Finding Influential Cores via Normalized Ricci Flows in Directed and Undirected Hypergraphs with Applications

Abstract: Many biological and social systems are naturally represented as edge-weighted directed or undirected hypergraphs since they exhibit group interactions involving three or more system units as opposed to pairwise interactions that can be incorporated in graph-theoretic representations. However, finding influential cores in hypergraphs is still not as extensively studied as their graph-theoretic counter-parts. To this end, we develop and implement a hypergraph-curvature guided discrete time diffusion process with suitable topological surgeries and edge-weight re-normalization procedures for both undirected and directed weighted hypergraphs to find influential cores. We successfully apply our framework for directed hypergraphs to seven metabolic hypergraphs and our framework for undirected hypergraphs to two social (co-authorship) hypergraphs to find influential cores, thereby demonstrating the practical feasibility of our approach. In addition, we prove a theorem showing that a certain edge weight re-normalization procedure in a prior research work for Ricci flows for edge-weighted graphs has the undesirable outcome of modifying the edge-weights to negative numbers, thereby rendering the procedure impossible to use. To the best of our knowledge, this seems to be one of the first articles that formulates algorithmic approaches for finding core(s) of (weighted or unweighted) directed hypergraphs.