Bakry-Émery curvature on graphs as an eigenvalue problem

Cushing, David; Kamtue, Supanat; Liu, Shiping; Peyerimhoff, Norbert

Abstract

In this paper, we reformulate the Bakry-Émery curvature on a weighted graph in terms of the smallest eigenvalue of a rank one perturbation of the so-called curvature matrix using Schur complement. This new viewpoint allows us to show various curvature function properties in a very conceptual way. We show that the curvature, as a function of the dimension parameter, is analytic, strictly monotone increasing and strictly concave until a certain threshold after which the function is constant. Furthermore, we derive the curvature of the Cartesian product using the crucial observation that the curvature matrix of the product is the direct sum of each component. Our approach of the curvature functions of graphs can be employed to establish analogous results for the curvature functions of weighted Riemannian manifolds. Moreover, as an application, we confirm a conjecture (in a general weighted case) of the fact that the curvature does not decrease under certain graph modifications.

Citation

Cushing, D., Kamtue, S., Liu, S., & Peyerimhoff, N. (2022). Bakry-Émery curvature on graphs as an eigenvalue problem. Calculus of Variations and Partial Differential Equations, 61, Article 62. https://doi.org/10.1007/s00526-021-02179-z

• Journal Article Type: Article
• Acceptance Date: Dec 27, 2022
• Online Publication Date: Feb 7, 2022
• Publication Date: 2022
• Deposit Date: May 16, 2022
• Publicly Available Date: Feb 7, 2023
• Journal: Calculus of Variations and Partial Differential Equations
• Print ISSN: 0944-2669
• Electronic ISSN: 1432-0835
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 61
• Article Number: 62
• DOI: https://doi.org/10.1007/s00526-021-02179-z
• Public URL: https://durham-repository.worktribe.com/output/1206094