Ricci Curvature and Ricci Flow for Graphs and Hypergraphs

Shape measures are fundamental to the analysis of mathematical structures, particularly in areas such as mathematics and mathematical physics. Concepts like gradients and curvatures are commonly employed to describe the "shape" of higher-dimensional objects. However, applying these ideas to discrete structures, such as networks, presents significant challenges, largely due to the lack of inherent geometric embeddings. This thesis explores the discretization of Ricci curvature, focusing specifically on Ollivier-Ricci curvature, and investigates its applications in understanding the structural properties of both graphs and hypergraphs. In the first part of the thesis, we focus on graphs and present original contributions related to Ollivier-Ricci curvature, establishing new theoretical bounds for both exact and approximate computations. This helps us in understanding how curvature can be leveraged to capture the underlying structure of complex networked systems, providing insight into various network phenomena that more conventional metrics fail to address. The mathematical challenges associated with adapting continuous curvature notions to discrete graphs are also explored, offering a refined framework for network analysis. In the second part of the thesis, we extend this analysis to hypergraphs, which provide a more accurate model for systems exhibiting higher-order interactions, such as biochemical and social networks, where multiple entities interact simultaneously. We develop a novel curvature-guided diffusion process, coupled with topological surgery and edge-weight re-normalization, to detect influential cores within these hypergraphs. This approach allows us to uncover key structural components in both directed and undirected hypergraphs. We apply our method to seven metabolic networks modeled as directed hypergraphs and two co-authorship networks modeled as undirected hypergraphs, demonstrating its effectiveness in identifying central modules and influential groups.