Iterative Generation and Generalized Degree Distribution of Higher-Order Fractal Scale-Free Networks

Fractals represent one of the fundamental manifestations of complexity, and fractal networks serve as tools for characterizing and investigating the fractal structures and properties of large-scale systems. Higher-order networks have emerged as a research hotspot due to their ability to express interactions among multiple nodes. This study proposes an iterative generation model for higher-order fractal networks. The iteration is controlled by three parameters: the dimension K of the simplicial complex, the multiplier m, and the iteration count t. The constructed network is a pure simplicial complex. Theoretical analysis using the similarity dimension and experimental verification using the box-counting dimension demonstrate that the generated networks exhibit fractal characteristics. When the multiplier m is large, the generalized degree distribution of the generated networks is characterized by its scale-free nature.