发表机构
Duke University; Harvard T.H. Chan School of Public Health（杜克大学; 哈佛大学陈曾熙公共卫生学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究社区数量对数增长时SDP松弛的紧性，推导渐近边界，并证明在谱稳定区域内单次SDP求解可实现精确恢复。
AI 中文摘要
我们研究了当社区数量呈对数增长时，用于社区恢复的半定规划（SDP）松弛。在平衡随机块模型中，设有$n=km$个顶点，我们考虑$k/\log m\to\gamma>0$的区间，社区内边概率为$\alpha\log m/m$，社区间边概率为$\beta\log m/m$，其中$\alpha>\beta>0$为固定常数。我们推导了远离临界情况时的尖锐渐近紧性边界。当稀有顶点导致紧性失效而主体部分仍保持谱稳定时，每个近优解的归一化矩阵误差仍然趋于零。我们证明了该误差以及归一化最优目标增益的匹配高概率指数，这些指数由决定紧性的同一局部修正所控制。在整个谱稳定区域内，一次SDP求解后接显式舍入和细化，即可在信息论阈值之上实现精确的社区恢复。即使当植入的社区矩阵不是SDP的最优解时，该保证仍然成立。
英文摘要
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.
Comments36 pages, 8 figures
