Ollivier-Ricci Curvature of Riemannian Manifolds and Directed Graphs with Applications to Graph Neural Networks
Read the original on arXiv AI →The Flow has not summarised this story yet — read it at arXiv AI.
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arXiv:2607. 22381v1 Announce Type: new Abstract: Curvature notions on graphs, particularly Ollivier-Ricci and Forman, have emerged as powerful tools for addressing fundamental issues in Graph Neural Networks (GNNs) such as oversmoothing and oversquashing, but rely almost exclusively on local edge-level comparisons and therefore fail to certify how information actually propagates over long distances.
arXiv:2604. 12211v2 Announce Type: replace Abstract: Ollivier-Ricci curvature (ORC), defined via the Wasserstein distance that captures rich geometric information, has received growing attention in both theory and applications.
The paper reviews the use of optimal transport for comparing undirected, unweighted graphs, focusing on three main distances: Wasserstein, Gromov-Wasserstein, and Bures-Wasserstein. It discusses closed-form solutions for the Wasserstein distance in one dimension, how transport plans identify influential nodes after perturbations, and derives spectral bounds for the Bures-Wasserstein distance to avoid full decompositions. The authors evaluate these distances on synthetic clustering data and a real-world time‑series network for anomaly detection.
arXiv:2606. 17185v1 Announce Type: new Abstract: Graph neural network architectures based on the graph Laplacian approximate the Laplace-Beltrami operator, thus limiting their application to isotropic operators.
arXiv:2203. 04711v2 Announce Type: replace Abstract: We present a framework for embedding graph structured data into a vector space, taking into account node features and topology of a graph into the optimal transport (OT) problem.
The paper introduces Ricci-Diffusion, a graph denoising technique that uses curvature-guided diffusion inspired by Ricci flow. Unlike traditional similarity-driven methods, it modulates local transport in the diffusion kernel based on edge-level curvature, steering edge-weight updates toward a more regular graph geometry. The authors provide theoretical analysis showing curvature’s ability to distinguish graph structures and induce first-order corrections, and demonstrate that the method converges to a stable denoised network, improving structure recovery and downstream performance on real-world and synthetic graphs.