Taylor's Law and the Spatial Distribution of Urban Facilities

Taylor's law is the footprint of ecosystems, which admits a power function relationship $S^{2}=am^{b}$ between the variance $S^{2}$ and mean number $m$ of organisms in an area. We examine the distribution of spatial coordinate data of seven urban facilities (beauty salons, banks, stadiums, schools, pharmacy, convenient stores and restaurants) in 37 major cities in China, and find that Taylor's law is validated among all 7 considered facilities, in the fashion that either all cities are combined together or each city is considered separately. Moreover, we find that the exponent $b$ falls between 1 and 2, which reveals that the distribution of urban facilities resembles that of the organisms in ecosystems. Furthermore, through decomposing the inverse of exponent $b$, we examine two different factors affecting\emph{ }the numbers of facilities in an area of a city respectively, which are the city-specific factor and the facility-specific factor. The city-specific factor reflects the overall density of all the facilities in a city, while the facility-specific factor indicates the overall aggregation level of each type of facility in all the cities. For example, Beijing ranks the first in the overall density, while restaurant tops the overall aggregation level.