Percolation transitions in the survival of interdependent agents on multiplex networks, catastrophic cascades, and SOS

TL;DR

This paper maps the survival of interdependent agents on multiplex networks onto an SOS surface growth model, revealing new phase transition behaviors, critical dimensions, and correcting previous literature findings.

Contribution

It introduces a novel SOS-based simulation algorithm and provides new insights into percolation transitions in multiplex networks across various dimensions.

Findings

d=4 is the upper critical dimension for the model

Percolation transition is continuous for d≤4 but not in ordinary percolation universality class

Cluster statistics in ER networks follow mean field theory, but cascade process does not

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 {\it 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 Erd\"os-R\'enyi (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 $d\leq 4$ but -- at least for $d\neq 3$ -- 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 {\it et al.}, PRL {\bf 108}, 228702 (2012).