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Community detection of hypergraphs by Ricci flow

Yulu Tian, Jicheng Ma, Yunyan Yang, Liang Zhao

TL;DR

This work addresses community detection in hypergraphs by introducing a curvature-driven framework: an Ollivier–Ricci curvature defined on hyperedges, a Ricci flow that evolves hyperedge weights, and HyperRCD, a detector that operates directly on hypergraphs without reducing to graphs. The authors prove long-time existence for the hypergraph Ricci flow and implement a discrete HyperRCD algorithm that updates weights via w'_h = −d(h) κ_α(h) and prunes edges to reveal communities, achieving robust, scalable performance. Across synthetic and real-world hypergraphs, HyperRCD demonstrates state-of-the-art or competitive results, outperforming several graph-based baselines and remaining competitive with hypergraph-specific methods; Mushroom data reveal a scalability limitation tied to LP-based Wasserstein computations. Overall, the approach preserves higher-order structure and provides a principled geometric method for hypergraph community detection with strong empirical validation.

Abstract

Community detection in hypergraphs is both instrumental for functional module identification and intricate due to higher-order interactions among nodes. We define a hypergraph Ricci flow that directly operates on higher-order interactions of hypergraphs and prove long-time existence of the flow. Building on this theoretical foundation, we develop HyperRCD-a Ricci-flow-based community detection approach that deforms hyperedge weights through curvature-driven evolution, which provides an effective mathematical representation of higher-order interactions mediated by weighted hyperedges between nodes. Extensive experiments on both synthetic and real-world hypergraphs demonstrate that HyperRCD exhibits remarkable enhanced robustness to topological variations and competitive performance across diverse datasets.

Community detection of hypergraphs by Ricci flow

TL;DR

This work addresses community detection in hypergraphs by introducing a curvature-driven framework: an Ollivier–Ricci curvature defined on hyperedges, a Ricci flow that evolves hyperedge weights, and HyperRCD, a detector that operates directly on hypergraphs without reducing to graphs. The authors prove long-time existence for the hypergraph Ricci flow and implement a discrete HyperRCD algorithm that updates weights via w'_h = −d(h) κ_α(h) and prunes edges to reveal communities, achieving robust, scalable performance. Across synthetic and real-world hypergraphs, HyperRCD demonstrates state-of-the-art or competitive results, outperforming several graph-based baselines and remaining competitive with hypergraph-specific methods; Mushroom data reveal a scalability limitation tied to LP-based Wasserstein computations. Overall, the approach preserves higher-order structure and provides a principled geometric method for hypergraph community detection with strong empirical validation.

Abstract

Community detection in hypergraphs is both instrumental for functional module identification and intricate due to higher-order interactions among nodes. We define a hypergraph Ricci flow that directly operates on higher-order interactions of hypergraphs and prove long-time existence of the flow. Building on this theoretical foundation, we develop HyperRCD-a Ricci-flow-based community detection approach that deforms hyperedge weights through curvature-driven evolution, which provides an effective mathematical representation of higher-order interactions mediated by weighted hyperedges between nodes. Extensive experiments on both synthetic and real-world hypergraphs demonstrate that HyperRCD exhibits remarkable enhanced robustness to topological variations and competitive performance across diverse datasets.
Paper Structure (11 sections, 4 theorems, 32 equations, 6 figures, 3 tables, 1 algorithm)

This paper contains 11 sections, 4 theorems, 32 equations, 6 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Ollivier–Ricci curvature of hypergraphs
  3. Ricci flow on hypergraphs
  4. Algorithm
  5. Experiments
  6. Datasets and comparison algorithms
  7. Experimental Setup
  8. Experimental Results
  9. Experimental Results for the Synthetic Networks.
  10. Experimental Results for the Real-World Networks
  11. Conclusion

Key Result

Lemma 2.1

For any two fixed vertices $u, v\in V$, the distance $d(u,v)$ is locally Lipschitz in $\mathbb{R}^m_+$ with respect to $\bm{w}=(w_{h_1},w_{h_2},\cdots,w_{h_m})$, i.e., for any $\bm{w}, \widetilde{\bm{w}}\in\mathbb{R}^m_+$, if $d$ and $\widetilde{d}$ are two distance functions determined by $\bm{w}$

Figures (6)

  • Figure 1: NMI performance on synthetic network D1
  • Figure 2: NMI performance on synthetic network D2
  • Figure 3: NMI performance on synthetic network D3
  • Figure 4: HyperRCD Community Detection on D1: Original vs. HyperRCD vs. Louvain
  • Figure 5: HyperRCD Community Detection on D3: Original vs. HyperRCD vs. Louvain
  • ...and 1 more figures

Theorems & Definitions (9)

  • Lemma 2.1
  • Definition 2.1
  • Remark 2.2
  • Definition 2.2
  • Proposition 2.3
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof