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Growing order-heterogeneous simplicial complexes.

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Summary
This summary is machine-generated.

This study introduces a growing simplicial complex model to capture complex systems with higher-order interactions. The model generates customizable structures with tunable power-law distributions for generalized degrees.

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Area of Science:

  • Complex Systems Science
  • Network Science
  • Mathematical Modeling

Background:

  • Traditional graph models capture pairwise interactions but fail to represent higher-order interactions in complex systems.
  • Higher-order interactions involve multiple entities simultaneously and require advanced mathematical frameworks like simplicial complexes.
  • Real-world systems often exhibit heterogeneity in interaction complexity, necessitating models that account for varying simplex dimensions.

Purpose of the Study:

  • To introduce a novel growing simplicial complex model that incorporates heterogeneous higher-order interactions.
  • To analyze the structural properties, specifically the generalized degree distribution, of the proposed model.
  • To demonstrate the model's flexibility in generating simplicial complexes with tunable characteristics.

Main Methods:

  • Developed a growing simplicial complex model where new simplices' dimensions are sampled from a probability distribution.
  • Performed theoretical analysis to derive the generalized degree distribution of faces within the model.
  • Conducted numerical simulations to validate theoretical predictions and explore emergent structural properties.

Main Results:

  • The generalized degree of faces in the growing simplicial complex model follows a power-law distribution.
  • The exponents of the power-law distribution can be precisely controlled by adjusting the simplex-dimension sampling distribution.
  • Numerical simulations confirmed the theoretical findings and demonstrated the model's ability to generate simplicial complexes with customizable structures.

Conclusions:

  • The proposed growing simplicial complex model provides a versatile framework for studying systems with heterogeneous higher-order interactions.
  • This model allows for the generation of simplicial complexes with tunable structural properties, facilitating the investigation of emergent phenomena.
  • The findings offer new theoretical tools for analyzing complex systems beyond pairwise interactions.