  Given that a group of cities follows a scaling law connecting urban
population with socio-economic or infrastructural metrics (transversal
scaling), should we expect that each city would follow the same behavior over
time (longitudinal scaling)? This assumption has important policy implications,
although rigorous empirical tests have been so far hindered by the lack of
suitable data. Here, we advance the debate by looking into the temporal
evolution of the scaling laws for 5507 municipalities in Brazil. We focus on
the relationship between population size and two urban variables, GDP and water
network length, analyzing the time evolution of the system of cities as well as
their individual trajectory. We find that longitudinal (individual) scaling
exponents are city-specific, but they are distributed around an average value
that approaches to the transversal scaling exponent when the data are
decomposed to eliminate external factors, and when we only consider cities with
a sufficiently large growth rate. Such results give support to the idea that
the longitudinal dynamics is a micro-scaling version of the transversal
dynamics of the entire urban system. Finally, we propose a mathematical
framework that connects the microscopic level to global behavior, and, in all
analyzed cases, we find good agreement between theoretical prediction and
empirical evidence.
