Abstract

We propose and analyze the problems of community goodness-of-fit and twosample testing for stochastic block models (SBM), where changes arise due to modification in community memberships of nodes. Motivated by practical applications, we consider the challenging sparse regime, where expected node degrees are constant, and the inter-community mean degree (b) scales proportionally to intracommunity mean degree (a). Prior work has sharply characterized partial or full community recovery in terms of a “signal-to-noise ratio” (SNR) based on a and b. For both problems, we propose computationally-efficient tests that can succeed far beyond the regime where recovery of community membership is even possible. Overall, for large changes, we need only SNR = O(1) whereas a naïve test based on community recovery with O(s) errors ? requires SNR = Conversely, in the small change regime, s n, via an information theoretic lower bound, we show that, surprisingly, no algorithm can do better than the naïve algorithm that first estimates the community up to O(s) errors and then detects changes. We validate these phenomena numerically on SBMs and on real-world datasets as well as Markov Random Fields where we only observe node data rather than the existence of links.