Growing order-heterogeneous simplicial complexes

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.