Tightness and Error Exponents of SDP with Logarithmically Many Communities

arXiv:2610.09303v1 Announce Type: cross 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 $k/\log m\to\gamma>0$, with edge probabilities $\alpha\log m/m$ within communities and $\beta\log m/m$ across them, for fixed $\alpha>\beta>0$. 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.