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Numerous urban indicators scale with population in a power law across cities,but whether the cross-sectional scaling law is applicable to the temporalgrowth of individual cities is unclear. Here we first find two paradoxicalscaling relationships that urban built-up area sub-linearly scales withpopulation across cities, but super-linearly scales with population over timein most individual cities because urban land expands faster than populationgrows. Different cities have diverse temporal scaling exponents and one cityeven has opposite temporal scaling regimes during two periods, stronglysupporting the absence of single temporal scaling and further illustrating thefailure of cross-sectional urban scaling in predicting temporal growth ofcities. We propose a conceptual model that can clarify the essential differenceand also connections between the cross-sectional scaling law and temporaltrajectories of cities. Our model shows that cities have an extra growth ofbuilt-up area over time besides the supposed growth predicted by thecross-sectional scaling law. Disparities of extra growth among different-sizedcities change the cross-sectional scaling exponent. Further analyses of GDP andother indicators confirm the contradiction between cross-sectional and temporalscaling relationships and the validity of the conceptual model. Our findingsmay open a new avenue towards the science of cities.

Cross-sectional Urban Scaling Fails in Predicting Temporal Growth of Cities