Static and dynamic behavior of multiplex networks under interlink strength variation

It has recently been suggested (Radicchi F. and Arenas A., Nat. Phys., 9 (2013) 717) 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 at which this happens is system dependent. In this article, we show in a similar way as in (Garrahan J. P. and Lesanovsky I., arXiv:1406.4706) that this is a consequence of the graph Laplacian formalism used in the above-mentioned paper by Radicchi and Arenas. We calculate the evolution of 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.