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

• URL: http://arxiv.org/abs/2604.14211v1
• Date: Mon, 06 Apr 2026 22:28:17 GMT
• Title: Ollivier-Ricci Curvature of Riemannian Manifolds and Directed Graphs with Applications to Graph Neural Networks
• Abstract summary: This thesis is an exposition of Ollivier-Ricci Curvature of metric spaces as introduced by Yann Ollivier.<n>We present some of the major results and proofs that connect Ollivier-Ricci curvature with classical Ricci curvature.<n>We present novel ideas and proofs regarding extensions of these results to directed graphs, and finally applications of graph-based Ollivier-Ricci curvature algorithms in network science and graph machine learning.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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