Community detection in sparse networks via Grothendieck's inequality

TL;DR

This paper introduces a flexible semidefinite programming approach for community detection in sparse networks, leveraging Grothendieck's inequality and randomness to achieve high accuracy in recovering community structures.

Contribution

It presents a novel method using Grothendieck's inequality to prove the consistency of semidefinite programs in sparse network community detection, allowing arbitrary accuracy.

Findings

Semidefinite programs can recover communities with minimal misclassification in sparse networks.

The method is applicable to various stochastic network models.

Achieves arbitrary small error rates in community detection.

Abstract

We present a simple and flexible method to prove consistency of semidefinite optimization problems on random graphs. The method is based on Grothendieck's inequality. Unlike the previous uses of this inequality that lead to constant relative accuracy, we achieve any given relative accuracy by leveraging randomness. We illustrate the method with the problem of community detection in sparse networks, those with bounded average degrees. We demonstrate that even in this regime, various simple and natural semidefinite programs can be used to recover the community structure up to an arbitrarily small fraction of misclassified vertices. The method is general; it can be applied to a variety of stochastic models of networks and semidefinite programs.