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Turing instabilities on Cartesian product networks.

Malbor Asllani1, Daniel M Busiello2, Timoteo Carletti3

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Summary
This summary is machine-generated.

This study analyzes Turing instabilities in reaction-diffusion systems on complex networks. Patterns can emerge if they are supported by individual network components, simplifying criteria for multiplex networks.

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Area of Science:

  • Mathematical modeling
  • Theoretical chemistry
  • Network science

Background:

  • Reaction-diffusion systems are fundamental to pattern formation in nature.
  • Understanding instabilities on complex network structures is crucial for predicting emergent behaviors.
  • Turing instabilities provide a mechanism for spontaneous pattern generation.

Purpose of the Study:

  • To investigate Turing instabilities in reaction-diffusion systems on Cartesian product networks.
  • To develop a theoretical framework for predicting pattern formation on these complex structures.
  • To simplify instability criteria for multiplex network configurations.

Main Methods:

  • Linear stability analysis of reaction-diffusion systems.
  • Expansion of perturbations using tensor products of eigenvectors from discrete Laplacian operators.
  • Analysis of dispersion relations dependent on discrete wavelengths (eigenvalues).

Main Results:

  • Pattern formation is possible if supported by at least one sub-graph of the Cartesian product network.
  • Specific prescriptions for multiplex networks yield simplified, explicit formulae for instability criteria.
  • The theoretical framework accurately predicts pattern formation, confirmed by numerical simulations.

Conclusions:

  • The proposed linear regime analysis effectively predicts Turing instabilities on Cartesian product networks.
  • The study provides a generalized method applicable to various network topologies.
  • The findings offer insights into pattern formation mechanisms in spatially extended systems and complex networks.