Abstract
Complex networks have attracted growing attention in many fields. As a generalization of fractal analysis, multifractal analysis (MFA) is a useful way to systematically describe the spatial heterogeneity of both theoretical and experimental fractal patterns. Some algorithms for MFA of unweighted complex networks have been proposed in the past a few years, including the sandbox (SB) algorithm recently employed by our group. In this paper, a modified SB algorithm (we call it SBw algorithm) is proposed for MFA of weighted networks. First, we use the SBw algorithm to study the multifractal property of two families of weighted fractal networks (WFNs): “Sierpinski” WFNs and “Cantor dust” WFNs. We also discuss how the fractal dimension and generalized fractal dimensions change with the edge-weights of the WFN. From the comparison between the theoretical and numerical fractal dimensions of these networks, we can find that the proposed SBw algorithm is efficient and feasible for MFA of weighted networks. Then, we apply the SBw algorithm to study multifractal properties of some real weighted networks — collaboration networks. It is found that the multifractality exists in these weighted networks, and is affected by their edge-weights.
Highlights
Out that a single fractal dimension is not enough to characterize the fractal property of a scale-free network when the network has a multifractal structure
In order to show that the SBw algorithm for MFA of weighted network is effective and feasible, we apply our method to study the multifractal behavior of the “Sierpinski” weighted fractal network (WFN) and the “Cantor dust” WFNs53
These WFNs are constructed by Iterated Function Systems (IFS)[55], whose Hausdorff dimension is completely characterized by two main parameters: the number of copies s > 1 and the scaling factor 0 < f < 1 of the IFS
Summary
Out that a single fractal dimension is not enough to characterize the fractal property of a scale-free network when the network has a multifractal structure They introduced a compact-box-burning (CBB) algorithm for MFA of complex networks. Wang et al.[42] proposed an improved fixed-size box-counting algorithm to study the multifractal behavior of complex networks. This algorithm was improved further by Li et al.[43]. They applied the improved fixed-size box-counting algorithm to study multifractal properties of a family of fractal networks proposed by Gallos et al.[47]. We apply the SBw algorithm to study multifractal properties of some real weighted networks — collaboration networks[54]
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