,h)\to \mathrm{\infty }$. A test is called 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.