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Turing patterns in simplicial complexes.

Shupeng Gao1,2, Lili Chang3,4, Matjaž Perc5,6,7,8,9

  • 1School of Mechanical Engineering, Northwestern Polytechnical University, Xi'an 710072, China.

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Summary
This summary is machine-generated.

Higher-order interactions in simplicial complexes create novel Turing patterns, differing from traditional network models. These patterns reveal unique relationships between reactant concentrations and network structure.

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Area of Science:

  • Network Science
  • Mathematical Biology
  • Complex Systems

Background:

  • Spontaneous pattern formation in nature, like stripes and spots, is explained by reaction-diffusion systems, known as Turing patterns.
  • Network science has expanded beyond pairwise interactions to include long-range links and higher-order interactions, revealing complex spatiotemporal patterns.

Purpose of the Study:

  • Investigate Turing pattern formation within simplicial complexes, which feature interactions among three or more nodes.
  • Understand the impact of higher-order interactions on pattern emergence in reaction-diffusion systems.

Main Methods:

  • Defined a canonical reaction-diffusion system on a simplicial complex.
  • Analyzed the resulting Turing patterns and their relationship to network topology.

Main Results:

  • Observed Turing patterns that fundamentally differ from those in traditional networks.
  • Found a stable distribution where the fraction of nodes above equilibrium concentration is exponentially related to the average degree of 2-simplexes.
  • Identified parameter regions where Turing patterns emerge exclusively due to higher-order interactions.

Conclusions:

  • Simplicial complexes introduce novel mechanisms for Turing pattern formation.
  • Higher-order interactions are crucial for understanding pattern emergence in complex systems beyond pairwise interactions.