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

2025-02-22

AI Summary

This paper addresses the problem of identifying core nodes in directed weighted hypergraphs under group interactions involving three or more nodes. Methodologically, it proposes the first discrete diffusion framework for directed weighted hypergraphs grounded in normalized Ricci curvature. It rectifies the negative-weight issue arising from conventional graph Ricci flow normalization by introducing hypergraph curvature-guided diffusion, topological surgery, and edge-weight adaptive normalization, and formulates a discrete-time dynamical system model. Contributions include: (i) the first algorithmic solution for core discovery in directed hypergraphs; (ii) a unified treatment of both directed and undirected, weighted hypergraphs; and (iii) empirical validation on seven metabolic hypergraphs and two collaboration-network hypergraphs, where the method robustly identifies biologically meaningful, high-influence core nodes—demonstrating effectiveness, robustness, and practical applicability.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Content-based information diffusion
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• Finding influential cores in hypergraphs.
• Developing hypergraph-curvature guided diffusion process.
• Applying framework to metabolic and social hypergraphs.

Innovation

• Normalized Ricci Flows
• Directed Hypergraphs Analysis
• Edge-weight Re-normalization Procedures