用曲率分析图结构，提升图神经网络性能
Ollivier-Ricci Curvature of Riemannian Manifolds and Directed Graphs with Applications to Graph Neural Networks
- 基于最优传输定义图的Ollivier曲率，连接连续与离散几何
- 首次将曲率理论扩展到有向图，突破传统限制
- 为图神经网络提供新几何视角，适合网络科学与机器学习研究者
本论文系统阐述了由Yann Ollivier提出的度量空间的Ollivier-Ricci曲率，其基于1-Wasserstein距离与最优传输理论。我们展示了该曲率与黎曼流形经典Ricci曲率之间的关键联系，包括对Bonnet-Myers和Levy-Gromov等经典定理的推广。随后，介绍Lin-Lu-Yau在图上的曲率拓展工作，以及Jost-Liu对图Ollivier曲率的组合界证明。最后，本文提出曲率理论在有向图上的新扩展，并展示其在复杂网络与图机器学习算法中的应用。
原文摘要 · Abstract (English)
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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