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.Comment: 10 pages, 9 figure