Node and Edge Nonlinear Eigenvector Centrality for Hypergraphs

TL;DR

This paper introduces a novel nonlinear eigenvector centrality measure for hypergraphs, capturing higher-order interactions and enabling efficient computation of important nodes and hyperedges.

Contribution

It generalizes existing network centrality concepts to hypergraphs using nonlinear Perron-Frobenius theory and provides an efficient iterative solution method.

Findings

Unique solutions for the nonlinear eigenvalue problems are guaranteed.

The proposed measures effectively identify important nodes and hyperedges.

Illustrations on real data demonstrate practical applicability.

Abstract

Network scientists have shown that there is great value in studying pairwise interactions between components in a system. From a linear algebra point of view, this involves defining and evaluating functions of the associated adjacency matrix. Recent work indicates that there are further benefits from accounting directly for higher order interactions, notably through a hypergraph representation where an edge may involve multiple nodes. Building on these ideas, we motivate, define and analyze a class of spectral centrality measures for identifying important nodes and hyperedges in hypergraphs, generalizing existing network science concepts. By exploiting the latest developments in nonlinear Perron-Frobenius theory, we show how the resulting constrained nonlinear eigenvalue problems have unique solutions that can be computed efficiently via a nonlinear power method iteration. We illustrate the measures on realistic data sets.