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Finding core subgraphs of directed graphs via discrete Ricci curvature flow

Juan Zhao, Jicheng Ma, Yunyan Yang, Liang Zhao

TL;DR

This work extends discrete Ricci curvature concepts to directed graphs by defining a curvature flow κ_e^α and proving a unique global solution for strongly connected graphs. By augmenting weakly connected graphs with artificial edges, the authors enable curvature flow-based core-subgraph detection, selecting the largest strongly connected component after edge deletions. Experiments on three real-world networks show the proposed method often outperforms traditional centrality-based baselines on core cohesiveness and structural metrics, and demonstrate robustness to edge perturbations. The approach broadens geometric network analysis to directed settings and suggests future extensions to directed hypergraphs and dynamic networks.

Abstract

Ricci curvature and its associated flow offer powerful geometric methods for analyzing complex networks. While existing research heavily focuses on applications for undirected graphs such as community detection and core extraction, there have been relatively less attention on directed graphs. In this paper, we introduce a definition of Ricci curvature and an accompanying curvature flow for directed graphs. Crucially, for strongly connected directed graphs, this flow admits a unique global solution. We then apply this flow to detect strongly connected subgraphs from weakly connected directed graphs. (A weakly connected graph is connected overall but not necessarily strongly connected). Unlike prior work requiring graphs to be strongly connected, our method loosens this requirement. We transform a weakly connected graph into a strongly connected one by adding edges with very large artificial weights. This modification does not compromise our core subgraph detection. Due to their extreme weight, these added edges are automatically discarded during the final iteration of the Ricci curvature flow. For core evaluation, our approach consistently surpasses traditional methods, achieving better results on at least two out of three key metrics. The implementation code is publicly available at https://github.com/12tangze12/Finding-core-subgraphs-on-directed-graphs.

Finding core subgraphs of directed graphs via discrete Ricci curvature flow

TL;DR

This work extends discrete Ricci curvature concepts to directed graphs by defining a curvature flow κ_e^α and proving a unique global solution for strongly connected graphs. By augmenting weakly connected graphs with artificial edges, the authors enable curvature flow-based core-subgraph detection, selecting the largest strongly connected component after edge deletions. Experiments on three real-world networks show the proposed method often outperforms traditional centrality-based baselines on core cohesiveness and structural metrics, and demonstrate robustness to edge perturbations. The approach broadens geometric network analysis to directed settings and suggests future extensions to directed hypergraphs and dynamic networks.

Abstract

Ricci curvature and its associated flow offer powerful geometric methods for analyzing complex networks. While existing research heavily focuses on applications for undirected graphs such as community detection and core extraction, there have been relatively less attention on directed graphs. In this paper, we introduce a definition of Ricci curvature and an accompanying curvature flow for directed graphs. Crucially, for strongly connected directed graphs, this flow admits a unique global solution. We then apply this flow to detect strongly connected subgraphs from weakly connected directed graphs. (A weakly connected graph is connected overall but not necessarily strongly connected). Unlike prior work requiring graphs to be strongly connected, our method loosens this requirement. We transform a weakly connected graph into a strongly connected one by adding edges with very large artificial weights. This modification does not compromise our core subgraph detection. Due to their extreme weight, these added edges are automatically discarded during the final iteration of the Ricci curvature flow. For core evaluation, our approach consistently surpasses traditional methods, achieving better results on at least two out of three key metrics. The implementation code is publicly available at https://github.com/12tangze12/Finding-core-subgraphs-on-directed-graphs.

Paper Structure

This paper contains 15 sections, 4 theorems, 43 equations, 8 figures, 8 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notations and main results
  3. Proof of Theorems \ref{['thm1']} and \ref{['discrete-thm']}
  4. Core subgraphs and Ricci curvature flow
  5. Core subgraphs and their metrics
  6. From weak connectivity to strong connectivity
  7. Finding core subgraphs via Ricci curvature flow
  8. Algorithms
  9. Experiments
  10. Real-world Datasets
  11. Effect of curvature parameter $\alpha$
  12. Core extraction via Ricci curvature flow
  13. Comparison with baseline centrality methods
  14. Robustness analysis under core edge deletion
  15. Conclusion

Key Result

Theorem 2.1

Assume $G=(V,E,\mathbf{w}_0)$ is a strongly connected directed graph, where $\mathbf{w}_0=(w_{0,e})_{e\in E}$ is the initial weights on $E$. Let $\alpha$ be a number in $[0,1]$, $\kappa_{e}^\alpha$ be the Ricci curvature defined as in (Olli-Ric), and $\rho_e=d(x,y)$ denotes the length of a directed

Figures (8)

  • Figure 1: Illustration of Example 1.
  • Figure 2: Illustration of Example 2. Red dashed edges indicate added edges.
  • Figure 3: Illustration of Example 3.
  • Figure 4: Illustration of Example 4.
  • Figure 5: Effect of curvature parameter $\alpha$ on the Physicians network.
  • ...and 3 more figures

Theorems & Definitions (4)

  • Theorem 2.1
  • Theorem 2.2
  • Lemma 3.1
  • Lemma 3.2