Son, Grassberger, and Paczuski reply

Percolation transitions in the survival of interdependent agents on multiplex networks, catastrophic cascades, and solid-on-solid surface growth

We present an efficient algorithm for simulating percolation transitions of mutually supporting viable clusters on multiplex networks (also known as "catastrophic cascades on interdependent networks"). This algorithm maps the problem onto a solid-on-solid-type model. We use this algorithm to study interdependent agents on duplex Erdös-Rényi (ER) networks and on lattices with dimensions 2, 3, 4, and 5. We obtain surprising results in all these cases, and we correct statements in the literature for ER networks and for two-dimensional lattices. In particular, we find that d=4 is the upper critical dimension and that the percolation transition is continuous for d≤4 but-at least for d≠3-not in the universality class of ordinary percolation. For ER networks we verify that the cluster statistics is exactly described by mean-field theory but find evidence that the cascade process is not. For d=5 we find a first-order transition as for ER networks, but we find also that small clusters have a nontrivial mass distribution that scales at the transition point. Finally, for d=2 with intermediate-range dependency links we propose a scenario that differs from that proposed in W. Li et al. [Phys. Rev. Lett. 108, 228702 (2012)].

The structure and dynamics of multilayer networks

In the past years, network theory has successfully characterized the interaction among the constituents of a variety of complex systems, ranging from biological to technological, and social systems. However, up until recently, attention was almost exclusively given to networks in which all components were treated on equivalent footing, while neglecting all the extra information about the temporal- or context-related properties of the interactions under study. Only in the last years, taking advantage of the enhanced resolution in real data sets, network scientists have directed their interest to the multiplex character of real-world systems, and explicitly considered the time-varying and multilayer nature of networks. We offer here a comprehensive review on both structural and dynamical organization of graphs made of diverse relationships (layers) between its constituents, and cover several relevant issues, from a full redefinition of the basic structural measures, to understanding how the multilayer nature of the network affects processes and dynamics.