Scaling laws in intra-urban systems and over time at the district level in Shanghai, China

Numerous urban indicators scale with urban population in a system of cities. However, whether the scaling law also exists in an intra-urban system and whether the scaling law across cities can be applied in the temporal growth of individual cities are not clear. Taking Shanghai, China, as an example, we collected urban population, building areas and other urban indicators from 2005 to 2017 in districts (the secondary administrative unit of Chinese cities). The building area has a robust scaling relationship with urban population among different districts in each year, indicating that the scaling law also exists in the intra-urban system composed by districts. The scaling exponent between building areas and population in the intra-urban system is less than one (sub-linear relationship), which means building areas increase more slowly than population across districts. Temporally, building areas increase faster than population in most individual districts, showing a super-linear relationship. The contrasting relationship between building area and population across districts and over time indicates that the scaling law cannot be applied in the temporal growth of individual districts or cities. In conclusion, the scaling law also exists in the intra-urban system, and the cross-sectional scaling law cannot be applied to individual parts over time. Our further analyses using GDP and other five urban indicators confirm the two findings.