Node and layer eigenvector centralities for multiplex networks

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 entrality measure that generalizes the 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

Multi-dimensional, multilayer, nonlinear and dynamic HITS

We introduce a ranking model for temporal multidimensional weighted and directed networks based on the Perron eigenvector of a multi-homogeneous order-preserving map. The model extends to the temporal multilayer setting the HITS algorithm and defines five centrality vectors: two for the nodes, two for the layers, and one for the temporal stamps. Nonlinearity is introduced in the standard HITS model in order to guarantee existence and uniqueness of these centrality vectors for any network, without any requirement on its connectivity structure. We introduce a globally convergent power iteration like algorithm for the computation of the centrality vectors. Numerical experiments on real-world networks are performed in order to assess the effectiveness of the proposed model and showcase the performance of the accompanying algorithm

A nonlinear spectral core-periphery detection method for multiplex networks

Core-periphery detection aims to separate the nodes of a complex network into two subsets: a core that is densely connected to the entire network and a periphery that is densely connected to the core but sparsely connected internally. The definition of core-periphery structure in multiplex networks that record different types of interactions between the same set of nodes but on different layers is nontrivial since a node may belong to the core in some layers and to the periphery in others. The current state-of-the-art approach relies on linear combinations of individual layer degree vectors whose layer weights need to be chosen a-priori. We propose a nonlinear spectral method for multiplex networks that simultaneously optimizes a node and a layer coreness vector by maximizing a suitable nonconvex homogeneous objective function by an alternating fixed point iteration. We prove global optimality and convergence guarantees for admissible hyper-parameter choices and convergence to local optima for the remaining cases. We derive a quantitative measure for the quality of a given multiplex core-periphery structure that allows the determination of the optimal core size. Numerical experiments on synthetic and real-world networks illustrate that our approach is robust against noisy layers and outperforms baseline methods with respect to a variety of core-periphery quality measures. In particular, all methods based on layer aggregation are improved when used in combination with the novel optimized layer coreness vector weights. As the runtime of our method depends linearly on the number of edges of the network it is scalable to large-scale multiplex networks