Paper · Sep 26, 2025

On the Ricci flow on Trees

Shuliang Bai, Bobo Hua, Yong Lin, Shuang Liu

math.DGarXiv:2509.22140v3
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Abstract

In this paper, we study the evolution of metrics on finite trees under continuous-time Ricci flows based on the Lin-Lu-Yau version of Ollivier Ricci curvature. We analyze long-time dynamics of edge weights and curvatures, providing precise characterizations of their limiting behaviors. We prove that the Ricci flow converges to metric with zero curvature on edges whose normalized weights converge to positive values only if the tree is a caterpillar tree.

Comments: 36

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 version of Bakry-Émery Ricci flow on a finite graph (Bobo Hua, Yong Lin and Tao Wang, 2024)5Hua 2024 Characterization of Infinite Ideal Polyhedra in Hyperbolic 3-Space via Combinatorial Ricci Flow (Huabin Ge, Bobo Hua, Hao Yu et al., 2025)8Ge 2025 Ricci Flow on Weighted Digraphs with Balancing Factor (Shuliang Bai, Rui Li, Shuang Liu et al., 2025)6Bai 2025 The weighted Forman and Lin-Lu-Yau Ricci flow on graphs (Shuliang Bai, Shuang Liu and Xin Lai, 2026)3Bai 2026 The Ricci flow with prescribed curvature on graphs (Yong Lin and Shuang Liu, 2026)4Lin 2026 The Ollivier Ricci flow with prescribed curvature on infinite graphs (Bobo Hua, Yong Lin and Shuang Liu, 2026)2Hua 2026 Evolving edge weights via local entropy flow and cohesion flow on graphs (Juan Zhao, Jicheng Ma, Yunyan Yang et al., 2026)7Zhao 2026 Curvature Diffusion of Inverse-weight Lin–Lu–Yau Ricci Flow on Finite Trees (Shuliang Bai and Bobo Hua, 2026)1Bai 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Curvature Diffusion of Inverse-weight Lin–Lu–Yau Ricci Flow on Finite Trees

    Shuliang Bai and Bobo Hua PDF

    Topics Discrete & graph Ricci · Same authors Shuliang Bai, Bobo Hua · Shared terms “finite trees”, “Lin”, “edge weights”

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

    Bobo Hua, Yong Lin and Shuang Liu PDF

    Topics Discrete & graph Ricci · Same authors Bobo Hua, Yong Lin, Shuang Liu · Shared terms “Ollivier Ricci”, “edge weights”, “Lin”

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

    Shuliang Bai, Shuang Liu and Xin Lai PDF

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

  4. The Ricci flow with prescribed curvature on graphs

    Yong Lin and Shuang Liu PDF

    Topics Discrete & graph Ricci · Same authors Yong Lin, Shuang Liu · Shared terms “Lin”, “edge weights”, “providing”

  5. A version of Bakry-Émery Ricci flow on a finite graph

    Bobo Hua, Yong Lin and Tao Wang PDF

    Topics Discrete & graph Ricci · Same authors Bobo Hua, Yong Lin · Shared terms “finite trees”, “long time”

  6. Ricci Flow on Weighted Digraphs with Balancing Factor

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

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

  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 · Shared terms “edge weights”, “Ollivier Ricci”, “Lin”

  8. Characterization of Infinite Ideal Polyhedra in Hyperbolic 3-Space via Combinatorial Ricci Flow

    Huabin Ge, Bobo Hua, Hao Yu et al. PDF

    Topics Discrete & graph Ricci · Same author Bobo Hua · Shared terms “characterization”

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.