Bakry-\'Emery curvature sharpness and curvature flow in finite weighted graphs. I. Theory

TL;DR

This paper introduces a generalized curvature flow on weighted graphs based on Bakry-Émery calculus, which evolves the weights to achieve curvature sharpness, extending previous concepts to non-reversible and degenerate cases.

Contribution

It develops a new curvature flow framework for mixed weighted graphs, accommodating non-reversible and degenerate edges, and extends Bakry-Émery curvature notions to this general setting.

Findings

Flow preserves Markovian property and converges to curvature sharp graphs.

Extended curvature concepts to non-reversible, degenerate weighted graphs.

Fundamental properties and examples of curvature sharp vertices and graphs.

Abstract

In this sequence of two papers, we introduce a curvature flow on (mixed) weighted graphs which is based on the Bakry-\'Emery calculus. The flow is described via a time-continuous evolution through the weighting schemes. By adapting this flow to preserve the Markovian property, its limits turn out to be curvature sharp. Our aim is to present the flow in the most general case of not necessarily reversible random walks allowing laziness, including vanishing transition probabilities along some edges ("degenerate" edges). This approach requires to extend all concepts (in particular, the Bakry-\'Emery curvature related notions) to this general case and it leads to a distinction between the underlying topology (a mixed combinatorial graph) and the weighting scheme (given by transition rates). We present various results about curvature sharp vertices and weighted graphs as well as some fundamental properties of this new curvature flow. This paper is accompanied by a second paper discussing the curvature flow implementation in Python for practical use. In this second paper we present examples and exhibit further properties of the flow.