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

Hierarchical networks as fundamental models to describe the complex networks, have many applications in networks science, engineering technology and so on. In this paper, we first propose a new class of hierarchical networks with fractal structure, which are the networks with triangles compared to traditional hierarchical networks. Second, we study the precise results of some structural properties to derive small-world effect and scale-free feature. Third, it is found that the constructed network is sparse through the average degree and density. Fourth, it is also demonstrated that 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 Formula: see text tends to stabilize at a lower bound as Formula: see text iterates to a certain number, and the average distance of Formula: see text has an increasing relationship along with the value of Formula: see text.