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Core detection via Ricci curvature flows on weighted graphs

Juan Zhao, Jicheng Ma, Yunyan Yang, Liang Zhao

Abstract

Graph Ricci curvature is crucial as it geometrically quantifies network structure. It pinpoints bottlenecks via negative curvature, identifies cohesive communities with positive curvature, and highlights robust hubs. This guides network analysis, resilience assessment, flow optimization, and effective algorithm design. In this paper, we derived upper and lower bounds for the weights along several kinds of discrete Ricci curvature flows. As an application, we utilized discrete Ricci curvature flows to detect the core subgraph of a finite undirected graph. The novelty of this work has two aspects. Firstly, along the Ricci curvature flow, the bounds for weights determine the minimum number of iterations required to ensure weights remain between two prescribed positive constants. In particular, for any fixed graph, we conclude weights can not overflow and can not be treated as zero, as long as the iteration does not exceed a certain number of times; Secondly, it demonstrates that our Ricci curvature flow method for identifying core subgraphs outperforms prior approaches, such as page rank, degree centrality, betweenness centrality and closeness centrality. The codes for our algorithms are available at https://github.com/12tangze12/core-detection-via-Ricci-flow.

Core detection via Ricci curvature flows on weighted graphs

Abstract

Graph Ricci curvature is crucial as it geometrically quantifies network structure. It pinpoints bottlenecks via negative curvature, identifies cohesive communities with positive curvature, and highlights robust hubs. This guides network analysis, resilience assessment, flow optimization, and effective algorithm design. In this paper, we derived upper and lower bounds for the weights along several kinds of discrete Ricci curvature flows. As an application, we utilized discrete Ricci curvature flows to detect the core subgraph of a finite undirected graph. The novelty of this work has two aspects. Firstly, along the Ricci curvature flow, the bounds for weights determine the minimum number of iterations required to ensure weights remain between two prescribed positive constants. In particular, for any fixed graph, we conclude weights can not overflow and can not be treated as zero, as long as the iteration does not exceed a certain number of times; Secondly, it demonstrates that our Ricci curvature flow method for identifying core subgraphs outperforms prior approaches, such as page rank, degree centrality, betweenness centrality and closeness centrality. The codes for our algorithms are available at https://github.com/12tangze12/core-detection-via-Ricci-flow.

Paper Structure

This paper contains 13 sections, 5 theorems, 69 equations, 4 figures, 9 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notations and main results
  3. Proofs of main theorems
  4. Core subgraphs and Ricci curvature flow
  5. Core subgraphs and their metrics
  6. Core detection via Ricci curvature flow
  7. Algorithms
  8. Experiments
  9. Real-world Datasets
  10. Sensitivity Analysis
  11. Comparison with baseline centrality methods
  12. Comparison with hypergraph Ricci curvature flow method
  13. Conclusion

Key Result

Theorem 2.1

Let $G=(V,E)$ be a finite graph, $E=\{e_1,e_2,\cdots,e_m\}$ be the edge set, and $\mathbf{w}_0$ be the initial weight on $E$. Then we have the following two conclusions: $(i)$ If $\kappa:E\rightarrow\mathbb{R}$ is Ollivier's Ricci curvature, then for any $0<s<1$, the flow (discrete-1) has a unique g $(ii)$ If $\kappa:E\rightarrow\mathbb{R}$ is Lin-Lu-Yau's Ricci curvature, then for any $0<s<1/2$,

Figures (4)

  • Figure 1: Finding a core subgraph
  • Figure 2: Finding a core subgraph
  • Figure 3: Finding a core subgraph
  • Figure 4: Finding a core subgraph

Theorems & Definitions (5)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5