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The theory of pattern formation on directed networks

Author

Listed:
  • Malbor Asllani

    (University of Insubria
    University of Florence, INFN and CSDC)

  • Joseph D. Challenger

    (University of Florence, INFN and CSDC)

  • Francesco Saverio Pavone

    (University of Florence, INFN and CSDC
    European Laboratory for Non-linear Spectroscopy
    National Institute of Optics, National Research Council)

  • Leonardo Sacconi

    (European Laboratory for Non-linear Spectroscopy
    National Institute of Optics, National Research Council)

  • Duccio Fanelli

    (University of Florence, INFN and CSDC)

Abstract

Dynamical processes on networks have generated widespread interest in recent years. The theory of pattern formation in reaction–diffusion systems defined on symmetric networks has often been investigated, due to its applications in a wide range of disciplines. Here we extend the theory to the case of directed networks, which are found in a number of different fields, such as neuroscience, computer networks and traffic systems. Owing to the structure of the network Laplacian, the dispersion relation has both real and imaginary parts, at variance with the case for a symmetric, undirected network. The homogeneous fixed point can become unstable due to the topology of the network, resulting in a new class of instabilities, which cannot be induced on undirected graphs. Results from a linear stability analysis allow the instability region to be analytically traced. Numerical simulations show travelling waves, or quasi-stationary patterns, depending on the characteristics of the underlying graph.

Suggested Citation

  • Malbor Asllani & Joseph D. Challenger & Francesco Saverio Pavone & Leonardo Sacconi & Duccio Fanelli, 2014. "The theory of pattern formation on directed networks," Nature Communications, Nature, vol. 5(1), pages 1-9, December.
    • Handle: RePEc:nat:natcom:v:5:y:2014:i:1:d:10.1038_ncomms5517
    DOI: 10.1038/ncomms5517
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    Cited by:

    1. Zheng, Qianqian & Shen, Jianwei, 2020. "Turing instability induced by random network in FitzHugh-Nagumo model," Applied Mathematics and Computation, Elsevier, vol. 381(C).
    2. Rahimabadi, Arsalan & Benali, Habib, 2023. "Extended fractional-polynomial generalizations of diffusion and Fisher–KPP equations on directed networks," Chaos, Solitons & Fractals, Elsevier, vol. 174(C).
    3. Chen, Mengxin & Zheng, Qianqian, 2023. "Diffusion-driven instability of a predator–prey model with interval biological coefficients," Chaos, Solitons & Fractals, Elsevier, vol. 172(C).
    4. Cencetti, Giulia & Battiston, Federico & Carletti, Timoteo & Fanelli, Duccio, 2020. "Generalized patterns from local and non local reactions," Chaos, Solitons & Fractals, Elsevier, vol. 134(C).
    5. Ide, Yusuke & Izuhara, Hirofumi & Machida, Takuya, 2016. "Turing instability in reaction–diffusion models on complex networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 457(C), pages 331-347.
    6. Song, Mingrui & Gao, Shupeng & Liu, Chen & Bai, Yue & Zhang, Lei & Xie, Beilong & Chang, Lili, 2023. "Cross-diffusion induced Turing patterns on multiplex networks of a predator–prey model," Chaos, Solitons & Fractals, Elsevier, vol. 168(C).
    7. Di Patti, Francesca & Fanelli, Duccio & Miele, Filippo & Carletti, Timoteo, 2017. "Benjamin–Feir instabilities on directed networks," Chaos, Solitons & Fractals, Elsevier, vol. 96(C), pages 8-16.
    8. Zheng, Qianqian & Shen, Jianwei & Pandey, Vikas & Guan, Linan & Guo, Yantao, 2023. "Turing instability in a network-organized epidemic model with delay," Chaos, Solitons & Fractals, Elsevier, vol. 168(C).
    9. Muolo, Riccardo & Gallo, Luca & Latora, Vito & Frasca, Mattia & Carletti, Timoteo, 2023. "Turing patterns in systems with high-order interactions," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).
    10. Riccardo Muolo & Joseph D. O’Brien & Timoteo Carletti & Malbor Asllani, 2024. "Persistence of chimera states and the challenge for synchronization in real-world networks," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 97(1), pages 1-16, January.
    11. Zheng, Qianqian & Shen, Jianwei & Xu, Yong & Pandey, Vikas & Guan, Linan, 2022. "Pattern mechanism in stochastic SIR networks with ER connectivity," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 603(C).

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