Paper · Jan 6, 2026

The weighted Forman and Lin-Lu-Yau Ricci flow on graphs

Shuliang Bai, Shuang Liu, Xin Lai

math.DGarXiv:2601.02673
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Abstract

In this paper, we propose a type of Ricci flow on graphs where the probability distribution for the Lin-Lu-Yau curvature remains constant over time, and also study the related Forman curvature flow. These two curvature flows coincide on trees. We first prove the existence and uniqueness of solutions for both curvature flows in general graphs. Then, we obtain that the normalized curvature flow on trees converges to a constant curvature metric, and under the uniform measure, a complete classification of trees can be obtained based on the convergence results.

Ricci flowOther curvature flowsStability & uniquenessDiscrete & 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)8Dawkins 2024 Ricci Flow on Weighted Digraphs with Balancing Factor (Shuliang Bai, Rui Li, Shuang Liu et al., 2025)5Bai 2025 On the Ricci flow on Trees (Shuliang Bai, Bobo Hua, Yong Lin et al., 2025)1Bai 2025 The Ricci flow with prescribed curvature on graphs (Yong Lin and Shuang Liu, 2026)3Lin 2026 The Ollivier Ricci flow with prescribed curvature on infinite graphs (Bobo Hua, Yong Lin and Shuang Liu, 2026)4Hua 2026 Evolving edge weights via local entropy flow and cohesion flow on graphs (Juan Zhao, Jicheng Ma, Yunyan Yang et al., 2026)7Zhao 2026 Equivalence of Lin–Lu–Yau curvature and 1/2-Ollivier curvature on weighted graphs (Shiping Liu and Yunyan Yang, 2026)6Liu 2026 Curvature Diffusion of Inverse-weight Lin–Lu–Yau Ricci Flow on Finite Trees (Shuliang Bai and Bobo Hua, 2026)2Bai 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the Ricci flow on Trees

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

    Topics Discrete & graph Ricci · Same authors Shuliang Bai, Shuang Liu · Shared terms “Lin”, “Yau”

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

    Shuliang Bai and Bobo Hua PDF

    Topics Discrete & graph Ricci · Same author Shuliang Bai · Shared terms “Yau Ricci”, “Lin”

  3. The Ricci flow with prescribed curvature on graphs

    Yong Lin and Shuang Liu PDF

    Topics Discrete & graph Ricci, Other curvature flows, Stability & uniqueness · Same author Shuang Liu · Shared terms “Lin”, “Yau Ricci”, “Yau curvature”

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

    Bobo Hua, Yong Lin and Shuang Liu PDF

    Topics Discrete & graph Ricci, Stability & uniqueness · Same author Shuang Liu · Shared terms “Yau Ricci”, “Lin”

  5. Ricci Flow on Weighted Digraphs with Balancing Factor

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

    Topics Discrete & graph Ricci, Stability & uniqueness · Same authors Shuliang Bai, Shuang Liu, Xin Lai

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

    Shiping Liu and Yunyan Yang PDF

    Topics Discrete & graph Ricci, Other curvature flows, Stability & uniqueness · Shared terms “Yau curvature”, “Lin”, “coincides”

  7. 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, Stability & uniqueness · Shared terms “Yau Ricci”, “Lin”

  8. A Ricci flow on graphs from effective resistance

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

    Topics Discrete & graph 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) Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) 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) 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.