Weak Multiplex Percolation

TL;DR

This paper introduces weak multiplex percolation, a new model for connectivity in multilayer networks, revealing complex phase transitions including continuous and discontinuous types depending on network parameters.

Contribution

It generalizes percolation theory to multilayer networks, describing critical phenomena and phase transitions unique to this framework.

Findings

Unusual continuous transition with quadratic growth in two-layer networks.

Discontinuous hybrid transition in three or more layers.

Critical behavior depends on degree distribution moments.

Abstract

In many systems consisting of interacting subsystems, the complex interactions between elements can be represented using multilayer networks. However percolation, key to understanding connectivity and robustness, is not trivially generalised to multiple layers. We describe a generalisation of percolation to multilayer networks: weak multiplex percolation. A node belongs to a connected component if at least one of its neighbours in each layer is in this component. We fully describe the critical phenomena of this process. In particular, in two layers, with finite second moments of the degree distributions, an unusual continuous transition with quadratic growth above the threshold. When the second moments diverge, the singularity is determined by the asymptotics of the degree distributions, creating a rich set of critical behaviours. In three or more layers we find a discontinuous hybrid transition which persists even in highly heterogeneous degree distributions, becoming continuous only when the powerlaw exponent reaches $1+ 1/(M-1)$ for $M$ layers.