Pattern invariance for reaction-diffusion systems on complex networks

TL;DR

This paper introduces two novel methods to modify complex network topologies in reaction-diffusion systems while preserving their dynamical patterns, enabling better control and understanding of pattern formation.

Contribution

The authors propose two techniques to alter network topology without changing the system's dynamical response, advancing the control of pattern formation in reaction-diffusion systems.

Findings

Both methods successfully preserve dynamical patterns on modified networks.

The first method offers a global redistribution of link weights, affecting overall network structure.

The second method provides localized control at the node level, allowing finer adjustments.

Abstract

Given a reaction-diffusion system interacting via a complex network, we propose two different techniques to modify the network topology while preserving its dynamical behaviour. In the region of parameters where the homogeneous solution gets spontaneously destabilized, perturbations grow along the unstable directions made available across the networks of connections, yielding irregular spatio-temporal patterns. We exploit the spectral properties of the Laplacian operator associated to the graph in order to modify its topology, while preserving the unstable manifold of the underlying equilibrium. The new network is isodynamic to the former, meaning that it reproduces the dynamical response (pattern) to a perturbation, as displayed by the original system. The first method acts directly on the eigenmodes, thus resulting in a general redistribution of link weights which, in some cases, can completely change the structure of the original network. The second method uses localization properties of the eigenvectors to identify and randomize a subnetwork that is mostly embedded only into the stable manifold. We test both techniques on different network topologies using the Ginzburg-Landau system as a reference model. Whereas the correlation between patterns on isodynamic networks generated via the first recipe is larger, the second method allows for a finer control at the level of single nodes. This work opens up a new perspective on the multiple possibilities for identifying the family of discrete supports that instigate equivalent dynamical responses on a multispecies reaction-diffusion system.