Combinatorial, Bakry-\'Emery, Ollivier's Ricci curvature notions and their motivation from Riemannian geometry

TL;DR

This survey explores three graph curvature notions—combinatorial, Bakry-Émery, and Ollivier's Ricci—highlighting their definitions, motivations from Riemannian geometry, and related global geometric results.

Contribution

It provides a comprehensive comparison of discrete curvature notions with their Riemannian counterparts, including definitions, motivations, and related global geometric theorems.

Findings

Comparison of discrete and Riemannian curvature concepts

Connections between graph curvature and global geometric results

Insights into the motivation behind different curvature notions

Abstract

In this survey, we study three different notions of curvature that are defined on graphs, namely, combinatorial curvature, Bakry-\'Emery curvature, and Ollivier's Ricci curvature. For each curvature notion, the definition and its motivation from Riemannian geometry will be explained. Moreover, we bring together some global results and geometric concepts in Riemannian geometry that are related to curvature (e.g. Bonnet-Myers theorem, Laplacian operator, Lichnerowicz theorem, Cheeger constant), and then compare them to the discrete analogues in some (if not all) of the discrete curvature notions. The structure of this survey is as follows: the first chapter is dedicated to relevant background in Riemannian geometry. Each following chapter is focussing on one of the discrete curvature notions. This survay is an MSc dissertation in Mathematical Sciences at Durham University.