title: Turing instabilities on Cartesian product networks.
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.
authors.affiliation: Dipartimento di Scienza e Alta Tecnologia, University of Insubria, via Valleggio 11, 22100 Como, Italy.
authors.forename: Malbor
authors.initials: M
authors.lastname: Asllani
authors.affiliation: Department of Physics and Astronomy G. Galilei, University of Padova, via Marzolo 8, 35131 Padova, Italy.
authors.forename: Daniel M
authors.initials: DM
authors.lastname: Busiello
authors.affiliation: Department of mathematics and Namur Center for Complex Systems - naXys, University of Namur, rempart de la Vierge 8, B 5000 Namur, Belgium.
authors.forename: Timoteo
authors.initials: T
authors.lastname: Carletti
authors.affiliation: Dipartimento di Fisica e Astronomia, University of Florence and INFN, Via Sansone 1, 50019 Sesto Fiorentino, Florence, Italy.
authors.forename: Duccio
authors.initials: D
authors.lastname: Fanelli
authors.affiliation: 1] Department of Physics and Astronomy G. Galilei, University of Padova, via Marzolo 8, 35131 Padova, Italy [2] Dipartimento di Fisica e Astronomia, University of Florence and INFN, Via Sansone 1, 50019 Sesto Fiorentino, Florence, Italy.
authors.forename: Gwendoline
authors.initials: G
authors.lastname: Planchon