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
Rd. 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