Tightness and Error Exponents of SDP with Logarithmically Many Communities

2026-10-06

AI Summary

This study addresses the unresolved tightness bounds and error exponents of semidefinite programming (SDP) for community recovery when the number of communities grows logarithmically. Focusing on the balanced stochastic block model, it integrates SDP relaxation, spectral analysis, and concentration inequalities to reveal that a local correction mechanism intrinsically governs both tightness and error exponents. The work derives sharp asymptotic tightness bounds alongside matching high-probability error exponents. Furthermore, it demonstrates that exact recovery is achievable through a single SDP solve even with a suboptimal planted matrix, establishing theoretical guarantees above the information-theoretic threshold and confirming robustness against perturbations from rare vertices.

Abstract

We study a semidefinite programming (SDP) relaxation for community recovery when the number of communities grows logarithmically. In the balanced stochastic block model with $n=km$ vertices, we consider the regime $k/\log m\toγ>0$, with edge probabilities $α\log m/m$ within communities and $β\log m/m$ across them, for fixed $α>β>0$. We derive the sharp asymptotic tightness boundary away from critical cases. When rare vertices cause tightness to fail while the bulk remains spectrally stable, the normalized matrix error of every near-optimal solution still vanishes. We prove matching high-probability exponents for this error and the normalized optimal objective gain, governed by the same local correction that determines tightness. Throughout this spectrally stable region, a single SDP solve followed by explicit rounding and refinement achieves exact community recovery above the information-theoretic threshold. This guarantee holds even when the planted community matrix is not an optimal solution to the SDP.

Problem

• Community Recovery
• Semidefinite Programming
• Stochastic Block Model
• Tightness
• Error Exponents

Innovation

• Semidefinite Programming
• Community Recovery
• Stochastic Block Model
• Tightness Boundary
• Error Exponents