How Robust are Reconstruction Thresholds for Community Detection?

TL;DR

This paper investigates the robustness of community detection thresholds in the stochastic block model under semirandom adversarial modifications, revealing that such changes can make detection strictly harder and that semidefinite programming algorithms remain effective.

Contribution

It introduces a semirandom model for community detection, demonstrating that helpful adversarial changes can shift thresholds and showing SDP-based algorithms' robustness in this setting.

Findings

Helpful modifications raise the detection threshold.

SDP algorithms remain effective in semirandom models.

Semirandom models explain practical algorithm preferences.

Abstract

The stochastic block model is one of the oldest and most ubiquitous models for studying clustering and community detection. In an exciting sequence of developments, motivated by deep but non-rigorous ideas from statistical physics, Decelle et al. conjectured a sharp threshold for when community detection is possible in the sparse regime. Mossel, Neeman and Sly and Massoulie proved the conjecture and gave matching algorithms and lower bounds. Here we revisit the stochastic block model from the perspective of semirandom models where we allow an adversary to make `helpful' changes that strengthen ties within each community and break ties between them. We show a surprising result that these `helpful' changes can shift the information-theoretic threshold, making the community detection problem strictly harder. We complement this by showing that an algorithm based on semidefinite programming (which was known to get close to the threshold) continues to work in the semirandom model (even for partial recovery). This suggests that algorithms based on semidefinite programming are robust in ways that any algorithm meeting the information-theoretic threshold cannot be. These results point to an interesting new direction: Can we find robust, semirandom analogues to some of the classical, average-case thresholds in statistics? We also explore this question in the broadcast tree model, and we show that the viewpoint of semirandom models can help explain why some algorithms are preferred to others in practice, in spite of the gaps in their statistical performance on random models.