Reaching Kesten-Stigum Threshold in the Stochastic Block Model under Node Corruptions

TL;DR

This paper introduces a polynomial-time algorithm for community detection in the stochastic block model that remains effective at the Kesten-Stigum threshold despite adversarial node corruptions, extending to the Z2 synchronization problem.

Contribution

The paper presents the first robust polynomial-time algorithm achieving the Kesten-Stigum threshold under node corruptions in stochastic block models and extends techniques to Z2 synchronization.

Findings

Achieves weak recovery at the Kesten-Stigum threshold with node corruptions.

Extends robustness to Z2 synchronization problem.

Introduces a novel identifiability proof using Grothendieck norm.

Abstract

We study robust community detection in the context of node-corrupted stochastic block model, where an adversary can arbitrarily modify all the edges incident to a fraction of the $n$ vertices. We present the first polynomial-time algorithm that achieves weak recovery at the Kesten-Stigum threshold even in the presence of a small constant fraction of corrupted nodes. Prior to this work, even state-of-the-art robust algorithms were known to break under such node corruption adversaries, when close to the Kesten-Stigum threshold. We further extend our techniques to the $Z_2$ synchronization problem, where our algorithm reaches the optimal recovery threshold in the presence of similar strong adversarial perturbations. The key ingredient of our algorithm is a novel identifiability proof that leverages the push-out effect of the Grothendieck norm of principal submatrices.