Exploring Ricci Curvature and Ricci Flow in Social and Biological Graphs and Hypergraphs

This thesis leverages algorithmic and graph-theoretic tools to gain new insights into complex networks, focusing on network shape measures, particularly Ricci curvature, and its discrete adaptations. The research spans multiple applications, including networks of functional correlations in brain regions, where we investigate structural changes in ADHD-diseased brain networks. By introducing and comparing Forman-Ricci and Ollivier-Ricci curvatures, we demonstrate their distinct contributions and limitations, showing that one cannot substitute for the other. In ADHD networks, for instance, we identify seven critical edges supported by neuroscience findings. The thesis also provides foundational work on applying Ollivier-Ricci curvature to complex networked systems, establishing theoretical bounds for exact and approximate computations. This analysis enhances our understanding of how curvature can capture underlying structures that elude more conventional metrics. Additionally, we generalize these approaches to hypergraphs, which model higher-order interactions in social and biological networks. Using a novel curvature-guided diffusion process coupled with topological surgery and edge-weight renormalization, we identify influential cores in directed and undirected hypergraphs, validated on metabolic and co-authorship networks.