Semidefinite Programming for Community Detection with Side Information

TL;DR

This paper develops an efficient semidefinite programming approach for community detection that leverages side information, achieving optimal recovery thresholds and demonstrating practical effectiveness through simulations.

Contribution

It introduces a novel SDP formulation for community detection with side information, matching the optimal recovery threshold of maximum likelihood methods.

Findings

SDP achieves the same exact recovery threshold as maximum likelihood with side information.

The proposed SDP is computationally efficient and asymptotically accurate.

Simulations confirm the method's effectiveness on modest-sized graphs.

Abstract

This paper produces an efficient Semidefinite Programming (SDP) solution for community detection that incorporates non-graph data, which in this context is known as side information. SDP is an efficient solution for standard community detection on graphs. We formulate a semi-definite relaxation for the maximum likelihood estimation of node labels, subject to observing both graph and non-graph data. This formulation is distinct from the SDP solution of standard community detection, but maintains its desirable properties. We calculate the exact recovery threshold for three types of non-graph information, which in this paper are called side information: partially revealed labels, noisy labels, as well as multiple observations (features) per node with arbitrary but finite cardinality. We find that SDP has the same exact recovery threshold in the presence of side information as maximum likelihood with side information. Thus, the methods developed herein are computationally efficient as well as asymptotically accurate for the solution of community detection in the presence of side information. Simulations show that the asymptotic results of this paper can also shed light on the performance of SDP for graphs of modest size.