Title: Power enhancement and phase transitions for global testing of the mixed membership stochastic block model
Authors: Tracy Ke - Harvard University (United States) [presenting]
Abstract: The mixed-membership stochastic block model (MMSBM) is a common model for social networks. We are interested in testing $K=1$ versus $K>1$, where $K$ is the number of communities. We first study the degree-based chi-square test and the orthodox Signed Quadrilateral (oSQ) test. We reveal that these two test statistics target to 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. Unfortunately, for each test, there exists a parameter regime where its power is unsatisfactory. Next, we propose a Power Enhancement (PE) test by combining these two test statistics. We show that the PE test statistic converges in law to a chi-square distribution with 2 degrees of freedom, and that it has a better power compared with both the degree-based chi-square test and the oSQ test. To assess the optimality of global testing, we adopt the phase transition framework. We consider a random-membership MMSBM and discover an explicit quantity that defines the Region of Possibility and the Region of Impossibility. A test is called optimally adaptive if it successfully distinguishes any alternative hypothesis in the Region of Impossibility from any null hypothesis. We show that the PE test is optimally adaptive, but the chi-square test and the oSQ test are not.