Average path length and Fermat distance in fractal networks composed of high-dimensional Sierpinski pyramids

Complex networks created by fractals have been associated with various applications such as social networks, modern communication, and fractal antennas. In this paper, we consider a series of scale-free networks based on the construction of high dimensional Sierpinski pyramids. Using self-similarity and elementary renewal theorem, we derive the asymptotic formulas of the average path length (APL) and the average Fermat distance (AFD) on our network sequence in turn. Our approaches are suitable for various networks modeled on self-similar fractals. It is known that the ratio of AFD to APL is between 3/2 and 2, which implies that AFD is also a key property to characterize small-world effect. Through an elaborate investigation of our networks, the limit of this ratio for our evolving networks is 3/2 and we analyze the result in the light of our interpretation. In fact, in some scale-free networks, the nodes with high degrees have more influence on the choice of Fermat points and thus Fermat distance. Our conclusions may lead to some interesting relations concerning the hyperbolicity, Laplacian spectrum and some multiparameter indices of networks.