How Optimal Transport Can Help Us to Determine Curvature of Complex Networks

Explore the concept of Ollivier Ricci curvature and its applications in network analysis through this 56-minute lecture from the Applied Algebraic Topology Network. Delve into the origins of this geometric notion and its adaptation to discrete structures like directed hypergraphs. Learn how optimal transport problems and Wasserstein distance contribute to defining this edge-based measure, and discover its effectiveness in detecting clustering and sparsity in network structures. Examine the implementation of Ollivier Ricci curvature in analyzing chemical reaction networks, and gain insights into metric measure spaces, undirected and directed graphs, and the properties of directed hypergraphs. Investigate the classification of directed hypergraphs, explore background in network analysis, and understand the significance of Ricci curvature in this context. Conclude with an examination of hyperloops, real network analysis, and random network analysis, followed by a Q&A session and summary.

Introduction
Theory
Metric measure space
Undirected graphs
Directed graphs
Properties of directed hyper graphs
Classification of directed hyper graphs
Background in network analysis
Why Ritchie curvature
Hyperloops
Real network analysis
Random network analysis
Questions
Summary