Exotic Critical Behavior of Weak Multiplex Percolation

TL;DR

This paper investigates the critical phenomena of weak multiplex percolation in interdependent networks, revealing unique phase transition behaviors and how they depend on the number of layers and degree distribution.

Contribution

It introduces the study of weak multiplex percolation, highlighting its distinct critical behavior and phase transition types compared to traditional multiplex percolation models.

Findings

In two layers, the giant component emerges with a continuous transition and quadratic growth.

In three or more layers, a discontinuous hybrid transition occurs.

Discontinuity vanishes at specific degree distribution exponents, depending on the number of layers.

Abstract

We describe the critical behavior of weak multiplex percolation, a generalization of percolation to multiplex or interdependent networks. A node can determine its active or inactive status simply by referencing neighboring nodes. This is not the case for the more commonly studied generalization of percolation to multiplex networks, the mutually connected clusters, which requires an interconnecting path within each layer between any two vertices in the giant mutually connected component. We study the emergence of a giant connected component of active nodes under the weak percolation rule, finding several non-typical phenomena. In two layers, the giant component emerges with a continuos phase transition, but with quadratic growth above the critical threshold. In three or more layers, a discontinuous hybrid transition occurs, similar to that found in the giant mutually connected component. In networks with asymptotically powerlaw degree distributions, defined by the decay exponent $\gamma$, the discontinuity vanishes but at $\gamma=1.5$ in three layers, more generally at $\gamma = 1+ 1/(M-1)$ in $M$ layers.