Bakry-Émery curvature sharpness and curvature flow in finite weighted graphs: theory

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

Abstract

In this sequence of two papers, we introduce a curvature flow on (mixed) weighted graphs which is based on the Bakry-Émery 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-Émery 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 another paper discussing the curvature flow implementation in Python for practical use, where we present various examples and exhibit further properties of the flow, like stability properties of curvature flow equilibria.

Citation

Cushing, D., Kamtue, S., Liu, S., Münch, F., Peyerimhoff, N., & Snodgrass, B. (2025). Bakry-Émery curvature sharpness and curvature flow in finite weighted graphs: theory. manuscripta mathematica, 176(1), Article 11. https://doi.org/10.1007/s00229-024-01606-7

• Journal Article Type: Article
• Acceptance Date: Dec 11, 2024
• Online Publication Date: Jan 24, 2025
• Publication Date: Feb 1, 2025
• Deposit Date: Jan 8, 2025
• Publicly Available Date: Jan 29, 2025
• Journal: manuscripta mathematica
• Print ISSN: 0025-2611
• Electronic ISSN: 1432-1785
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 176
• Issue: 1
• Article Number: 11
• DOI: https://doi.org/10.1007/s00229-024-01606-7
• Keywords: 60J27, Secondary: 53A70, 05C82, Primary: 53E20
• Public URL: https://durham-repository.worktribe.com/output/3327476