Vector Centrality in Hypergraphs

TL;DR

This paper introduces a novel vector-based centrality measure for hypergraphs that captures multi-order interactions and reveals diverse node roles, outperforming traditional scalar metrics in synthetic and real-world networks.

Contribution

It proposes a new vectorial centrality measure for hypergraphs, extending eigenvector centrality to higher-order interactions with enhanced interpretability.

Findings

The measure effectively identifies influential nodes in hypergraphs.

It uncovers different roles of nodes at various interaction orders.

Compared to scalar measures, it provides richer insights into network structure.

Abstract

Identifying the most influential nodes in networked systems is of vital importance to optimize their function and control. Several scalar metrics have been proposed to that effect, but the recent shift in focus towards network structures which go beyond a simple collection of dyadic interactions has rendered them void of performance guarantees. We here introduce a new measure of node's centrality, which is no longer a scalar value, but a vector with dimension one lower than the highest order of interaction in a hypergraph. Such a vectorial measure is linked to the eigenvector centrality for networks containing only dyadic interactions, but it has a significant added value in all other situations where interactions occur at higher-orders. In particular, it is able to unveil different roles which may be played by the same node at different orders of interactions -- information that is otherwise impossible to retrieve by single scalar measures. We demonstrate the efficacy of our measure with applications to synthetic networks and to three real world hypergraphs, and compare our results with those obtained by applying other scalar measures of centrality proposed in the literature.