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Turing patterns mediated by network topology in homogeneous active systems.

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Summary
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Turing instability can create patterns in complex networks by adjusting species diffusion or network connections. Network structure influences pattern stability, with heterogeneous networks showing more stable patterns than homogenous ones.

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Area of Science:

  • Dynamical systems theory
  • Network science
  • Mathematical biology

Background:

  • Turing instability is a key mechanism for pattern formation in continuous systems.
  • Recent work extends Turing instability to discrete media like complex networks.
  • Understanding the link between network topology and pattern induction is crucial for socioeconomic applications.

Purpose of the Study:

  • To mathematically describe a two-species reaction-diffusion process on various network topologies.
  • To investigate how network topology influences Turing-induced pattern formation.
  • To explore the potential of Turing instability as a generative mechanism in socioeconomic contexts.

Main Methods:

  • Developed a general mathematical model for a two-species reaction-diffusion system on networks.
  • Analyzed predator-prey dynamics applied to competing features in social contexts.
  • Investigated the role of diffusion coefficients and graph Laplacian eigenvectors in pattern emergence.

Main Results:

  • Turing instability can be induced in any network topology by manipulating diffusion or connectivity.
  • Pattern formation is influenced by the interplay between diffusion rates and network structure.
  • Networks with high degree heterogeneity exhibit more stable patterns compared to homogeneous networks.

Conclusions:

  • Network topology significantly impacts Turing-induced pattern formation.
  • Diffusion and connectivity are key parameters for controlling pattern emergence.
  • Heterogeneous networks offer greater stability for emergent patterns, suggesting potential for predictable social dynamics.