Turing instabilities on Cartesian product networks

Malbor Asllani, Daniel M. Busiello, Timoteo Carletti, Duccio Fanelli, Gwendoline Planchon

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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.
Original languageEnglish
Article number5:12927
Pages (from-to)1
Number of pages10
JournalScientific Reports
Issue number12927
Publication statusPublished - 6 Aug 2015


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


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