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

Abstract

The "SOS" in the title does not refer to the international distress signal, but to "solid-on-solid" (SOS) surface growth. The catastrophic cascades are those observed by Buldyrev et al. in interdependent networks, which we re-interpret as multiplex networks with agents that can only survive if they mutually support each other, and whose survival struggle we map onto an SOS type growth model. This mapping not only reveals non-trivial structures in the phase space of the model, but also leads to a new and extremely efficient simulation algorithm. 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 new and surprising results in all these cases, and we correct statements in the literature for ER networks and for 2-d lattices. In particular, we find that d = 4 is the upper critical dimension, 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 different from that proposed in W. Li et al., PRL 108, 228702 (2012).

Universality of Critically Pinned Interfaces in Two-Dimensional Isotropic Random Media

“…Together with analytical arguments this gives strong support to our more general claim. It also suggests that a percolative phase transition in a 2-d model with intermediate length dependency links [44] is continuous and in the OP class, as claimed in [45] and in contrast to claims made in [44].…”

Critical Dynamics of thek-Core Pruning Process

“…Understanding the k-core pruning process and accompanying structural changes can shed light on various critical phenomena, such as the jamming transition, the rigidity percolation, glassy dynamics [15,16], complex contagion [17], mass extinction [18], avalanches in neuronal networks [19], and many others. Furthermore, the k-core pruning process is one of the simplest examples of dynamic processes associated with hybrid phase transitions, sharing, for example, some common properties with cascade failures in interdependent networks that have recently received significant attention in the literature [6,12,[20][21][22][23][24].…”

Universality and scaling laws in the cascading failure model with healing

“…Phase transitions are often accompanied by scaling laws but the numerical test of the relevant quantities for the CF model had been a challenging task. Recently, however, efficient algorithms have been developed, that allow for large scale simulation of the mutually connected components (MCC) in the CF model with computational time of O(N 1.2 ) [10,12].…”

“…The critical values m 0 and q c have been published [12] with great accuracy for the square lattice without healing (w = 0). We measured p c and q c too and used this case as a benchmark test.…”