Vector Centrality in Hypergraphs

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

The transition to synchronization of networked systems

Abstract We study the synchronization properties of a generic networked dynamical system, and show that, under a suitable approximation, the transition to synchronization can be predicted with the only help of eigenvalues and eigenvectors of the graph Laplacian matrix. The transition comes out to be made of a well defined sequence of events, each of which corresponds to a specific clustered state. The network’s nodes involved in each of the clusters can be identified, and the value of the coupling strength at which the events are taking place can be approximately ascertained. Finally, we present large-scale simulations which show the accuracy of the approximation made, and of our predictions in describing the synchronization transition of both synthetic and real-world large size networks, and we even report that the observed sequence of clusters is preserved in heterogeneous networks made of slightly non-identical systems