### Abstract

The problem of Turing instabilities for a reaction-diffusion system defined on a complex Cartesian product network 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 wavelengths, the eigenvalues of the aforementioned Laplacians. Patterns can develop on the Cartesian network, if they are supported on at least one of its constitutive 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.

The dispersion relation which controls the onset of the instability depends on a set of discrete wavelengths, the eigenvalues of the aforementioned Laplacians. Patterns can develop on the Cartesian network, if they are supported on at least one of its constitutive 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.

Original language | English |
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Publisher | Namur center for complex systems |

Number of pages | 10 |

Volume | 13 |

Edition | 14 |

Publication status | Published - 1 Dec 2014 |

### Publication series

Name | naXys Technical Report Series |
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Publisher | University of Namur |

No. | 14 |

Volume | 13 |

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### Keywords

- complex networks
- turing patterns
- non linear dynamics
- cartesian product networks
- reaction-diffusion
- spatio-temporal patterns

### Cite this

Asllani, M., Busiello, D. M., Carletti, T., Fanelli, D., & Planchon, G. (2014).

*Turing instabilities on Cartesian product networks*. (14 ed.) (naXys Technical Report Series; Vol. 13, No. 14). Namur center for complex systems.