A Perron-Frobenius theory for block matrices associated to a multiplex network

TL;DR

This paper extends Perron-Frobenius theory to block matrices of multiplex networks, establishing conditions for the uniqueness of the Perron vector and expressing it via layer-specific Perron vectors.

Contribution

It introduces a novel theoretical framework linking the Perron vectors of multiplex networks to those of individual layers and their adjacency matrices.

Findings

Conditions for Perron vector uniqueness in multiplex networks

Relations between irreducibility of block matrices and layer matrices

Method to compute the Perron vector from layer Perron vectors

Abstract

The uniqueness of the Perron vector of a nonnegative block matrix associated to a multiplex network is discussed. The conclusions come from the relationships between the irreducibility of some nonnegative block matrix associated to a multiplex network and the irreducibility of the corresponding matrices to each layer as well as the irreducibility of the adjacency matrix of the projection network. In addition the computation of that Perron vector in terms of the Perron vectors of the blocks is also addressed. Finally we present the precise relations that allow to express the Perron eigenvector of the multiplex network in terms of the Perron eigenvectors of its layers.