Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Guo, Shengtao, Fang, Ethan X., Lu, Junwei
Institutions: Duke University, Harvard University
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7220961156