Turing patterns mediated by network topology in homogeneous active systems

Abstract

Mechanisms of pattern formation-of which the Turing instability is an archetype-constitute an important class of dynamical processes occurring in biological, ecological and chemical systems. Recently, it has been shown that the Turing instability can induce pattern formation in discrete media such as complex networks, opening up the intriguing possibility of exploring it as a generative mechanism in a plethora of socioeconomic contexts. Yet, much remains to be understood in terms of the precise connection between network topology and its role in inducing the patterns. Here, we present a general mathematical description of a two-species reaction-diffusion process occurring on different flavors of network topology. The dynamical equations are of the predator-prey class, that while traditionally used to model species population, has also been used to model competition between antagonistic ideas in social systems. We demonstrate that the Turing instability can be induced in any network topology, by tuning the diffusion of the competing species, or by altering network connectivity. The extent to which the emergent patterns reflect topological properties is determined by a complex interplay between the diffusion coefficients and the localization properties of the eigenvectors of the graph Laplacian. We find that networks with large degree fluctuations tend to have stable patterns over the space of initial perturbations, whereas patterns in more homogenous networks are purely stochastic. arXiv:1903.03845v1 [cond-mat.dis-nn]

Cross-diffusion on multiplex networks

“…They found that the dynamics of reaction-diffusion systems organized in large network had obvious differences from those coupled in small networks, yielding distinct non-uniform patterns. Inspired by these advances, great efforts have been made in the exploration of such network instability [24][25][26][27][28][29][30][31]. Moreover, Turing instabilities have also been explored in more complicated systems, such as [32,33].…”

A theory of pattern formation for reaction–diffusion systems on temporal networks

“…There is a small yet growing literature suggesting that patterns on networks may be adjusted or prevented by changing the network topology, yet all of these draw their conclusions by comparing two different static network topologies [21,23] or through specific simulations of particular models [24,65–67]. Indeed, our instability criteria involve effective diffusion rates scriptDufalse(tfalse)ρℓfalse(tfalse) and scriptDvfalse(tfalse)ρℓfalse(tfalse), which are highly dependent on the network topology at time t , and hence show in general that network topology strongly influences both the onset of diffusive instability and the actual patterns which result.…”

“…As in the PDE case, dynamic processes which lead to patterns arise in a variety of applications on networks [8–15], with a similar Turing mechanism yielding a collection of unstable modes, which correspond to eigenvectors of the Laplacian matrix corresponding to the underlying network structure [13,16–22]. A common theme to the literature is that the network topology can influence the emergence or suppression of certain patterns [21,23–25].…”

Turing Instability of Brusselator in the Reaction-Diffusion Network

“…In [20], they inferred a general theoretical proof that the Turing system's three key features are directly determined by the topology. Recently, Mimar et al [21] proved that the degree of Laplace can reflect the system's topological characteristics under different networks, which is related to the local characteristics of the diffusion coefficient.…”