  Numerous urban indicators scale with population in a power law across cities,
but whether the cross-sectional scaling law is applicable to the temporal
growth of individual cities is unclear. Here we first find two paradoxical
scaling relationships that urban built-up area sub-linearly scales with
population across cities, but super-linearly scales with population over time
in most individual cities because urban land expands faster than population
grows. Different cities have diverse temporal scaling exponents and one city
even has opposite temporal scaling regimes during two periods, strongly
supporting the absence of single temporal scaling and further illustrating the
failure of cross-sectional urban scaling in predicting temporal growth of
cities. We propose a conceptual model that can clarify the essential difference
and also connections between the cross-sectional scaling law and temporal
trajectories of cities. Our model shows that cities have an extra growth of
built-up area over time besides the supposed growth predicted by the
cross-sectional scaling law. Disparities of extra growth among different-sized
cities change the cross-sectional scaling exponent. Further analyses of GDP and
other indicators confirm the contradiction between cross-sectional and temporal
scaling relationships and the validity of the conceptual model. Our findings
may open a new avenue towards the science of cities.
