Efficient Near-Optimal Testing of Community Changes in Balanced Stochastic Block Models

Aditya Gangrade, Praveen Venkatesh, Bobak Nazer, Venkatesh Saligrama

Abstract: We propose and analyze the problems of community goodness-of-fit and two-sample 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 SBM regime, where degree-per-node is constant, and the inter-community mean degree (b) scales proportionally to intra-community mean degree (b). Prior work has sharply characterized partial or full recovery of SBM community 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, s >> sqrt(n), we need only SNR = O(1) whereas a na\"ive test based on community recovery with O(s) errors requires SNR = Theta(log n). Conversely, in the small change regime, s << sqrt(n), via an information theoretic lower bound, we show surprisingly that no algorithm can do better than the na\"ive algorithm that first recovers the full community 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.

CMT Num: 5460