Analyses of Some Structural Properties on a Class of Hierarchical Scale-free Networks

TL;DR

This paper introduces a new class of hierarchical, scale-free, and fractal networks with triangles, analyzing their structural properties to demonstrate small-world effects, power-law degree distribution, and network sparsity.

Contribution

It proposes a novel hierarchical network model with fractal and scale-free features, providing detailed analysis of its structural properties and behaviors.

Findings

Networks exhibit small-world effect and scale-free features.

Degree distribution follows power law for hubs and exponential for bottom nodes.

Clustering coefficient stabilizes, and average distance increases with ln of network size.

Abstract

Hierarchical networks actually have many applications in the real world. Firstly, we propose a new class of hierarchical networks with scale-free and fractal structure, which are the networks with triangles compared to traditional hierarchical networks. Secondly, we study the precise results of some structural properties to derive small-world effect and scale-free feature. Thirdly, it is found that the constructed network is sparse through the average degree and density. Fourthly, it is also demonstrated the degree distributions of hub nodes and the bottom nodes are the power law and exponential, respectively. Finally, we prove that clustering coefficient with a definite value z tends to stabilize at a lower bound as t iterates to a certain number, and the average distance of G_{t}^{z} has a increasing relationship along with the value of lnN_{t}.