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

Varun Kotharkar

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.

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

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.
Paper Structure (59 sections, 44 theorems, 195 equations, 1 table)

This paper contains 59 sections, 44 theorems, 195 equations, 1 table.

Table of Contents

  1. Introduction
  2. Model.
  3. Contributions.
  4. Standing moderately dense window.
  5. Organization.
  6. Scope and non-scope.
  7. Related work.
  8. Preliminaries
  9. Asymptotic regime and constants.
  10. Indexing and asymptotic notation.
  11. Indicators and all-ones objects.
  12. Normalized Laplacian as an operator.
  13. A standing SBM window used throughout
  14. Rates.
  15. Graphs, measures, and transportation distance
  16. ...and 44 more sections

Key Result

Lemma 3.1

Let $X_1,\ldots,X_m$ be independent with $X_i\sim \mathrm{Ber}(p_i)$ and set $S=\sum_{i=1}^m X_i$ and $\mu=\mathbb{E} S=\sum_i p_i$. Then for any $\lambda\ge 0$,

Theorems & Definitions (87)

  • Remark 1: Why Assumption \ref{['ass:BC']} is imposed
  • Remark 2: Immediate consequences of Assumption \ref{['ass:MDT']}
  • Lemma 3.1: Bernstein--Chernoff for Bernoulli sums
  • proof
  • Lemma 3.2: Degree concentration
  • proof
  • Lemma 3.3: Co-degree concentration
  • proof
  • Lemma 3.4: Blockwise degree concentration
  • proof
  • ...and 77 more