Fractal and multifractal properties of a family of fractal networks

Abstract

In this work, we study the fractal and multifractal properties of a family of fractal networks introduced by Gallos et al. ( Proc. Natl. Acad. Sci. U.S. A., 2007, 104: 7746). In this fractal network model, there is a parameter e 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, dependence relationship of fractal dimension and the multifractal parameters on the parameter e. First, we find that the empirical fractal dimensions of these networks obtained by our program coincide with the theoretical formula given by Song et al. ( Nat. Phys, 2006, 2: 275). Then from the shape of the τ (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.

Fractal analysis of the 3D microstructure of porous materials

“…In contract to a "monofractal" with geometrical parameters determined by only one fractal dimension d, the spectrum of generalized fractal dimensions of a multifractal (inhomogeneous fractal) determines some structural and statistical properties of the system. The theory of multifractals attracts attention of researches engaged in various fields of physics and mathematics because analysis of experimental results suggests that many systems exhibit multifractal behav ior [16][17][18][19][20][21][22][23][24][25][26][27].…”

Fractal and multifractal analysis of complex networks: Estonian network of payments

“…Song et al [30] developed a method for calculating the fractal dimension of a complex network by using a box-covering algorithm and identified self-similarity as a property of complex networks [31]. Moreover, a myriad of algorithms and studies on networks' multifractal analysis have been proposed and developed lately [32][33][34][35][36][37].…”

Determination of multifractal dimensions of complex networks by means of the sandbox algorithm

“…Lee and Jung [43] found that MFA is the best tool to describe the probability distribution of the clustering coefficient of a complex network. Some algorithms have been proposed to calculate the mass exponents τ (q) and to study the multifractal properties of complex networks [44][45][46][47]. Based on the compact-box-burning algorithm for fractal analysis of complex networks which is introduced by Song et al [5], Furuya and Yakubo [44] proposed a compact-box-burning (CBB) algorithm for MFA of complex networks and applied it to show that some networks have multifractal structures.…”

“…Wang et al [45] proposed a modified fixed-size boxcounting algorithm to detect the multifractal behavior of some theoretical and real networks. Li et al [46] improved the modified fixed-size box-counting algorithm [45] and used it to investigate the multifractal properties of a family of fractal networks introduced by Gallos et al [48]. We call the algorithm in Ref.…”

“…We call the algorithm in Ref. [46] the improved BC algorithm. Recently, we adopted the improved BC to study the multifractal properties of the recurrence networks constructed from fractional Brownian motions [47].…”