Hidden transition in multiplex networks

TL;DR

This paper uncovers a novel discontinuous phase transition in weak multiplex percolation on multi-layer networks, where small perturbations can cause abrupt collapse or recovery without typical singularities, revealing new critical dynamics.

Contribution

It introduces a previously unknown discontinuous transition in multiplex networks with three or more layers, characterized by unique collapse and relaxation times near the critical point.

Findings

Discontinuous transition occurs without giant component singularity.

Collapse time scales as 1/(Pi - Pi_c) above criticality.

Relaxation time is exponential below criticality, proportional to 1/(Pi_c - Pi).

Abstract

Weak multiplex percolation generalizes percolation to multi-layer networks, represented as networks with a common set of nodes linked by multiple types (colors) of edges. We report a novel discontinuous phase transition in this problem. This anomalous transition occurs in networks of three or more layers without unconnected nodes, $P(0)=0$. Above a critical value of a control parameter, the removal of a tiny fraction $\Delta$ of nodes or edges triggers a failure cascade which ends either with the total collapse of the network, or a return to stability with the system essentially intact. The discontinuity is not accompanied by any singularity of the giant component, in contrast to the discontinuous hybrid transition which usually appears in such problems. The control parameter is the fraction of nodes in each layer with a single connection, $\Pi=P(1)$. We obtain asymptotic expressions for the collapse time and relaxation time, above and below the critical point $\Pi_c$, respectively. In the limit $\Delta\to0$ the total collapse for $\Pi>\Pi_\text{c}$ takes a time $T \propto 1/(\Pi-\Pi_\text{c})$, while there is an exponential relaxation below $\Pi_\text{c}$ with a relaxation time $\tau \propto 1/[\Pi_\text{c}-\Pi]$.