Paper · Dec 23, 2022

Bakry-Émery curvature sharpness and curvature flow in finite weighted graphs. II. Implementation

David Cushing, Supanat Kamtue, Shiping Liu, Florentin Münch, Norbert Peyerimhoff, Ben Snodgrass

math.CAWider flowsarXiv:2212.12401
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Abstract

In this second part of a sequence of two papers, we discuss the implementation of a curvature flow on weighted graphs based on the Bakry-Émery calculus. This flow can be adapted to preserve the Markovian property and its limits as time goes to infinity turn out to be curvature sharp weighted graphs. After reviewing some of the main results of the first paper concerned with the theoretical aspects, we present various examples (random graphs, paths, cycles, complete graphs, wedge sums and Cartesian products of complete graphs, hypercubes) and exhibit further properties of this flow. One particular aspect in our investigations is asymptotic stability and instability of curvature flow equilibria. The paper ends with a description of the available Python functions and routines available in the ancillary file. We hope that the explanations of the Python implementation via examples will help users to carry out their own curvature flow experiments.

Other curvature flowsLimit spaces & RCDStability & uniqueness

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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 Compactness and rigidity of self-shrinking surfaces (Tang-Kai Lee, 2021)8Lee 2021 Bakry-Émery curvature sharpness and curvature flow in finite weighted graphs. I. Theory (David Cushing, Supanat Kamtue, Shiping Liu et al., 2022)1Cushing 2022 Time analyticity for the heat equation under Bakry-Émery Ricci curvature condition (Ling Wu, 2022)5Wu 2022 A version of Bakry-Émery Ricci flow on a finite graph (Bobo Hua, Yong Lin and Tao Wang, 2024)3Hua 2024 Rigidity of weighted manifolds via classification results for semilinear equations (Giulio Ciraolo, Alberto Farina and Troy Petitt, 2025)4Ciraolo 2025 Weighted volume comparison and monotonicity for L^p-bound of Bakry-Émery Ricci curvature (Jintao Ye and Xiaohua Zhu, 2026)6Ye 2026 Equivalence of Lin–Lu–Yau curvature and 1/2-Ollivier curvature on weighted graphs (Shiping Liu and Yunyan Yang, 2026)2Liu 2026 On the geometry of asymptotically conical Ricci shrinkers (I): weighted analysis (Yu Li, Longhuan Yan and Wenjia Zhang, 2026)7Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Bakry-Émery curvature sharpness and curvature flow in finite weighted graphs. I. Theory

    David Cushing, Supanat Kamtue, Shiping Liu et al.Wider flows PDF

    Topics Limit spaces & RCD, Other curvature flows · Same authors David Cushing, Supanat Kamtue, Shiping Liu · Shared terms “weighted graphs”, “Émery curvature”, “curvature sharpness”

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

    Shiping Liu and Yunyan Yang PDF

    Topics Other curvature flows, Stability & uniqueness · Same author Shiping Liu · Shared terms “weighted graphs”

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

    Bobo Hua, Yong Lin and Tao Wang PDF

    Topics Limit spaces & RCD, Stability & uniqueness · Shared terms “Bakry Émery”

  4. Rigidity of weighted manifolds via classification results for semilinear equations

    Giulio Ciraolo, Alberto Farina and Troy Petitt PDF

    Topics Limit spaces & RCD, Stability & uniqueness · Shared terms “Bakry Émery”, “sharpness”

  5. Time analyticity for the heat equation under Bakry-Émery Ricci curvature condition

    Ling Wu PDF

    Topics Limit spaces & RCD · Shared terms “Bakry Émery”

  6. Weighted volume comparison and monotonicity for -bound of Bakry-Émery Ricci curvature

    Jintao Ye and Xiaohua Zhu PDF

    Topics Limit spaces & RCD · Shared terms “Bakry Émery”

  7. On the geometry of asymptotically conical Ricci shrinkers (I): weighted analysis

    Yu Li, Longhuan Yan and Wenjia Zhang PDF

    Topics Limit spaces & RCD, Stability & uniqueness

  8. Compactness and rigidity of self-shrinking surfaces

    Tang-Kai LeeWider flows PDF

    Topics Limit spaces & RCD, Stability & uniqueness · Shared terms “theoretical”

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.

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) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997) 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)

In the guidebook

  1. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  2. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

  3. 11B.3
    Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities

    Hamilton’s compactness theorem and the missing injectivity radius bound.