Exact Recovery in the Geometric SBM
Julia Gaudio, Andrew Jin
TL;DR
This work characterizes the exact recovery threshold in the symmetric Geometric Stochastic Block Model (GSBM) with distance-dependent edge probabilities via a generalized CH-divergence: $I(f_{\text{in}}, f_{\text{out}}) = \lambda \nu_d r^d \int_0^r \left( 1 - \sqrt{f_{\text{in}}(t) f_{\text{out}}(t)} - \sqrt{(1 - f_{\text{in}}(t))(1 - f_{\text{out}}(t))} \right) t^{d-1} dt / r^d$. Exact recovery is achievable in polynomial time when $I(f_{\text{in}}, f_{\text{out}}) > 1$ (with the 1D extra condition $\lambda r > 1$) and impossible otherwise. The authors provide a two-phase algorithm: Phase I builds an almost-exact estimator by partitioning space into blocks and labeling a root block via Pairwise Classify, followed by Propagate to other blocks; Phase II refines using a likelihood-ratio test to attain exact recovery in $O(n \log n)$ time. The results extend previous SBM findings to the geometric setting and general spatial dimensions, clarifying how CH-divergence governs distinguishability under spatially embedded models. The work lays groundwork for future extensions to multiple communities and weaker regularity assumptions on the edge functions $f_{\text{in}}$ and $f_{\text{out}}$.
Abstract
Community detection is the problem of identifying dense communities in networks. Motivated by transitive behavior in social networks ("thy friend is my friend"), an emerging line of work considers spatially-embedded networks, which inherently produce graphs containing many triangles. In this paper, we consider the problem of exact label recovery in the Geometric Stochastic Block Model (GSBM), a model proposed by Baccelli and Sankararaman as the spatially-embedded analogue of the well-studied Stochastic Block Model. Under mild technical assumptions, we completely characterize the information-theoretic threshold for exact recovery, generalizing the earlier work of Gaudio, Niu, and Wei.