Eigenvector-based centrality measures are among the most popular centrality measures in network science. The underlying idea is intuitive and the mathematical description is extremely simple in the framework of standard, mono-layer networks. Moreover, several efficient computational tools are available for their computation. Moving up in dimensionality, several efforts have been made in the past to describe an eigenvector-based centrality measure that generalizes Bonacich index to the case of multiplex networks. In this work, we propose a new definition of eigenvector centrality that relies on the Perron eigenvector of a multi-homogeneous map defined in terms of the tensor describing the network. We prove that existence and uniqueness of such centrality are guaranteed under very mild assumptions on the multiplex network. Extensive numerical studies are proposed to test the newly introduced centrality measure and to compare it to other existing eigenvector-based centralities.
Following @cite_29 , in order to compute @math , we build the supra-adjacency matrix associated with the multilayer adjacency tensor @math , which is @math block matrix of the form: [ B = [ ] ] where the @math , for @math , are the adjacency matrices of the graphs appearing on each layer, and @math , for @math , are the inter-layer adjacency tensors. If we now let @math be the spectral radius of @math , then @math and the associated eigenvector (if uniquely determined) is @math , where vec is the standard vectorization operator, and @math and @math are as in .
@cite_29
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A challenging problem is to identify the most central agents in interconnected multilayer networks. Here, De present a mathematical framework to calculate centrality in such networks—versatility—and rank nodes accordingly.