Minimax Rates for Robust Community Detection

TL;DR

This paper introduces an efficient, robust algorithm for community detection in stochastic block models that tolerates adversarial corruptions, achieving near-optimal error bounds similar to the uncorrupted case.

Contribution

It presents a novel semidefinite programming approach that enhances robustness against adversarial corruptions in community detection.

Findings

Algorithm tolerates an $$-fraction of corruptions with error $O() + e^{-rac{C}{2}}$

Bounds match minimax rates for SBM without corruptions

Algorithms are doubly-robust under semi-random noise models

Abstract

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 $\epsilon$-fraction of corruptions and achieves error $O(\epsilon) + e^{-\frac{C}{2} (1 \pm o(1))}$ where $C = (\sqrt{a} - \sqrt{b})^2$ 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 $\mathbb{Z}_2$-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.