Enhanced Forman curvature and its relation to Ollivier curvature

¹ The Beyond Center for Fundamental Science, Arizona State University - Tempe AZ, USA
² Department of Informatics, University of Sussex - Falmer, UK
³ SwissScientific Technologies SA - rue du Rhone 59, CH-1204 Geneva, Switzerland

Received: 27 February 2021
Accepted: 17 March 2021

Abstract

Recent advances in emergent geometry and discretized approaches to quantum gravity have relied upon the notion of a discrete measure of graph curvature. We focus on the two main measures that have been studied, the so-called Ollivier-Ricci and Forman-Ricci curvatures. These two approaches have a very different origin, and both have advantages and disadvantages. In this work we study the relationship between the two measures for a class of graphs that are important in quantum gravity applications. We discover that under a specific set of circumstances they are equivalent, opening up the possibility of replacing the more fundamental Ollivier-Ricci curvature by the computationally more accessible Forman-Ricci curvature in certain applications to models of emergent spacetime and quantum gravity.