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This study reveals that the average degree of a network quantitatively influences Turing pattern formation. An exponential decay relationship was found, enabling prediction and control of patterns in activator-inhibitor systems.

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Area of Science:

  • Complex Systems
  • Network Science
  • Mathematical Biology

Background:

  • Turing patterns arise from activator-inhibitor systems, traditionally studied in continuous domains.
  • Networked reaction-diffusion systems have advanced Turing pattern research, but network topology's precise influence remains unclear.
  • Average degree is a critical network topological feature.

Purpose of the Study:

  • To investigate the precise influence of network topology, specifically average degree, on Turing pattern formation.
  • To establish a quantitative relationship between average degree and pattern formation in networked systems.

Main Methods:

  • Qualitative analysis of average degree's influence on pattern formation.
  • Nonlinear regression to derive a quantitative relationship between pattern formation and average degree.
  • Validation across diverse activator-inhibitor systems (biology, ecology, chemistry).

Main Results:

  • Average degree significantly impacts Turing pattern formation.
  • An "exponential decay" relationship quantitatively describes pattern formation's dependence on average degree.
  • This finding is consistent across biological, ecological, and chemical models.

Conclusions:

  • The average degree is a key determinant of Turing pattern formation in networked systems.
  • The identified exponential decay provides a predictive and controllable model for pattern formation.
  • This research offers new insights into understanding and manipulating pattern formation in complex networks.