Semidefinite Programming for the Asymmetric Stochastic Block Model

TL;DR

This paper investigates the limitations of existing semidefinite programming approaches for community detection in asymmetric stochastic block models and proposes a new SDP formulation tailored for the asymmetric case.

Contribution

It demonstrates the failure of symmetric SDP in asymmetric models and introduces a new SDP formulation designed to handle asymmetry in community detection.

Findings

Sym-SDP fails in certain asymmetric cases despite information-theoretic feasibility.

A geometric interpretation explains sym-SDP's failure in asymmetric settings.

Proposes a new SDP formulation for asymmetric stochastic block models.

Abstract

We consider semidefinite programming (SDP) for the binary stochastic block model with equal-sized communities. Prior work of Hajek, Wu, and Xu proposed an SDP (sym-SDP) for the symmetric case where the intra-community edge probabilities are equal, and showed that the SDP achieves the information-theoretic threshold for exact recovery under the symmetry assumption. A key open question is whether SDPs can be used to achieve exact recovery for non-symmetric block models. In order to inform the design of a new SDP for the non-symmetric setting, we investigate the failure of sym-SDP when it is applied to non-symmetric settings. We formally show that sym-SDP fails to return the correct labeling of the vertices in some information-theoretically feasible, asymmetric cases. In addition, we give an intuitive geometric interpretation of the failure of sym-SDP in asymmetric settings, which in turn suggests an SDP formulation to handle the asymmetric setting. Still, this new SDP cannot be readily analyzed by existing techniques, suggesting a fundamental limitation in the design of SDPs for community detection.