Linear Programming and Community Detection

TL;DR

This paper compares linear programming and semidefinite programming relaxations for community detection in random graphs, showing LP's limitations in certain regimes where SDP succeeds.

Contribution

It provides a theoretical analysis of LP relaxation performance, revealing its phase transition behavior and limitations compared to SDP in community detection.

Findings

SDP recovers communities in logarithmic degree regime

LP fails to recover communities in the same regime

LP exhibits a phase transition from recovery to non-recovery

Abstract

The problem of community detection with two equal-sized communities is closely related to the minimum graph bisection problem over certain random graph models. In the stochastic block model distribution over networks with community structure, a well-known semidefinite programming (SDP) relaxation of the minimum bisection problem recovers the underlying communities whenever possible. Motivated by their superior scalability, we study the theoretical performance of linear programming (LP) relaxations of the minimum bisection problem for the same random models. We show that unlike the SDP relaxation that undergoes a phase transition in the logarithmic average-degree regime, the LP relaxation exhibits a transition from recovery to non-recovery in the linear average-degree regime. We show that in the logarithmic average-degree regime, the LP relaxation fails in recovering the planted bisection with high probability.