Enhanced Forman curvature and its relation to Ollivier curvature

TL;DR

This paper investigates the relationship between Forman-Ricci and Ollivier-Ricci curvatures in graphs relevant to quantum gravity, revealing conditions under which they are equivalent, which could enhance models of emergent spacetime.

Contribution

It establishes conditions for the equivalence of two key graph curvature measures, enabling combined use in quantum gravity models.

Findings

Identifies circumstances where Forman and Ollivier curvatures are equivalent.

Provides insights into applying curvature measures in quantum gravity.

Suggests potential for improved emergent spacetime models.

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, potentially opening up the possibility of exploiting the relative strengths of both approaches in models of emergent spacetime and quantum gravity.