Community Detection with a Subsampled Semidefinite Program

TL;DR

This paper investigates a subsampled semidefinite programming approach for community detection in stochastic block models, confirming its statistical effectiveness and computational advantages.

Contribution

It provides a positive resolution to a conjecture about the statistical limits of subsampled SDP methods for two-community stochastic block models.

Findings

Subsampled SDP achieves optimal statistical detection thresholds.

The method offers significant computational savings over full SDP.

Theoretical analysis confirms the conjecture for balanced two-community models.

Abstract

Semidefinite programming is an important tool to tackle several problems in data science and signal processing, including clustering and community detection. However, semidefinite programs are often slow in practice, so speed up techniques such as sketching are often considered. In the context of community detection in the stochastic block model, Mixon and Xie \cite{mixon2020sketching} have recently proposed a sketching framework in which a semidefinite program is solved only on a subsampled subgraph of the network, giving rise to significant computational savings. In this short paper, we provide a positive answer to a conjecture of Mixon and Xie about the statistical limits of this technique for the stochastic block model with two balanced communities.