Ollivier-Ricci Curvature of Riemannian Manifolds and Directed Graphs with Applications to Graph Neural Networks
A new mathematical framework could improve how AI models understand complex, non-symmetric networks like social media.
A new academic thesis by researcher Eleanor Wiesler provides a significant bridge between pure mathematics and applied AI, focusing on the Ollivier-Ricci curvature. This concept, rooted in optimal transport theory and the 1-Wasserstein distance, measures how the geometry of a space (like a manifold or a graph) deviates from being 'flat'. Wiesler's work first solidifies the connection between this modern curvature and the classical Ricci curvature from Riemannian geometry, extending foundational theorems like Bonnet-Myers and Levy-Gromov to this broader context.
The core innovation lies in the thesis's second half, where Wiesler presents novel ideas for extending the Ollivier-Ricci curvature framework to directed graphs. Most real-world networks—from social media followings to citation trails—are directed, meaning relationships aren't mutual. Traditional graph analysis often loses this directional data. By creating a rigorous mathematical formalism for curvature on directed graphs, this research provides new tools for quantifying the complex, non-symmetric structure of these networks.
Finally, the thesis directly targets applications in graph machine learning, specifically Graph Neural Networks (GNNs). By integrating these advanced curvature-based metrics, GNN algorithms could gain a more profound, geometrically-informed understanding of network topology. This could lead to improvements in tasks like node classification, link prediction, and community detection, especially for messy, real-world data where directionality and local structure are critical.
- Extends the mathematical framework of Ollivier-Ricci curvature from undirected to directed graphs, capturing asymmetric relationships.
- Bridges high-level differential geometry (theorems like Bonnet-Myers) with practical network science and graph machine learning algorithms.
- Provides a novel geometric lens for Graph Neural Networks (GNNs) to improve learning on complex, real-world network data.
Why It Matters
It provides a rigorous mathematical backbone for building better AI models that understand the directional, messy connections in real-world networks.