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.
All the centrality measures listed so far are described as @math non-negative matrices. To retrieve a proper centrality score, i.e., a non-negative real number, one has to aggregate the values appearing in the centrality vectors according to some heuristics. @cite_8 the authors propose to use weighted sums, so that the centrality of a node will be the corresponding entry of the vector @math , where @math is a vector of weights. Different choices of @math provide different rankings. If no intuition information is available concerning the importance of layers, then the (possibly) most appropriate choice is to pick @math , thus deriving
@cite_8
2086243891
Many real-world complex systems consist of a set of elementary units connected by relationships of different kinds. All such systems are better described in terms of multiplex networks, where the links at each layer represent a different type of interaction between the same set of nodes, rather than in terms of (single-layer) networks. In this paper we present a general framework to describe and study multiplex networks, whose links are either unweighted or weighted. In particular we propose a series of measures to characterize the multiplexicity of the systems in terms of: i) basic node and link properties such as the node degree, and the edge overlap and reinforcement, ii) local properties such as the clustering coefficient and the transitivity, iii) global properties related to the navigability of the multiplex across the different layers. The measures we introduce are validated on a genuine multiplex data set of Indonesian terrorists, where information among 78 individuals are recorded with respect to mutual trust, common operations, exchanged communications and business relationships.