Discrete curvature on graphs from the effective resistance* *Funding: KD was supported by The Alan Turing Institute under the EPSRC Grant EP/N510129/1. RL acknowledges support from the EPSRC Grant Nos. EP/V013068/1 and EP/V03474X/1.

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 approximate 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.