Achieving Exact Cluster Recovery Threshold via Semidefinite Programming: Extensions

TL;DR

This paper extends the proven effectiveness of semidefinite programming (SDP) relaxations for exact community recovery in various stochastic block models, demonstrating their versatility and sharp threshold achievement.

Contribution

The paper generalizes previous results by showing SDP relaxations achieve sharp recovery thresholds in multiple complex community detection models, including unequal cluster sizes and censored models.

Findings

SDP achieves sharp thresholds in models with unequal cluster sizes.

SDP successfully recovers communities in models with multiple equal-sized clusters.

A sufficient condition for SDP to recover communities with outliers is provided.

Abstract

Resolving a conjecture of Abbe, Bandeira and Hall, the authors have recently shown that the semidefinite programming (SDP) relaxation of the maximum likelihood estimator achieves the sharp threshold for exactly recovering the community structure under the binary stochastic block model of two equal-sized clusters. The same was shown for the case of a single cluster and outliers. Extending the proof techniques, in this paper it is shown that SDP relaxations also achieve the sharp recovery threshold in the following cases: (1) Binary stochastic block model with two clusters of sizes proportional to network size but not necessarily equal; (2) Stochastic block model with a fixed number of equal-sized clusters; (3) Binary censored block model with the background graph being Erd\H{o}s-R\'enyi. Furthermore, a sufficient condition is given for an SDP procedure to achieve exact recovery for the general case of a fixed number of clusters plus outliers. These results demonstrate the versatility of SDP relaxation as a simple, general purpose, computationally feasible methodology for community detection.