Scaling properties of urban facilities

Two measurements are employed to quantitatively investigate the scaling properties of the spatial distribution of urban facilities: the $K$ function [whose derivative gives the radial distribution function $\rho \left(t\right)=K^{\prime }\left(t\right)/2\pi t$] by number counting and the variance-mean relationship by the method of expanding bins. The $K$ function and the variance-mean relationship are both power functions. This means that the spatial distributions of urban facilities are scaling invariant. Further analysis of more data (which includes eight types of facilities in 37 major Chinese cities) shows that the the power laws broadly hold for all combinations of facilities and cities. A double stochastic process (DSP) model is proposed as a mathematical mechanism by which spatial point patterns can be generated that resemble the actual distribution of urban facilities both qualitatively and quantitatively. Simulation of the DSP yields a better agreement with the urban data than the correlated percolation model.

Figure 1

Scaling properties of three facilities (banks, convenience stores, and restaurants) in Beijing and Guangzhou measured by two methods: (a) the $K$ function in a log-log plot and (b) the variance-mean relationship in a log-log plot by the method of expanding bins.

Figure 2

Exponent $b$ of the variance-mean relationship in 29 Chinese cities (8 cities are excluded, as there were not enough sample data for at least one facility) for (a) beauty salons, banks, stadiums, and schools and (b) pharmacy, convenience stores, restaurants, and tea houses. Error ranges indicate only statistical errors from regression.

Figure 3

Double stochastic process (DSP) modeling of banks in a large Beijing area ($2^{15}\times 2^{15}$ ${\mathrm{m}}^2$): (a) original coordinate data on banks in Beijing mapped to a ($2^{10}\times 2^{10}$) lattice; (b) a random sample of the point pattern generated by the DSP; (c) fitting of the power-law covariance; (d) variance-mean relationship. In both (c) and (d), dashed lines represent the real data, and solid lines are the simulation results averaged over 50 samples.

Figure 4

Double stochastic process (DSP) modeling of banks in a metropolitan area of Beijing ($2^{14}\times 2^{14}$ ${\mathrm{m}}^2$): (a) original coordinate data on banks in Beijing mapped to a ($2^{10}\times 2^{10}$) lattice; (b) a random sample of the point pattern generated from the DSP; (c) fitting of the power-law covariance; (d) variance-mean relationship. In both (c) and (d), dashed lines represent results from the real data, and solid lines are those from simulation averaged over 50 samples.

Figure 5

Correlated percolation model (CPM) of banks in a metropolitan area of Beijing ($2^{14}\times 2^{14}$ ${\mathrm{m}}^2$), (a) the original coordinate data of banks in Beijing mapped to a ($2^{10}\times 2^{10}$) lattice; (b) a random sample of the point pattern from the CPM; (c) fitting of the power-law covariance; (d) variance-mean relationship. In both (c) and (d), dashed lines represent results from the real data; solid lines, those from simulation averaged over 50 samples.