Discrete curvature on graphs from the effective resistance

TL;DR

This paper introduces a new discrete curvature measure for graphs based on effective resistance, connecting it to existing curvatures and demonstrating convergence to continuous curvature in Euclidean random graphs.

Contribution

It proposes a novel resistance-based curvature for graphs that is computationally efficient and relates to established discrete curvatures, with potential broad applications.

Findings

Relation to Ollivier, Forman, and combinatorial curvatures

Convergence to continuous curvature in Euclidean random graphs

Efficient computation and theoretical analysis

Abstract

This article introduces a new approach to discrete curvature based on the concept of effective resistances. We propose a curvature on the nodes and links of a graph and present the evidence for their interpretation as a curvature. Notably, we find a relation to a number of well-established discrete curvatures (Ollivier, Forman, combinatorial curvature) and show evidence for convergence to continuous curvature in the case of Euclidean random graphs. Being both efficient to calculate and highly amenable to theoretical analysis, these resistance curvatures have the potential to shed new light on the theory of discrete curvature and its many applications in mathematics, network science, data science and physics.