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

Lin Qi, Jiaxin Zhang

TL;DR

The paper tackles the problem of generating higher-order fractal networks with tunable dimensionality. It introduces an iterative construction that produces pure $K$-dimensional simplicial complexes $\mathcal{K}_{t}(K,m)$ by transforming each $K$-simplex into a larger structure via edge midpoints, bottoms, and multiplier nodes, and analyzes the 1-skeleton $\mathcal{G}_{t}(K,m)$. Fractal characteristics are established analytically through the similarity dimension $d_s = \log S / \log 2$, with $S$ determined by $(K,m)$, and are corroborated by box-counting estimates that match $d_s$, confirming fractality. For large $m$, the generalized degree distributions $P_{K,l}(k)$ with $1 \le l \le K-1$ exhibit approximate power-law scaling, $P_{K,l}(k) \sim k^{-\gamma}$, with an explicit exponent $\gamma$ that depends on $(K,m)$, indicating scale-free behavior in higher-order interactions. Overall, the framework provides a controllable method to generate fractal higher-order networks with potential applications in modeling multilateral interactions in complex systems.

Abstract

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.

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

TL;DR

The paper tackles the problem of generating higher-order fractal networks with tunable dimensionality. It introduces an iterative construction that produces pure -dimensional simplicial complexes by transforming each -simplex into a larger structure via edge midpoints, bottoms, and multiplier nodes, and analyzes the 1-skeleton . Fractal characteristics are established analytically through the similarity dimension , with determined by , and are corroborated by box-counting estimates that match , confirming fractality. For large , the generalized degree distributions with exhibit approximate power-law scaling, , with an explicit exponent that depends on , indicating scale-free behavior in higher-order interactions. Overall, the framework provides a controllable method to generate fractal higher-order networks with potential applications in modeling multilateral interactions in complex systems.

Abstract

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.

Paper Structure

This paper contains 11 sections, 7 theorems, 27 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Model Setup
  3. Fundamental Concepts of Higher-order Networks
  4. Model
  5. Properties
  6. The Number of $K$-Dimensional Simplices
  7. Fractal Dimension
  8. Similarity Dimension
  9. Box-counting Dimension
  10. Generalized Degree Distribution
  11. Conclusion

Key Result

Proposition 3.1

When $t \ge K$, the generalized degree $k_{K,l}$ of an $l$-simplex in the network may take the value $(m + 1)^r$ ($r = 0, 1, 2, \cdots, K - L$), and the first occurrence time of an $l$-dimensional face whose generalized degree equals to $(m + 1)^r$ is $r$. Let $Y_{K}(l, t, r)$ denote the number of $ For $t \ge 0$ and $l = K$, For $t \ge 0$ and $l = 1, 2, 3, \dots, K - 1$, or, equivalently,

Figures (6)

  • Figure 1: Simplicial Complex, 1-Skeleton, and Clique Complex. (a) A two-dimensional simplicial complex, with blue triangles representing the 2-simplices it contains, edges denoting the 1-simplices it encompasses, and points signifying the 0-simplices it holds; (b) The 1-skeleton corresponding to the two-dimensional simplicial complex in (a) is also the 1-skeleton of the network in (c); (c) The clique complex generated by (b) is a simplicial complex composed of the 3-simplex $\{1,5,6,7\}$, the 2-simplex $\{1,2,3\}$, the 1-simplex $\{1,4\}$, and their faces.
  • Figure 2: Schematic Diagram of the Iterative Construction Process. The grey nodes in the diagram represent nodes present at the outset, the blue nodes denote midpoints inserted along edges during iteration, and the green nodes signify multiplier nodes. (a) Schematic of one iteration when $K = 1$ and $m = 2$, where the blue node form bottom; (b) Schematic of one iteration when $K = 2$ and $m = 1$, and the blue edges form bottoms; (c)Schematic of one iteration when $K = 3$ and $m = 1$, and the blue triangles form bottoms.
  • Figure 3: Networks Obtained After Multiple Iterations. (a)-(c) Networks after four iterations with $K = 1$ for $m = 1, 2, 3$, respectively; (d)-(f) Networks after four iterations with $K = 2$ for $m = 1, 2, 3$, respectively; (g)-(i) Networks after three iterations with $m = 1$ for $K = 2, 3, 4$, respectively.
  • Figure 4: Plot of the fractal dimension as a function of dimension $K$ and multiplier $m$. (a) The fractal dimension of the graph generated with $m = 3$ after two iterations, for $K$ ranging from 1 to 10; (b) The fractal dimension of the graph generated with $K = 2$ after two iterations, for $m$ ranging from 0 to 10.
  • Figure 5: Plot of the 1-dimensional generalized degree distribution $P_{K, 1}(k)$ for networks under different combinations of parameters $K$ and $m$ in a double-logarithmic coordinate system.
  • ...and 1 more figures

Theorems & Definitions (14)

  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • proof
  • ...and 4 more