Communication in complex networks

Abstract

One of the properties of interest in the analysis of networks is global communicability, i.e., how easy or difficult it is, generally, to reach nodes from other nodes by following edges. Different global communicability measures provide quantitative assessments of this property, emphasizing different aspects of the problem. This paper investigates the sensitivity of global measures of communicability to local changes. In particular, for directed, weighted networks, we study how different global measures of communicability change when the weight of a single edge is changed; or, in the unweighted case, when an edge is added or removed. The measures we study include the total network communicability, based on the matrix exponential of the adjacency matrix, and the Perron network communicability, defined in terms of the Perron root of the adjacency matrix and the associated left and right eigenvectors. Finding what local changes lead to the largest changes in global communicability has many potential applications, including assessing the resilience of a system to failure or attack, guidance for incremental system improvements, and studying the sensitivity of global communicability measures to errors in the network connection data.

Perron communicability and sensitivity of multilayer networks

“…Several of the techniques discussed evaluate the exponential of the adjacency matrix of the network, or the exponential of adjacency matrix determined by the line graph associated with the given network. The present paper extends the communicability and sensitivity analysis in [8,19] to multilayer networks. Since multilayer networks typically have a large number of nodes and edges, we focus on techniques that are well suited for large-scale networks.…”

“…Matrix function-based communicability measures have been generalized in [3] to the case of layer-coupled multiplex networks that can be represented by a supra-adjacency matrix B of the form (6), i.e., by A defined by (5). Following the argument in [8], assume that the Perron root ρ of a supra-adjacency matrix B of the form ( 6) is significantly larger than the magnitude of its the other eigenvalues. Then…”

“…The investigation of the importance of nodes and edges, as well as the sensitivity of the communicability within a network to changes in the edge weights of the network with only one kind of nodes and edges has received considerable attention in the literature; see, e.g., [4,5,8,9,10,11,12,17,19] and references therein. Several of the techniques discussed evaluate the exponential of the adjacency matrix of the network, or the exponential of adjacency matrix determined by the line graph associated with the given network.…”

The Fréchet derivative of the tensor t-function

“…Consider again (20), specifically the argument of f . Thanks to (8) and Theorem 1(iii), we can write…”

“…For functions of matrices, the Fréchet derivative is a well-established object with applications in, e.g., condition number estimation [1], analysis of complex networks [8], and the solution of matrix optimization problems [30]. In this work, we consider the Fréchet derivative of functions of tensors, in order to generalize the above techniques to the tensor setting.…”