Asymptotic formula on APL of fractal evolving networks generated by Durer Pentagon

Complex networks constructed by fractals have many applications in many fields, such as the data center networks and fractal antennas. In this paper, we consider a kind of evolving networks modeled on the classical fractal, Durer Pentagon, whose nodes are all the solid pentagons in the construction of Durer Pentagon up to stage t. In this network, two nodes are neighbors if and only if the intersection of their corresponding pentagons is a line segment. Using self-similarity and renewal theorem, we obtain the asymptotic formula on average path length (APL) of our evolving network.