Bakry-\'Emery curvature on graphs as an eigenvalue problem

TL;DR

This paper reformulates Bakry-Émery curvature on graphs as an eigenvalue problem, revealing its properties and behavior under graph operations, and extends these insights to Riemannian manifolds.

Contribution

It introduces a novel eigenvalue-based formulation of graph curvature, enabling conceptual analysis and extension to manifolds, and confirms a conjecture on curvature behavior under graph modifications.

Findings

Curvature as a function of dimension is analytic, increasing, and concave until a threshold.

Curvature of Cartesian products is derived from component curvatures.

Curvature does not decrease under certain graph modifications.

Abstract

In this paper, we reformulate the Bakry-\'Emery 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.