We extend the concept of eigenvector centrality to multiplex networks, and
introduce several alternative parameters that quantify the importance of nodes
in a multi-layered networked system, including the definition of vectorial-type
centralities. In addition, we rigorously show that, under reasonable
conditions, such centrality measures exist and are unique. Computer experiments
and simulations demonstrate that the proposed measures provide substantially
different results when applied to the same multiplex structure, and highlight
the non-trivial relationships between the different measures of centrality
introduced
The uniqueness of the Perron vector of a nonnegative block matrix associated to a multiplex network is discussed. The conclusions come from the relationships between the irreducibility of some nonnegative block matrix associated to a multiplex network and the irreducibility of the corresponding matrices to each layer as well as the irreducibility of the adjacency matrix of the projection network. In addition the computation of that Perron vector in terms of the Perron vectors of the blocks is also addressed. Finally we present the precise relations that allow to express the Perron eigenvector of the multiplex network in terms of the Perron eigenvectors of its layers.
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