Empirical analysis on the connection between power-law distributions and allometries for urban indicators

TL;DR

This study investigates the relationship between power-law distributions and allometries among urban indicators in Brazilian cities, revealing consistent patterns and dependencies that enhance understanding of urban dynamics.

Contribution

It provides an extensive empirical analysis demonstrating the connection between power-law distributions and allometries across multiple urban indicators.

Findings

Urban indicators follow power-law distributions.

Allometries are correlated with power-law behaviors.

Residuals exhibit log-normal distribution with constant variance.

Abstract

We report on the existing connection between power-law distributions and allometries. As it was first reported in [PLoS ONE 7, e40393 (2012)] for the relationship between homicides and population, when these urban indicators present asymptotic power-law distributions, they can also display specific allometries among themselves. Here, we present an extensive characterization of this connection when considering all possible pairs of relationships from twelve urban indicators of Brazilian cities (such as child labor, illiteracy, income, sanitation and unemployment). Our analysis reveals that all our urban indicators are asymptotically distributed as power laws and that the proposed connection also holds for our data when the allometric relationship displays enough correlations. We have also found that not all allometric relationships are independent and that they can be understood as a consequence of the allometric relationship between the urban indicator and the population size. We further show that the residuals fluctuations surrounding the allometries are characterized by an almost constant variance and log-normal distributions.