Matrix function-based centrality measures for layer-coupled multiplex networks.

Centrality measures identify the most important nodes in a complex network. In recent years, multilayer networks have emerged as a flexible tool to create increasingly realistic models of complex systems. In this paper, we generalize matrix function-based centrality and communicability measures to the case of layer-coupled multiplex networks. We use the supra-adjacency matrix as the network representation, which has already been used to generalize eigenvector centrality to temporal and multiplex networks. With this representation, the definition of single-layer matrix function-based centrality measures in terms of walks on the networks carries over naturally to the multilayer case. Several aggregation techniques allow the ranking of nodes, layers, as well as node-layer pairs in terms of their importance in the network. We present efficient and scalable numerical methods based on Krylov subspace techniques and Gauss quadrature rules, which provide a high accuracy in only a few iterations and which scale linearly in the network size under the assumption of sparsity in the supra-adjacency matrix. Finally, we present extensive numerical studies for both directed and undirected as well as weighted and unweighted multiplex networks. While we focus on social and transportation applications the networks' size ranges between $89$ and $2.28 \cdot 10^6$ nodes and between $3$ and $37$ layers.