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 {\it et al.} ({\it Proc. Natl. Acad. Sci. U.S.A.}, 2007, {\bf 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 {\it et al.} ( {\it Nat. Phys}, 2006, {\bf 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 $e$.

12 pages, 7 figures, accepted by J. Stat. Mech

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• Multifractal Characterization of Protein Contact Networks
• A Fixed-Mass multifractal approach for unweighted complex networks
• Multifractality in random networks with power-law decaying bond strengths
• A computationally-efficient sandbox algorithm for multifractal analysis of large-scale complex networks with tens of millions of nodes
• Multifractality of complex networks is also due to geometry. The Geometric SandBox algorithm
• Bifractality of fractal scale-free networks
• Random walks on bifractal networks