Table 1.
Geographical meanings of capacity dimension, information dimension, and correlation dimension for traffic network.
Fig 1.
The singularity spectrums of two different multifractal growth modes: Spatial concentration and deconcentration.
(A) Spatial aggregation mode; (B) Singularity spectrum for spatial aggregation; (C) Spatial diffusion mode; (D) Singularity spectrum for spatial diffusion. Note: The squares with larger size represent network intensive areas, while the smaller squares represent network sparser areas. The local parameter f(α) curve stands for the local fractal dimension of the sets of units with the singularity exponent α.
Fig 2.
Schematic representation of rescaling probability distribution for multifractal computation based on box-counting method.
The probability distribution of traffic links is calculated over all boxes and then weighted by moment order q. Dark red boxes represent dominant structures with higher weighted probability, and dark blue boxes represent lightweight regions with lower weighted probability. (A) Partial traffic networks in Shenzhen. (B) When q = 0, all nonempty boxes are equally weighted (gray). (C) When q = 1, the nonempty boxes are weighted by real growth probability. (D)-(E) For q>0, the boxes with relative high-density measures gradually gain more importance and their contribution to the entropy will dominate. (F) For q<0, the boxes with relatively low-density measures gradually gain more importance and their contribution to the entropy will dominate.
Fig 3.
The spatial distribution of twelve megacities of China.
All of them are the most representative cities in typical regions of China, including North China (Beijing, Tianjin) drawn in red, Northeast China drawn in orange (Harbin, Shenyang), Central China drawn in brown (Zhengzhou, Wuhan), West China drawn in purple (Xi’an, Chengdu), East China drawn in blue (Shanghai, Nanjing), South China drawn in green (Shenzhen, Guangzhou).
Fig 4.
The distribution of street networks in twelve representative cities.
The black line is the city boundary identified, which is smaller than the municipal area for most cities; the grey lines represent the street network.
Table 2.
Basic fractal parameters of street network for 12 Chinese cities.
Fig 5.
The generalized correlation dimension spectrums of street networks of 12 Chinese cities.
The Dq curve of Beijing appears in both (A) and (B) as a reference. They are monotonic decreasing functions of q, indicating multifractal property. The right tails of Dq curves and convergence values of q>0 are very close to each other for most cities. While the left tails of Dq curves exhibit wider gaps. This means that the development of traffic networks in the central areas of different cities is similar, but there are significant differences in the suburbs from city to city.
Fig 6.
The singularity exponent spectrums of street networks for different cities.
The α(q) curve of Beijing appears in both (A) and (B) as a reference. The singularity exponent α(q) curve is a monotonic decreasing function of q. Generally, the left tails of curves exhibit distinct differences among cities.
Table 3.
Multifractal parameters of street network for each city by OLS regression method.
Fig 7.
The local fractal dimension spectrums of street networks for different cities.
The f(q) curve of Beijing appears in both (A) and (B) as a reference. The local fractal dimension f(q) curve is a distinct non-symmetric shape curve, high on the left and low on the right. Besides, the f(q) spectrums of most cities show an abnormal decrease when q<-2. Guangzhou and Shenzhen are exceptions.
Fig 8.
The local singularity spectrums of street networks for different cities.
The singularity spectrum f(α) is an asymmetric unimodal curve, low on the left tails and high on the right tails. The f(α) spectrum inclines to the left.
Fig 9.
The log-log plots for estimating the local fractal dimension f(q) of Beijing’s traffic network with changes of q.
With the absolute value of q increases, the scattered points in log-log plots become more and more disordered, and the scaling relationships are broken seriously.
Table 4.
The statistic thresholds of the moment order q based on significance level α = 0.05.
Table 5.
The main results and inferences of calculations and analyses on multifractal traffic networks of 12 Chinese cities.