Fractal And Multifractal Properties And Laplace Of Two Families Of Fractal Networks

In this work, first we study the fractal and multifractal properties of a family of fractal networks which are resistant to disease spread introduced by Zhang et al. form Fudan University(J. Stat. Mech.: Theor. Exp., 2008, P00008). In this fractal network model, there is a parameter e which is between 0 and 1, and allowing for tuning the level of fractality in the network. Here we also examine the dependence relationship of the fractal dimension and the multifractal parameters on parameter e. First, we find that empirical fractal dimensions of these network obtained by our program coincide with the theoretical values given by Zhang et al. Then from the shape of?(q) and D(q) curves, we find the existence of multifractality in these network. Last,we use the new Laplace operator introduced by professor Lin Yong from Renmin University of China to calculate Laplace eigenvalue and energy of the network model and another network model introduced by Gallos et al(Proc. Nat. Acad. Sci. USA,2007, 104, 7746-7751). We find that the second Laplace eigenvalue of these two kinds of networks are almost the same with the increase of e(close to zero); there exists a linear regressive relationship between the parameter e and energy.