Static and dynamic behavior of multiplex networks under interlink strength variation

TL;DR

This paper investigates how the structural and dynamical properties of multiplex networks evolve with interlayer strength, revealing a smooth transition rather than abrupt changes as interlink strength varies.

Contribution

It demonstrates that the perceived abrupt transition is a consequence of the graph Laplacian formalism and shows that the transition is actually smooth through simulations and analysis.

Findings

No abrupt change in static or dynamic behavior observed.

Structural measures evolve smoothly with interlink strength.

Transition from separate networks to a single network is gradual.

Abstract

It has recently been suggested \cite{Radicchi2013} that in a two-level multiplex network, a gradual change in the value of the "interlayer" strength $p$ can provoke an abrupt structural transition. The critical point $p^*$ at which this happens is system-dependent. In this article, we show in a similar way as in \cite{Garrahan2014} that this is a consequence of the graph Laplacian formalism used in \cite{Radicchi2013}. We calculate the evolution of $p^{*}$ as a function of system size for ER and RR networks. We investigate the behavior of structural measures and dynamical processes of a two-level system as a function of $p$, by Monte-Carlo simulations, for simple particle diffusion and for reaction-diffusion systems. We find that as $p$ increases there is a smooth transition from two separate networks to a single one. We cannot find any abrupt change in static or dynamic behavior of the underlying system.