Semidefinite Programming for the Asymmetric Stochastic Block Model

2025-06-23

AI Summary

This paper addresses exact community recovery in the asymmetric binary Stochastic Block Model (SBM). While existing semidefinite programming (SDP) methods achieve the information-theoretic limit under symmetry, their failure mechanism in the asymmetric setting remains unclear, and no principled SDP design exists for this case. First, the authors uncover the geometric root of symmetric SDP failure: it provably fails to recover communities exactly in certain information-theoretically solvable regimes, exposing a fundamental limitation of standard SDP formulations. Second, they propose a novel adaptive SDP framework that explicitly incorporates asymmetry via prior constraints on community sizes and a weighted objective function reflecting heterogeneous edge probabilities. Theoretically, the new SDP achieves exact recovery over a strictly broader parameter regime and attains the tight information-theoretic threshold. This work extends the applicability boundary of SDP in statistical inference and establishes a new paradigm for learning from structurally imbalanced graphs.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Security and Privacy: Large-scale security measurements

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.

Problem

• SDP fails in non-symmetric stochastic block models
• Investigates sym-SDP failure in asymmetric cases
• Proposes new SDP for asymmetric community detection

Innovation

• SDP for asymmetric stochastic block model
• Geometric interpretation of sym-SDP failure
• New SDP formulation for non-symmetric cases