Turing instabilities on Cartesian product networks

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

Research output: Book/Report/JournalOther report

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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.
Original languageEnglish
PublisherNamur center for complex systems
Number of pages10
Volume13
Edition14
Publication statusPublished - 1 Dec 2014

Publication series

NamenaXys Technical Report Series
PublisherUniversity of Namur
No.14
Volume13

Fingerprint

Turing Instability
Cartesian product
Discrete Laplacian
Discrete Operators
Reaction Kinetics
Dispersion Relation
Reaction-diffusion System
Cartesian
Tensor Product
Eigenvector
Expand
Explicit Formula
Subgraph
Wavelength
Eigenvalue
Perturbation
Numerical Simulation

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.
Asllani, Malbor ; Busiello, Daniel M. ; Carletti, Timoteo ; Fanelli, Duccio ; Planchon, Gwendoline. / Turing instabilities on Cartesian product networks. 14 ed. Namur center for complex systems, 2014. 10 p. (naXys Technical Report Series; 14).
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Asllani, M, Busiello, DM, Carletti, T, Fanelli, D & Planchon, G 2014, Turing instabilities on Cartesian product networks. naXys Technical Report Series, no. 14, vol. 13, vol. 13, 14 edn, Namur center for complex systems.

Turing instabilities on Cartesian product networks. / Asllani, Malbor; Busiello, Daniel M.; Carletti, Timoteo; Fanelli, Duccio; Planchon, Gwendoline.

14 ed. Namur center for complex systems, 2014. 10 p. (naXys Technical Report Series; Vol. 13, No. 14).

Research output: Book/Report/JournalOther report

TY - BOOK

T1 - Turing instabilities on Cartesian product networks

AU - Asllani, Malbor

AU - Busiello, Daniel M.

AU - Carletti, Timoteo

AU - Fanelli, Duccio

AU - Planchon, Gwendoline

PY - 2014/12/1

Y1 - 2014/12/1

N2 - 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.

AB - 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.

KW - complex networks

KW - turing patterns

KW - non linear dynamics

KW - cartesian product networks

KW - reaction-diffusion

KW - spatio-temporal patterns

M3 - Other report

VL - 13

T3 - naXys Technical Report Series

BT - Turing instabilities on Cartesian product networks

PB - Namur center for complex systems

ER -

Asllani M, Busiello DM, Carletti T, Fanelli D, Planchon G. Turing instabilities on Cartesian product networks. 14 ed. Namur center for complex systems, 2014. 10 p. (naXys Technical Report Series; 14).