Paper · Aug 2, 2025

Core detection via Ricci curvature flows on weighted graphs

Juan Zhao, Jicheng Ma, Yunyan Yang, Liang Zhao

math.AParXiv:2508.01400v2Physica A 692 (2026) 131525
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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.

Comments: 21 pages

Ricci flowDiscrete & graph Ricci

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Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper A Ricci flow on graphs from effective resistance (Aleyah Dawkins, Vishal Gupta, Mark Kempton et al., 2024)5Dawkins 2024 A modified Ricci flow on arbitrary weighted graph (Jicheng Ma and Yunyan Yang, 2024)1Ma 2024 Evolution of weights on a connected finite graph (Jicheng Ma and Yunyan Yang, 2024)3Ma 2024 Ricci Flow on Weighted Digraphs with Balancing Factor (Shuliang Bai, Rui Li, Shuang Liu et al., 2025)6Bai 2025 On the Ricci flow on Trees (Shuliang Bai, Bobo Hua, Yong Lin et al., 2025)7Bai 2025 Evolving edge weights via local entropy flow and cohesion flow on graphs (Juan Zhao, Jicheng Ma, Yunyan Yang et al., 2026)2Zhao 2026 Equivalence of Lin–Lu–Yau curvature and 1/2-Ollivier curvature on weighted graphs (Shiping Liu and Yunyan Yang, 2026)4Liu 2026 Curvature Diffusion of Inverse-weight Lin–Lu–Yau Ricci Flow on Finite Trees (Shuliang Bai and Bobo Hua, 2026)8Bai 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A modified Ricci flow on arbitrary weighted graph

    Jicheng Ma and Yunyan Yang PDF

    Topics Discrete & graph Ricci · Same authors Jicheng Ma, Yunyan Yang · Shared terms “weighted graphs”, “detection”, “algorithm”

  2. Evolving edge weights via local entropy flow and cohesion flow on graphs

    Juan Zhao, Jicheng Ma, Yunyan Yang et al. PDF

    Topics Discrete & graph Ricci · Same authors Juan Zhao, Jicheng Ma, Yunyan Yang · Shared terms “network analysis”, “weighted graphs”, “detection”

  3. Evolution of weights on a connected finite graph

    Jicheng Ma and Yunyan Yang PDF

    Topics Discrete & graph Ricci · Same authors Jicheng Ma, Yunyan Yang · Shared terms “detection”, “algorithm”, “iteration”

  4. Equivalence of Lin–Lu–Yau curvature and 1/2-Ollivier curvature on weighted graphs

    Shiping Liu and Yunyan Yang PDF

    Topics Discrete & graph Ricci · Same author Yunyan Yang · Shared terms “weighted graphs”

  5. A Ricci flow on graphs from effective resistance

    Aleyah Dawkins, Vishal Gupta, Mark Kempton et al. PDF

    Topics Discrete & graph Ricci · Shared terms “effective”

  6. Ricci Flow on Weighted Digraphs with Balancing Factor

    Shuliang Bai, Rui Li, Shuang Liu et al. PDF

    Topics Discrete & graph Ricci · Shared terms “weighted graphs”, “detection”, “undirected graph”

  7. On the Ricci flow on Trees

    Shuliang Bai, Bobo Hua, Yong Lin et al. PDF

    Topics Discrete & graph Ricci

  8. Curvature Diffusion of Inverse-weight Lin–Lu–Yau Ricci Flow on Finite Trees

    Shuliang Bai and Bobo Hua PDF

    Topics Discrete & graph Ricci

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Discrete & graph Ricci Curvature and Ricci flow for graphs, networks and triangulated surfaces. It runs from Chow–Luo's combinatorial flow on circle packings to Ollivier's… Start with: Combinatorial Ricci Flows on Surfaces (Chow et al., 2002)

In the guidebook

  1. 6D.11
    Computing Geometric FlowsBook 6D · Diffusions, Finance and Computation

    Numerical curve shortening and mean curvature flow, level sets and phase fields, discrete Ricci flow on surfaces and in conformal mapping, rotationally symmetric neckpinches, and what computation has taught about singularities.

  2. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  3. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.