Paper · Oct 5, 2020

Ollivier Ricci-flow on weighted graphs

Shuliang Bai, Yong Lin, Linyuan Lu, Zhiyu Wang, Shing-Tung Yau

math.DGarXiv:2010.01802v9
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Abstract

We study the existence of solutions of Ricci flow equations of Ollivier-Lin-Lu-Yau curvature defined on weighted graphs. Our work is motivated by in which the discrete time Ricci flow algorithm has been applied successfully as a discrete geometric approach in detecting complex networks. Our main result is the existence and uniqueness theorem for solutions to a continuous time normalized Ricci flow. We also display possible solutions to the Ricci flow on path graph and prove the Ricci flow on finite star graph with at least three leaves converges to constant-weighted star.

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Submitted
Oct 5, 2020
Last revised
Jun 20, 2025 (v9)
Subjects
math.DG

Comments: 21 pages

Ricci flowStability & uniquenessDiscrete & graph Ricci

Its neighbourhood

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 2016, 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)5Ma 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)1Bai 2025 The weighted Forman and Lin-Lu-Yau Ricci flow on graphs (Shuliang Bai, Shuang Liu and Xin Lai, 2026)4Bai 2026 The Ricci flow with prescribed curvature on graphs (Yong Lin and Shuang Liu, 2026)7Lin 2026 The Ollivier Ricci flow with prescribed curvature on infinite graphs (Bobo Hua, Yong Lin and Shuang Liu, 2026)2Hua 2026 Equivalence of Lin–Lu–Yau curvature and 1/2-Ollivier curvature on weighted graphs (Shiping Liu and Yunyan Yang, 2026)3Liu 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. On the Ricci flow on Trees

    Shuliang Bai, Bobo Hua, Yong Lin et al.

    Topics Discrete & graph Ricci · Same authors Shuliang Bai, Yong Lin · Shared terms “Ollivier Ricci”, “continuous time”, “time Ricci”

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

    Bobo Hua, Yong Lin and Shuang Liu

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

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

    Shiping Liu and Yunyan Yang

    Topics Discrete & graph Ricci, Stability & uniqueness · Shared terms “weighted graphs”, “Ollivier”, “Yau curvature”

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

    Shuliang Bai, Shuang Liu and Xin Lai

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

  5. A modified Ricci flow on arbitrary weighted graph

    Jicheng Ma and Yunyan Yang

    Topics Discrete & graph Ricci, Stability & uniqueness · Shared terms “weighted graphs”, “Ollivier Ricci”, “complex networks”

  6. Ricci Flow on Weighted Digraphs with Balancing Factor

    Shuliang Bai, Rui Li, Shuang Liu et al.

    Topics Discrete & graph Ricci, Stability & uniqueness · Same author Shuliang Bai · Shared terms “weighted graphs”, “flow algorithm”, “continuous”

  7. The Ricci flow with prescribed curvature on graphs

    Yong Lin and Shuang Liu

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

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

    Shuliang Bai and Bobo Hua

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

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) 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.