Bakry-\'Emery curvature sharpness and curvature flow in finite weighted graphs. II. Implementation

TL;DR

This paper details the implementation of a curvature flow on weighted graphs based on Bakry-Émery calculus, demonstrating its properties, examples, and providing Python tools for experimentation.

Contribution

It introduces a practical implementation of curvature flow on graphs, including stability analysis and Python routines, complementing theoretical results from the first paper.

Findings

Curvature flow preserves Markovian property.

Flow limits are curvature sharp graphs.

Implementation includes Python routines for experiments.

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-\'Emery 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.