Weighted average geodesic distance of Vicsek network in three-dimensional space

Abstract

Fractal generally has self-similarity. Using the self-similarity of fractal, we can obtain some important theories about complex networks. In this paper, we concern the Vicsek fractal in three-dimensional space, which provides a natural generalization of Vicsek fractal. Concretely, the Vicsek fractal in three-dimensional space is obtained by repeatedly removing equilateral cubes from an initial equilateral cube of unit side length, at each stage each remaining cube is divided into [Formula: see text] smaller cubes of which [Formula: see text] are kept and the rest discarded, where [Formula: see text] is odd. In addition, we obtain the skeleton network of the Vicsek fractal in three-dimensional space. Then we focus on weighted average geodesic distance of the Vicsek fractal in three-dimensional space. Take [Formula: see text] as an example, we define a similar measure on the Vicsek fractal in three-dimensional space by weight vector and calculate the weighted average geodesic distance. At the same time, asymptotic formula of weighted average geodesic distance on the skeleton network is also obtained. Finally, the general formula of weighted average geodesic distance should be applicable to the models when [Formula: see text], the base of a power, is odd.