Abstract
Fractal and self-similarity are important characteristics of complex networks. The correlation dimension is one of the measures implemented to characterize the fractal nature of unweighted structures, but it has not been extended to weighted networks. In this paper, the correlation dimension is extended to the weighted networks. The proposed method uses edge-weights accumulation to obtain scale distances. It can be used not only for weighted networks but also for unweighted networks. We selected six weighted networks, including two synthetic fractal networks and four real-world networks, to validate it. The results show that the proposed method was effective for the fractal scaling analysis of weighted complex networks. Meanwhile, this method was used to analyze the fractal properties of the Newman–Watts (NW) unweighted small-world networks. Compared with other fractal dimensions, the correlation dimension is more suitable for the quantitative analysis of small-world effects.
Highlights
IntroductionRevealing and characterizing complex systems from a complex networks perspective has attracted attention
Correlation Dimension of ComplexRecently, revealing and characterizing complex systems from a complex networks perspective has attracted attention
The results show that compared with other fractal dimensions, the correlation dimension of weighted fractal network (WFN) is close to the theoretical value
Summary
Revealing and characterizing complex systems from a complex networks perspective has attracted attention. Song et al extended the fractal dimension to complex networks and found that many real-world networks have self-repetitive structures at all scales [11,12]. Lacasa et al proposed a method for correlation dimension in complex networks, and it is only applicable to the networks in coordinate space [27,28]. The edge-weights of complex networks exhibit the strength of the correlation among its components and are coupled into a topology for more accurately representing the network structure. Wei’s method was denoted by BCANw and proven to be valid for calculating the information dimensions [14,34] and volume dimensions [17] in weighted complex networks. The correlation dimension should be extended to adapt to the real-world weighted networks and contribute to the studies of chaotic signals from the perspective of complex networks
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