Turing instabilities on Cartesian product networks

TL;DR

This paper analyzes Turing instabilities on Cartesian product networks using a linear approach, deriving explicit criteria for pattern formation and validating results with numerical simulations of reaction kinetics.

Contribution

It introduces a novel framework for understanding Turing instabilities on Cartesian product networks, including explicit formulas for instability conditions and applications to multiplex networks.

Findings

Patterns form if supported on sub-graphs of the Cartesian network.

Explicit instability criteria are derived for multiplex networks.

Numerical simulations confirm the theoretical predictions.

Abstract

The problem of Turing instabilities for a reaction-diffusion system defined on a complex Cartesian product networks is considered. To this end we operate in the linear regime and expand the time dependent perturbation on a basis formed by the tensor product of the eigenvectors of the discrete Laplacian operators, associated to each of the individual networks that build the Cartesian product. The dispersion relation which controls the onset of the instability depends on a set of discrete wave- lenghts, the eigenvalues of the aforementioned Laplacians. Patterns can develop on the Cartesian network, if they are supported on at least one of its constituive sub-graphs. Multiplex networks are also obtained under specific prescriptions. In this case, the criteria for the instability reduce to compact explicit formulae. Numerical simulations carried out for the Mimura-Murray reaction kinetics confirm the adequacy of the proposed theory.