We study the problem of exact community recovery in the Geometric Stochastic Block Model (GSBM), where each vertex has an unknown community label as well as a known position, generated according to a Poisson point process in $\mathbb{R}^d$. Edges are formed independently conditioned on the community labels and positions, where vertices may only be connected by an edge if they are within a prescribed distance of each other. The GSBM thus favors the formation of dense local subgraphs, which commonly occur in real-world networks, a property that makes the GSBM qualitatively very different from the standard Stochastic Block Model (SBM). We propose a linear-time algorithm for exact community recovery, which succeeds down to the information-theoretic threshold, confirming a conjecture of Abbe, Baccelli, and Sankararaman. The algorithm involves two phases. The first phase exploits the density of local subgraphs to propagate estimated community labels among sufficiently occupied subregions, and produces an almost-exact vertex labeling. The second phase then refines the initial labels using a Poisson testing procedure. Thus, the GSBM enjoys local to global amplification just as the SBM, with the advantage of admitting an information-theoretically optimal, linear-time algorithm.

翻译：我们研究了几何随机块模型（GSBM）中精确社区恢复的问题，其中每个顶点具有未知的社区标签以及根据 $\mathbb{R}^d$ 中的泊松点过程生成的已知位置。边是基于社区标签和位置条件独立形成的，仅当顶点彼此处于规定距离内时才能通过边连接。因此，GSBM 倾向于形成密集的局部子图，这常见于现实网络，该特性使得 GSBM 在性质上与标准随机块模型（SBM）截然不同。我们提出了一种用于精确社区恢复的线性时间算法，该算法成功达到信息论阈值，证实了 Abbe、Baccelli 和 Sankararaman 的猜想。该算法包含两个阶段：第一阶段利用局部子图的密度在充分占据的子区域中传播估计的社区标签，并生成近乎精确的顶点标签；第二阶段随后通过泊松检验程序优化初始标签。因此，GSBM 如同 SBM 一样实现了从局部到全局的放大，且其优势在于能够容纳信息论最优的线性时间算法。