LLY Ricci Reweighting in Stochastic Block Models: Uniform Curvature Concentration and Finite-Horizon Tracking
This addresses community detection in random graphs, offering a principled curvature-based method with theoretical guarantees, though it appears incremental as it builds on existing curvature concepts.
The paper tackles community recovery in stochastic block models by using Lin-Lu-Yau Ricci curvature to reweight edges, proving that this amplifies within-block connectivity and improves spectral clustering with a larger eigengap and misclustering guarantees.
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.