Message passing theory for percolation models on multiplex networks with link overlap

TL;DR

This paper develops a message passing framework to analyze percolation transitions in multiplex networks with link overlap, providing new algorithms and phase diagrams for understanding their connectivity properties.

Contribution

It introduces two generalized message passing algorithms for multiplex networks with link overlap, extending existing methods to include directed percolation and mutual connectivity.

Findings

Two algorithms successfully model percolation and epidemic spreading.

Discontinuous percolation transitions occur in non-trivial cases.

Tricritical points influence the nature of phase transitions.

Abstract

Multiplex networks describe a large variety of complex systems including infrastructures, transportation networks and biological systems. Most of these networks feature a significant link overlap. It is therefore of particular importance to characterize the mutually connected giant component in these networks. Here we provide a message passing theory for characterizing the percolation transition in multiplex networks with link overlap and an arbitrary number of layers $M$. Specifically we propose and compare two message passing algorithms, that generalize the algorithm widely used to study the percolation transition in multiplex networks without link overlap. The first algorithm describes a directed percolation transition and admits an epidemic spreading interpretation. The second algorithm describes the emergence of the mutually connected giant component, that is the percolation transition, but does not preserve the epidemic spreading interpretation. We obtain the phase diagrams for the percolation and directed percolation transition in simple representative cases. We demonstrate that for the same multiplex network structure, in which the directed percolation transition has non-trivial tricritical points, the percolation transition has a discontinuous phase transition, with the exception of the trivial case in which all the layers completely overlap.