Fast computation of matrix function-based centrality measures for layer-coupled multiplex networks

Abstract

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 • 10 6 nodes and between 3 and 37 layers.

Orientations and matrix function-based centralities in multiplex network analysis of urban public transport

“…Some recent works provide generalizations of centrality measures well-studied on single-layer graphs to the case of different multilayer architectures [29,63,69,67,64,72,65,16]. Most importantly for this paper, the class of matrix functionbased centrality measures has very recently been generalized to the case of layercoupled multiplex networks.…”

“…, L, and an inter-layer edge set Ẽ. Note that similar networks have been considered before, e.g., in [63,64,65,16] but in this paper we employ a different notion of inter-layer edges, which is determined by the data.…”

“…We choose a supra-adjacency matrix representation as the linear algebraic formulation of these multiplex networks [47,63,64,65,16]. The two different types of edges are represented by two separate matrices.…”

“…On the one hand, the above definition of A inter is more general than those in [63,64,65,16], where each E (lk) is a full identity matrix, as it includes all combinations of up to n zero diagonal entries. On the other hand, the construction in [63,64,65,16] allows each layer-layer pair to be coupled with a different weight, while A inter defined in Equation (4.2) is unweighted.…”

“…On the one hand, the above definition of A inter is more general than those in [63,64,65,16], where each E (lk) is a full identity matrix, as it includes all combinations of up to n zero diagonal entries. On the other hand, the construction in [63,64,65,16] allows each layer-layer pair to be coupled with a different weight, while A inter defined in Equation (4.2) is unweighted. Later in this section we will discuss an approach to weight the inter-layer adjacency matrix by means of a parameter ω, which can be interpreted as modeling a constant transfer time between all lines at all stops of the network.…”