Pattern formation in multiplex networks

TL;DR

This paper explores how pattern formation occurs in multiplex networks where activator and inhibitor species occupy separate layers, revealing unique instability conditions driven by network topology differences.

Contribution

It introduces a novel framework for pattern formation in multiplex networks, showing topology-driven instabilities that differ from single-layer networks.

Findings

Diffusion-induced instability can occur even with equal mobility rates.

Heterogeneous patterns emerge due to multiplex network structure.

Instability conditions depend on degrees across layers.

Abstract

The advances in understanding complex networks have generated increasing interest in dynamical processes occurring on them. Pattern formation in activator-inhibitor systems has been studied in networks, revealing differences from the classical continuous media. Here we study pattern formation in a new framework, namely multiplex networks. These are systems where activator and inhibitor species occupy separate nodes in different layers. Species react across layers but diffuse only within their own layer of distinct network topology. This multiplicity generates heterogeneous patterns with significant differences from those observed in single-layer networks. Remarkably, diffusion-induced instability can occur even if the two species have the same mobility rates; condition which can never destabilize single-layer networks. The instability condition is revealed using perturbation theory and expressed by a combination of degrees in the different layers. Our theory demonstrates that the existence of such topology-driven instabilities is generic in multiplex networks, providing a new mechanism of pattern formation.