Enhanced Forman curvature and its relation to Ollivier curvature

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.

The birth of geometry in exponential random graphs

“…To efficiently employ these notions, however, one must have at one's disposal graphs much bigger than what straightforward Monte Carlo simulations allow (we have considered graphs with a few thousand vertices). Another important quantity that connects graphs and manifolds, frequently seen in the recent literature, is the Ollivier curvature [35][36][37][38][39][40]. (The Ollivier curvature plays a central role in the random graph ensembles of [7][8][9], where it is used for the construction of the graph Hamiltonian, in analogy to how the Ricci curvature is used in the Einstein-Hilbert action.…”

Quantum Flatness in Two-Dimensional CDT Quantum Gravity

“…Our philosophy here is different from that of "combinatorial quantum gravity"[33,34,35,36], which investigates the emergence of geometry from specific ensembles of random graphs, with the ambition of having its Lorentzian structure emerge alongside, something not yet achieved by any model as far as we know. In this approach, one uses the Ollivier-Ricci curvature in its original form for small values of δ, and has recently also compared it to another discrete notion of so-called Forman-Ricci curvature[37].…”