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.
Highlights
The advances in understanding complex networks have generated increasing interest in dynamical processes occurring on them
Our theory demonstrates that the existence of such topology-driven instabilities is generic in multiplex networks, providing a new mechanism of pattern formation
Complex networks are ubiquitous in nature[12]; two typical examples are epidemics spreading over transportation systems[13] and ecological systems where distinct habitats communicate through dispersal connections[14,15,16,17]
Summary
The advances in understanding complex networks have generated increasing interest in dynamical processes occurring on them. Such media are often described by reaction-diffusion systems and consist of elements obeying an activator-inhibitor dynamics with local coupling In his pioneering paper[1], Turing showed that a uniform steady state can be spontaneously destabilized, leading to a spontaneous formation of a periodic spatial pattern, when reacting species diffuse with different mobilities. It was later proposed by Gierer and Meinhardt[4] that an activator-inhibitor chemical reaction is a typical example achieving Turing’s scenario. Asllani et al studied Turing patterns in the context of multiplex networks[36], where it was found that an additional inter-layer diffusion process can induce instabilities even if they are prevented in the isolated layers
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