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Hypergraph clustering using Ricci curvature: an edge transport perspective

Olympio Hacquard

TL;DR

The paper addresses clustering in hypergraphs by extending Ollivier-Ricci curvature through two transport schemes: node-Ricci flow on the clique expansion and edge-Ricci flow on the line expansion. It defines probability measures on nodes or edges, computes curvature via $W(\cdot,\cdot)$, and evolves weights with $w^{(l+1)}(x,y)=(1-\kappa^{(l)}(x,y))\,w^{(l)}(x,y)$ to reveal communities, with edge-transport exploiting hyperedge structure. Through synthetic hypergraph SBMs and real datasets, the work demonstrates that edge-Ricci flow is particularly effective for large hyperedges and that node- and edge-centric flows are complementary in capturing hypergraph structure, supported by practical guidelines on hyperparameters and computational costs. The findings offer an interpretable, geometry-inspired framework for hypergraph clustering that compares favorably with neural embeddings on several benchmarks while providing clearer theoretical intuition and scalability insights.

Abstract

In this paper, we introduce a novel method for extending Ricci flow to hypergraphs by defining probability measures on the edges and transporting them on the line expansion. This approach yields a new weighting on the edges, which proves particularly effective for community detection. We extensively compare this method with a similar notion of Ricci flow defined on the clique expansion, demonstrating its enhanced sensitivity to the hypergraph structure, especially in the presence of large hyperedges. The two methods are complementary and together form a powerful and highly interpretable framework for community detection in hypergraphs.

Hypergraph clustering using Ricci curvature: an edge transport perspective

TL;DR

The paper addresses clustering in hypergraphs by extending Ollivier-Ricci curvature through two transport schemes: node-Ricci flow on the clique expansion and edge-Ricci flow on the line expansion. It defines probability measures on nodes or edges, computes curvature via , and evolves weights with to reveal communities, with edge-transport exploiting hyperedge structure. Through synthetic hypergraph SBMs and real datasets, the work demonstrates that edge-Ricci flow is particularly effective for large hyperedges and that node- and edge-centric flows are complementary in capturing hypergraph structure, supported by practical guidelines on hyperparameters and computational costs. The findings offer an interpretable, geometry-inspired framework for hypergraph clustering that compares favorably with neural embeddings on several benchmarks while providing clearer theoretical intuition and scalability insights.

Abstract

In this paper, we introduce a novel method for extending Ricci flow to hypergraphs by defining probability measures on the edges and transporting them on the line expansion. This approach yields a new weighting on the edges, which proves particularly effective for community detection. We extensively compare this method with a similar notion of Ricci flow defined on the clique expansion, demonstrating its enhanced sensitivity to the hypergraph structure, especially in the presence of large hyperedges. The two methods are complementary and together form a powerful and highly interpretable framework for community detection in hypergraphs.

Paper Structure

This paper contains 28 sections, 1 theorem, 10 equations, 11 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model
  3. Hypergraphs and associated graphs
  4. Ricci curvature and flows on graphs
  5. Two complementary notions of Ricci curvature on hypergraphs
  6. Nodes transport on the clique graph
  7. Edges transport on the line graph
  8. A toy clustering example
  9. Experiments
  10. Hypergraphs stochastic block models
  11. Real data
  12. Hyperparameter discussion
  13. The number of flow iterations $N$
  14. Parameters $\alpha$ and $p$ for $\nu^{\alpha, p}$
  15. Aggregation and graph weightings
  16. ...and 13 more sections

Key Result

Proposition 1

We keep the same notation as above and consider the edge-Ricci flow with measures $\tilde{\mu}$ defined by Equation eq:edges_meas_alt and aggregation $Agg$ with the maximum function. At the $l$-th iteration, the Ricci flow can only take three possible values: $w_{\mathcal{N}, 1}^{(l)}$ for the edge

Figures (11)

  • Figure 1: Examples of hypergraphs $H_i$ with their clique $\mathbf{C}(H_i)$ and line $\mathbf{L}(H_i)$ expansions. $H_1$, $H_2$ and $H_3$ share the same clique but have different line expansions.
  • Figure 2: Ollivier-Ricci curvature on a graph with a clear two-communities structure.
  • Figure 3: Edge-Ricci transport between nodes $x$ and $y$. A measure on $St(x)$ is transported onto a measure on $St(y)$ via the line graph.
  • Figure 4: Example of hypergraph $H(a,b)$ with $a=6$ and $b=4$. The gateway nodes are represented in blue.
  • Figure 5: Hypergraph stochastic block model reconstruction using hypergraph notions of Ricci flow.
  • ...and 6 more figures

Theorems & Definitions (1)

  • Proposition 1