Minimax Rates for Robust Community Detection
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In this work, we study the problem of community detection in the stochastic block model with adversarial node corruptions. Our main result is an efficient algorithm that can tolerate an ϵ-fraction of corruptions and achieves error O(ϵ) + e^-C/2 (1 ± o(1)) where C = (√(a) - √(b))^2 is the signal-to-noise ratio and a/n and b/n are the inter-community and intra-community connection probabilities respectively. These bounds essentially match the minimax rates for the SBM without corruptions. We also give robust algorithms for ℤ_2-synchronization. At the heart of our algorithm is a new semidefinite program that uses global information to robustly boost the accuracy of a rough clustering. Moreover, we show that our algorithms are doubly-robust in the sense that they work in an even more challenging noise model that mixes adversarial corruptions with unbounded monotone changes, from the semi-random model.
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