Paper · Mar 2, 2024

A Ricci flow on graphs from effective resistance

Aleyah Dawkins, Vishal Gupta, Mark Kempton, William Linz, Jeremy Quail, Harry Richman, Zachary Stier

math.COarXiv:2403.01151
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Abstract

In this paper, we introduce a new notion of curvature on the edges of a graph that is defined in terms of effective resistances. We call this the Ricci–Foster curvature. We study the Ricci flow resulting from this curvature. We prove the existence of solutions to Ricci flow on short time intervals, and prove that Ricci flow preserves graphs with nonnegative (resp. positive) curvature.

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Submitted
Mar 2, 2024
Subjects
math.CO, math.DG

Comments: 14 pages, 8 figures, comments welcome!

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 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)3Bai 2025 The weighted Forman and Lin-Lu-Yau Ricci flow on graphs (Shuliang Bai, Shuang Liu and Xin Lai, 2026)5Bai 2026 The Ollivier Ricci flow with prescribed curvature on infinite graphs (Bobo Hua, Yong Lin and Shuang Liu, 2026)1Hua 2026 Evolving edge weights via local entropy flow and cohesion flow on graphs (Juan Zhao, Jicheng Ma, Yunyan Yang et al., 2026)4Zhao 2026 Equivalence of Lin–Lu–Yau curvature and 1/2-Ollivier curvature on weighted graphs (Shiping Liu and Yunyan Yang, 2026)8Liu 2026 Curvature Diffusion of Inverse-weight Lin–Lu–Yau Ricci Flow on Finite Trees (Shuliang Bai and Bobo Hua, 2026)6Bai 2026 Local Combinatorial Criteria for Geometric Hyper-ideal Triangulations via Combinatorial Ricci Flow (Soichiro Uemura, 2026)7Uemura 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The Ollivier Ricci flow with prescribed curvature on infinite graphs

    Bobo Hua, Yong Lin and Shuang Liu

    Topics Discrete & graph Ricci

  2. Core detection via Ricci curvature flows on weighted graphs

    Juan Zhao, Jicheng Ma, Yunyan Yang et al.

    Topics Discrete & graph Ricci · Shared terms “effective”

  3. On the Ricci flow on Trees

    Shuliang Bai, Bobo Hua, Yong Lin et al.

    Topics Discrete & graph Ricci

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

    Juan Zhao, Jicheng Ma, Yunyan Yang et al.

    Topics Discrete & graph Ricci · Shared terms “resulting”

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

    Shuliang Bai, Shuang Liu and Xin Lai

    Topics Discrete & graph Ricci

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

    Shuliang Bai and Bobo Hua

    Topics Discrete & graph Ricci

  7. Local Combinatorial Criteria for Geometric Hyper-ideal Triangulations via Combinatorial Ricci Flow

    Soichiro Uemura

    Topics Discrete & graph Ricci · Shared terms “interval”

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

    Shiping Liu and Yunyan Yang

    Topics Discrete & graph 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) 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.