Ollivier-Ricci Curvature of Riemannian Manifolds and Directed Graphs with Applications to Graph Neural Networks

Abstract:This thesis is an exposition of Ollivier-Ricci Curvature of metric spaces as introduced by Yann Ollivier, which is based upon the 1-Wasserstein Distance and optimal transport theory. We present some of the major results and proofs that connect Ollivier-Ricci curvature with classical Ricci curvature of Riemannian manifolds, including extensions of various theoretical bounds and theorems such as Bonnet-Myers and Levy-Gromov. Then we shift to results introduced by Lin-Lu-Yau on an extension of Ollivier-Ricci curvature on graphs, as well as the work of Jost-Liu on proving various combinatorial bounds for graph Ollivier-Ricci curvature. At the end of this thesis we present novel ideas and proofs regarding extensions of these results to directed graphs, and finally applications of graph-based Ollivier-Ricci curvature to various algorithms in network science and graph machine learning.

Differential Geometry (math.DG); Artificial Intelligence (cs.AI); Social and Information Networks (cs.SI); Combinatorics (math.CO)

arXiv:2604.14211 [math.DG]

(or arXiv:2604.14211v2 [math.DG] for this version)

https://doi.org/10.48550/arXiv.2604.14211 Focus to learn more arXiv-issued DOI via DataCite

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