Three hypergraph eigenvector centralities

TL;DR

This paper extends eigenvector centrality to hypergraphs using tensor eigenvectors, providing new tools to analyze complex systems with multi-way interactions, demonstrated on real-world datasets.

Contribution

It introduces three novel tensor eigenvector centralities for hypergraphs, expanding the analysis of multi-way relationships beyond traditional graph methods.

Findings

Different centralities reveal diverse information in real-world hypergraph data.

Tensor eigenvector centralities can be applied to various complex systems.

Analysis of datasets shows the effectiveness of the proposed methods.

Abstract

Eigenvector centrality is a standard network analysis tool for determining the importance of (or ranking of) entities in a connected system that is represented by a graph. However, many complex systems and datasets have natural multi-way interactions that are more faithfully modeled by a hypergraph. Here we extend the notion of graph eigenvector centrality to uniform hypergraphs. Traditional graph eigenvector centralities are given by a positive eigenvector of the adjacency matrix, which is guaranteed to exist by the Perron-Frobenius theorem under some mild conditions. The natural representation of a hypergraph is a hypermatrix (colloquially, a tensor). Using recently established Perron-Frobenius theory for tensors, we develop three tensor eigenvectors centralities for hypergraphs, each with different interpretations. We show that these centralities can reveal different information on real-world data by analyzing hypergraphs constructed from n-gram frequencies, co-tagging on stack exchange, and drug combinations observed in patient emergency room visits.