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.
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