Fractal and multifractal properties of a family of fractal networks

In this work, we study the fractal and multifractal properties of a family of fractal networks introduced by Callus et al (2007 Proc. Nat. Acad. Sci. USA 104 774(1). In this fractal network model, there is a parameter which is between 0 and 1, and allows for tuning, the level of fractality in the network. Here we examine the multifractal behavior of these networks, the dependence relationship of the fractal dimension and the multifractal parameters on parameter c. First, we find that the empirical tract al dimensions of these networks obtained by our program coincide with the theoretical formula given by Song et of (2006 Nature Phys. 2 275). Then from the shape of the tau(q) and D(q) curves, we find the existence of multifractality in these networks. Last, we find that there exists a linear relationship between the average information dimension < D(1)> and the parameter e.