Paper · Jun 8, 2026

The Ollivier Ricci flow with prescribed curvature on infinite graphs

Bobo Hua, Yong Lin, Shuang Liu

math.DGarXiv:2606.09017
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Abstract

In this paper, we consider the Ricci flow with prescribed curvature on infinite graphs, which reads as \beginequation* \fracddtω(t)=-(κ(t)-κ^*)ω(t), t>0, \endequation* where is the edge weight, and are Lin-Lu-Yau Ricci curvature and the prescribed curvature on the set of edges, respectively. First, we establish the existence and uniqueness of the solution to the Ricci flow. Furthermore, we prove the convergence of the Ricci flow for graphs with girth at least 6 under two different conditions. Our convergence result aligns with the conclusion of Rodin and Sullivan (J Differ Geom, 26(2) 1987) that a circle packing in the plane with the hexagonal pattern is the regular hexagonal packing.

Ricci flowStability & 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 version of Bakry-Émery Ricci flow on a finite graph (Bobo Hua, Yong Lin and Tao Wang, 2024)4Hua 2024 Combinatorial Ricci flows on infinite disk triangulations (Huabin Ge, Bobo Hua and Puchun Zhou, 2025)6Ge 2025 On the Ricci flow on Trees (Shuliang Bai, Bobo Hua, Yong Lin et al., 2025)2Bai 2025 The weighted Forman and Lin-Lu-Yau Ricci flow on graphs (Shuliang Bai, Shuang Liu and Xin Lai, 2026)5Bai 2026 The Ricci flow with prescribed curvature on graphs (Yong Lin and Shuang Liu, 2026)1Lin 2026 Evolving edge weights via local entropy flow and cohesion flow on graphs (Juan Zhao, Jicheng Ma, Yunyan Yang et al., 2026)8Zhao 2026 Equivalence of Lin–Lu–Yau curvature and 1/2-Ollivier curvature on weighted graphs (Shiping Liu and Yunyan Yang, 2026)7Liu 2026 Curvature Diffusion of Inverse-weight Lin–Lu–Yau Ricci Flow on Finite Trees (Shuliang Bai and Bobo Hua, 2026)3Bai 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The Ricci flow with prescribed curvature on graphs

    Yong Lin and Shuang Liu PDF

    Topics Discrete & graph Ricci, Stability & uniqueness · Same authors Yong Lin, Shuang Liu · Shared terms “prescribed curvature”, “Differ Geom”, “Lin”

  2. On the Ricci flow on Trees

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

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

  3. 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 Bobo Hua · Shared terms “Yau Ricci”, “Lin”, “edge weights”

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

    Bobo Hua, Yong Lin and Tao Wang PDF

    Topics Discrete & graph Ricci, Stability & uniqueness · Same authors Bobo Hua, Yong Lin

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

    Shuliang Bai, Shuang Liu and Xin Lai PDF

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

  6. Combinatorial Ricci flows on infinite disk triangulations

    Huabin Ge, Bobo Hua and Puchun Zhou PDF

    Topics Discrete & graph Ricci, Stability & uniqueness · Same author Bobo Hua · Shared terms “circle packings”

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

    Shiping Liu and Yunyan Yang PDF

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

  8. 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 “edge weights”, “Ollivier Ricci”, “Yau 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) 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.