Paper · Jun 25, 2026

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

Juan Zhao, Jicheng Ma, Yunyan Yang, Liang Zhao

math.CAarXiv:2606.26750
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Abstract

In this paper, we first propose two different quantities on graphs, namely local entropy and cohesion, then design two corresponding flows for edge weights: the local entropy flow and the cohesion flow. We establish the global existence and uniqueness of solutions for both flows and investigate their asymptotic behaviors, including the case that the limit goes to positive infinity. Moreover, they can be applied to fundamental network analysis tasks, including community detection and node classification. Empirical evaluations demonstrate that our method achieves performance competitive with Ollivier Ricci flow and Lin-Lu-Yau Ricci flow on benchmark network analysis tasks. In experimental scenarios, we first apply the cohesion flow to evolve the edge weights of the graph, and then apply the local entropy flow to further update the resulting weighted graph. Both flows are computationally efficient, leading to a significant reduction in overall computational cost and improved scalability.

Comments: 29 pages, 8 figures

Ricci flowKähler geometryStability & uniquenessLong-time existence & convergenceDiscrete & 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 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 Core detection via Ricci curvature flows on weighted graphs (Juan Zhao, Jicheng Ma, Yunyan Yang et al., 2025)2Zhao 2025 On the Ricci flow on Trees (Shuliang Bai, Bobo Hua, Yong Lin et al., 2025)6Bai 2025 The Ricci flow with prescribed curvature on graphs (Yong Lin and Shuang Liu, 2026)8Lin 2026 The Ollivier Ricci flow with prescribed curvature on infinite graphs (Bobo Hua, Yong Lin and Shuang Liu, 2026)5Hua 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)7Bai 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, Long-time existence & convergence, Stability & uniqueness · Same authors Jicheng Ma, Yunyan Yang · Shared terms “weighted graphs”, “community detection”, “Ollivier Ricci”

  2. Core detection via Ricci curvature flows on weighted 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, Long-time existence & convergence · Same authors Jicheng Ma, Yunyan Yang · Shared terms “community detection”, “Ollivier Ricci”, “Yau Ricci”

  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, Long-time existence & convergence, Stability & uniqueness · Same author Yunyan Yang · Shared terms “weighted graphs”, “Ollivier”, “Lin”

  5. The Ollivier Ricci flow with prescribed curvature on infinite graphs

    Bobo Hua, Yong Lin and Shuang Liu PDF

    Topics Discrete & graph Ricci, Stability & uniqueness · Shared terms “edge weights”, “Ollivier Ricci”, “Yau Ricci”

  6. On the Ricci flow on Trees

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

    Topics Discrete & graph Ricci · Shared terms “edge weights”, “Ollivier Ricci”, “Lin”

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

    Shuliang Bai and Bobo Hua PDF

    Topics Discrete & graph Ricci · Shared terms “edge weights”, “Yau Ricci”, “Lin”

  8. The Ricci flow with prescribed curvature on graphs

    Yong Lin and Shuang Liu PDF

    Topics Discrete & graph Ricci, Long-time existence & convergence, Stability & uniqueness · Shared terms “edge weights”, “Lin”, “Yau Ricci”

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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) Kähler geometry Complex manifolds with compatible Riemannian metrics. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

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.