LLY Ricci Reweighting in Stochastic Block Models: Uniform Curvature Concentration and Finite-Horizon Tracking

TL;DR

This paper introduces a curvature-based edge reweighting method for community detection in stochastic block models, demonstrating improved spectral clustering performance through uniform curvature concentration and finite-horizon analysis.

Contribution

It develops a Ricci curvature reweighting technique that enhances community detection, providing theoretical guarantees and a finite-horizon curvature flow interpretation.

Findings

Uniform concentration of edge curvatures in moderate-density regimes

Enhanced spectral clustering with larger eigengaps after reweighting

Finite-horizon reweighting tracks a deterministic curvature flow

Abstract

We study curvature-driven edge reweighting for community recovery in the balanced two-block stochastic block model. Given a graph G with initial weights equal to the adjacency matrix, we iteratively update edge weights using Lin-Lu-Yau (Ollivier-type) Ricci curvature, while all transportation costs are computed in the unweighted graph metric. In a moderate-density regime we prove uniform concentration of edge curvatures and show that a single Ricci reweighting step produces a two-level weighting that amplifies within-block connectivity relative to across-block connectivity. As a consequence, spectral clustering on the reweighted graph has a strictly larger population eigengap, and we obtain corresponding non-asymptotic perturbation bounds and Davis-Kahan misclustering guarantees. We further analyze a fixed finite horizon of iterated reweighting, where the random iterates track a deterministic two-weight recursion uniformly over the time horizon. This yields a principled finite-horizon curvature flow interpretation for community detection in a canonical random graph model.