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On the Relation between Graph Ricci Curvature and Community Structure

Sathyanarayanan Rengaswami, Theodora Bourni, Vasileios Maroulas

TL;DR

The paper addresses the relationship between graph curvature and community structure using Ollivier-Ricci curvature. It harnesses optimal transport on graphs to bound the Wasserstein distance $W(m_x,m_y)$ between edge-endpoint distributions, yielding a theorem that constrains the curvature of intercommunity edges in terms of community sizes $m,n$ and the count of intercommunity edges $k$. A precise inequality $k \le \frac{-m+\sqrt{m^2+4(m-1)(2n-1)}}{2}$ guarantees $\kappa(e) \le 0$ for all intercommunity edges between two communities, and the paper demonstrates sharpness via zero- and positive-curvature configurations alongside empirical tests. The results provide guidance for curvature-based community detection and deepen understanding of how intercommunity connectivity shapes local geometric properties on graphs.

Abstract

The connection between curvature and topology is a very well-studied theme in the subject of differential geometry. By suitably defining curvature on networks, the study of this theme has been extended into the domain of network analysis as well. In particular, this has led to curvature-based community detection algorithms. In this paper, we reveal the relation between community structure of a network and the curvature of its edges. In particular, we give apriori bounds on the curvature of intercommunity edges of a graph.

On the Relation between Graph Ricci Curvature and Community Structure

TL;DR

The paper addresses the relationship between graph curvature and community structure using Ollivier-Ricci curvature. It harnesses optimal transport on graphs to bound the Wasserstein distance between edge-endpoint distributions, yielding a theorem that constrains the curvature of intercommunity edges in terms of community sizes and the count of intercommunity edges . A precise inequality guarantees for all intercommunity edges between two communities, and the paper demonstrates sharpness via zero- and positive-curvature configurations alongside empirical tests. The results provide guidance for curvature-based community detection and deepen understanding of how intercommunity connectivity shapes local geometric properties on graphs.

Abstract

The connection between curvature and topology is a very well-studied theme in the subject of differential geometry. By suitably defining curvature on networks, the study of this theme has been extended into the domain of network analysis as well. In particular, this has led to curvature-based community detection algorithms. In this paper, we reveal the relation between community structure of a network and the curvature of its edges. In particular, we give apriori bounds on the curvature of intercommunity edges of a graph.
Paper Structure (9 sections, 7 theorems, 25 equations, 6 figures, 2 tables)

This paper contains 9 sections, 7 theorems, 25 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions
  4. Optimal Transportation on Graphs
  5. The Ollivier-Ricci Curvature
  6. A Zero-Curvature Example
  7. Proof of Main Theorem
  8. Empirical Results
  9. Conclusion

Key Result

Theorem 1.1

Suppose $\mathcal{G}$ is a graph comprised of several communities. Let $C_i,C_j$ be distinguished communities in $\mathcal{G}$ whose sizes are $m$ and $n$. Let $k$ be the total number of edges that are either intercommunity edges between $C_i$ and $C_j$, or from any other community to $C_i$ or $C_j$ we have $\kappa(e) \leq 0$ for every intercommunity edge $e$ between $C_i$ and $C_j$. In particular

Figures (6)

  • Figure 1: A configuration with zero curvature on all intercommunity edges
  • Figure 2: Another configuration with zero curvature on intercommunity edges
  • Figure 3: Three communities, with a potential function
  • Figure 4: Two communities with all $n$ intercommunity edges positively curved
  • Figure 5: Distribution of Proportion of Nonpositively Curved Intercommunity Edges
  • ...and 1 more figures

Theorems & Definitions (12)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Proposition 2.1
  • Proposition 2.2
  • Example
  • Remark
  • Theorem 3.1
  • proof
  • Remark
  • ...and 2 more