A two-steps tensor eigenvector centrality for nodes and hyperedges in hypergraphs

TL;DR

This paper introduces a novel tensor-based centrality measure for hypergraphs that captures higher-order interactions and mutual reinforcement between nodes and hyperedges, with proven existence, uniqueness, and practical validation.

Contribution

It proposes a new tensor eigenvector centrality for hypergraphs, extending traditional measures to account for higher-order relationships and providing theoretical guarantees.

Findings

The centrality measure is well-defined and unique due to Perron-Frobenius theorem.

It captures mutual reinforcement between nodes and hyperedges.

Validated on real-world hypergraph datasets.

Abstract

Hypergraphs have been a powerful tool to represent higher-order interactions, where hyperedges can connect an arbitrary number of nodes. Quantifying the relative importance of nodes and hyperedges in hypergraphs is a fundamental problem in network analysis. In this paper, we propose a new tensor-based centrality measure for general hypergraphs. We use a third-order tensor to represent the relationship between nodes and hyperedges. The tensor's positive Perron vector is defined as the centrality vector of the hypergraph. The existence and uniqueness of this centrality vector are guaranteed by the Perron-Frobenius theorem for tensors. This new centrality measure captures a higher-order mutual reinforcement mechanism: a node's importance is determined by the importance of its incident hyperedges and the other nodes within these hyperedges; symmetrically, a hyperedge's importance is determined by the importance of its constituent nodes and the other hyperedges containing these nodes. We further provide a combinatorial interpretation by proving that the centrality vector represents the limit geometric capacity of two-steps expansion trees. We illustrate the centrality measure on real-world hypergraph datasets.