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.

24 pages

References in corpus (27)

• Emergence of scaling in random networks
• Statistical mechanics of complex networks
• The structure and function of complex networks
• Community structure in social and biological networks
• Finding and evaluating community structure in networks
• Modularity and community structure in networks
• The large-scale organization of metabolic networks
• Dynamic reconfiguration of human brain networks during learning
• Networks beyond pairwise interactions: structure and dynamics
• Community structure in directed networks
• Detecting rich-club ordering in complex networks
• The physics of higher-order interactions in complex systems
• Forman curvature for complex networks
• Comparative analysis of two discretizations of Ricci curvature for complex networks
• Community Detection in Large Hypergraphs
• Topological implications of negative curvature for biological and social networks
• Mapping flows on hypergraphs
• Integrality Gaps for Random Integer Programs via Discrepancy
• Hyper-cores promote localization and efficient seeding in higher-order processes
• Systematic evaluation of a new combinatorial curvature for complex networks
• On the Complexity of Newman's Community Finding Approach for Biological and Social Networks
• A New Computationally Efficient Measure of Topological Redundancy of Biological and Social Networks
• Finding Densest $k$-Connected Subgraphs
• The structural evolution of temporal hypergraphs through the lens of hyper-cores
• Effective and Efficient Core Computation in Signed Networks
• On the Simple Quasi Crossing Number of $K_{11}$
• A new transport distance and its associated Ricci curvature of hypergraphs