Abstract
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.
Highlights
One of the main goals of network analysts and data scientists is to identify relevant components in a network
The last eigenvector-based centrality measure that we found in the literature is described in [16, 17] and relies on the use of the multilayer adjacency tensor B
In this paper we have introduced the f eigenvector centrality: a new multidimensional eigenvector-based centrality measure for nodes and layers in multiplex networks with undirected layers
Summary
One of the main goals of network analysts and data scientists is to identify relevant components in a network. In order to exploit the Perron–Frobenius theory for matrices for describing eigenvector-based centrality measures in this framework, a popular idea in the current literature is to work with suitable matrix eigenvector equations. This approach is a form of “linearization” of the higher-order structure which, as we shall further discuss later in the paper, might lead to a loss of information. Our main contribution is the introduction of a novel centrality measure based on the Perron eigenvector of a multi-homogeneous map These maps generalize the concept of homogenous functions and their definition is recalled at the beginning of section 4.1 below.
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