Abstract
Traffic networks have been proved to be fractal systems. However, previous studies mainly focused on monofractal networks, while complex systems are of multifractal structure. This paper is devoted to exploring the general regularities of multifractal scaling processes in the street network of 12 Chinese cities. The city clustering algorithm is employed to identify urban boundaries for defining comparable study areas; box-counting method and the direct determination method are utilized to extract spatial data; the least squares calculation is employed to estimate the global and local multifractal parameters. The results showed multifractal structure of urban street networks. The global multifractal dimension spectrums are inverse S-shaped curves, while the local singularity spectrums are asymmetric unimodal curves. If the moment order q approaches negative infinity, the generalized correlation dimension will seriously exceed the embedding space dimension 2, and the local fractal dimension curve displays an abnormal decrease for most cities. The scaling relation of local fractal dimension gradually breaks if the q value is too high, but the different levels of the network always keep the scaling reflecting singularity exponent. The main conclusions are as follows. First, urban street networks follow multifractal scaling law, and scaling precedes local fractal structure. Second, the patterns of traffic networks take on characteristics of spatial concentration, but they also show the implied trend of spatial deconcentration. Third, the development space of central area and network intensive areas is limited, while the fringe zone and network sparse areas show the phenomenon of disordered evolution. This work may be revealing for understanding and further research on complex spatial networks by using multifractal theory.
Highlights
Scientific research includes two processes: one is description, and the other is understanding
[Result] At the global level, the Dq spectrums take on inverse S-shaped curves; at the local level, the f(α) spectrums take on unimodal curves. [Explanation] For monofractals, the Dq spectrums become horizontal straight lines, and each f(α) spectrum condenses into a point
The generalized correlation dimension spectrums take on inverse S-shaped curves, and the local fractal dimension spectrums take on unimodal curves
Summary
Scientific research includes two processes: one is description, and the other is understanding. Traffic networks proved to be typical complex spatial systems with no characteristic scale [7,8,9,10,11,12]. It is hard to model network structure mathematically, but it is relatively easy to model hierarchical structure using proper mathematical tools In this regard, multifractal theory may provide an advisable approach to studying self-organized complex networks such as urban transportation through self-similar hierarchical networks. Spatial analysis of multifractal systems are based on multifractal parameter spectrums, including global multifractal spectrum, i.e., Dq-q spectrum, and local parameter spectrum, i.e., f(α)-α spectrum The latter is termed f(α) curve and represents the basic multifractal spectrum. This is the simplest approach to distinguishing monofractal from multifractals
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