Static and dynamic behavior of multiplex networks under interlink strength variation

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It has recently been suggested 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 Garrahan2014 that this is a consequence of the graph Laplacian formalism used in 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.