On the relation between Transversal and Longitudinal Scaling in Cities

TL;DR

This study investigates the relationship between transversal and longitudinal scaling laws in cities, analyzing data from 5507 Brazilian municipalities to understand how individual city growth relates to overall urban system patterns.

Contribution

It provides empirical evidence that longitudinal city-specific scaling exponents are aligned with transversal exponents when external factors are controlled and large-growth cities are considered.

Findings

Longitudinal exponents are city-specific but centered around the transversal exponent.

Scaling behavior of individual cities approaches the system-wide pattern under certain conditions.

A mathematical framework links micro-level city dynamics to macro-level urban scaling laws.

Abstract

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.