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Title: Optimal adaptivity of signed-polygon statistics for network testing Authors:  Tracy Ke - Harvard University (United States) [presenting]
Abstract: Given a symmetric social network, the focus is on testing whether it has only one community or multiple communities. The desired tests should (a) accommodate severe degree heterogeneity, (b) accommodate mixed-memberships, (c) have a tractable null distribution, and (d) adapt automatically to different levels of sparsity, and achieve the optimal phase diagram. How to find such a test is a challenging problem. We propose the Signed Polygon as a class of new tests. Fixing $m>2$, for each $m$-gon in the network, we define a score using the centered adjacency matrix. The sum of such scores is then the m-th order Signed Polygon statistic. The Signed Triangle (SgnT) and the Signed Quadrilateral (SgnQ) are special examples of the Signed Polygon. We show that both the SgnT and SgnQ tests satisfy (a)-(d), and especially, they work well for both very sparse and less sparse networks. The analysis of the SgnT and SgnQ tests is delicate and tedious, and the main reason is that we need a unified proof that covers a wide range of sparsity levels and a wide range of degree heterogeneity. For lower bound theory, we use a phase transition framework, which includes the standard minimax argument, but is more informative.