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Bakry–Émery Curvature Sharpness and Curvature Flow in Finite Weighted Graphs. Implementation

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

Bakry–Émery Curvature Sharpness and Curvature Flow in Finite Weighted Graphs. Implementation Thumbnail


Authors

David Cushing

Supanat Kamtue

Shiping Liu

Florentin Münch



Abstract

In this paper, 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 corresponding paper concerned with the theoretical aspects, we present various examples (random graphs, paths, cycles, complete graphs, wedge sums and Cartesian products of complete graphs, and hypercubes) and exhibit various properties of this flow. One particular aspect of our investigations is asymptotic stability and instability of curvature flow equilibria. The paper ends with a description of the Python functions and routines freely available in an ancillary file on arXiv or via github. We hope that the explanations of the Python implementation via examples will help users to carry out their own curvature flow experiments.

Citation

Cushing, D., Kamtue, S., Liu, S., Münch, F., Peyerimhoff, N., & Snodgrass, B. (2023). Bakry–Émery Curvature Sharpness and Curvature Flow in Finite Weighted Graphs. Implementation. Axioms, 12(6), Article 577. https://doi.org/10.3390/axioms12060577

Journal Article Type Article
Acceptance Date Jun 7, 2023
Online Publication Date Jun 11, 2023
Publication Date 2023-06
Deposit Date Nov 21, 2023
Publicly Available Date Nov 21, 2023
Journal Axioms
Publisher MDPI
Peer Reviewed Peer Reviewed
Volume 12
Issue 6
Article Number 577
DOI https://doi.org/10.3390/axioms12060577
Keywords Geometry and Topology; Logic; Mathematical Physics; Algebra and Number Theory; Analysis
Public URL https://durham-repository.worktribe.com/output/1945917

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