Fig 1.
Trends in the evolution of per capita values of Brazilian urban indicators.
The bar plots show the temporal evolution of eight urban indicators in the years 1991, 2000 and 2010 in which the national census took place. The bar plots are average values over Brazilian cities, where each urban indicator for each city have been divided by the corresponding city population, that is, made per capita. The definition of each indicator is provided in the Methods Section. The tiny error bars are 95% bootstrap confidence intervals for the average values. We note that elderly population, female population, homicides and family income display an increasing tendency; whereas child labor, illiteracy and male population have decreased over time. Despite having increased, the unemployment rates show a not very clear trend.
Table 1.
Allometric relationships between urban indicators and population size.
Values of parameters 𝓐i and βi obtained via orthogonal distance regression on the relationship between logYi(t) and logN(t) for each urban indicator in the year t (see Methods Section). The values inside the brackets are the standard errors (SE) in the last decimal of the estimated parameters. The last column shows the values of the Pearson linear correlation coefficient ρ for each allometry in log-log scale.
Fig 2.
Allometric laws and the definition of the scale-adjusted metric DYi(t).
The scatter plots show the allometric relationships between number of homicides and population size for the years t = 1991, 2000 and 2010 in log-log scale (see S1 and S2 Figs for all other indicators). The allometric exponents βi (see Methods Section for details on the calculation of βi) and Pearson correlation coefficient ρ are shown in the figures. We highlight a particular city (Peixoto de Azevedo) in the three years to illustrate the definition and the evolution DYi(t). For this city, the number homicides was quite above the allometric law in the year t = 1991; however, it has approached the expected value by the allometric law over the years.
Fig 3.
Trends in the evolution of the average values of the scale-adjusted metrics.
The bar plots show the evolution of the average DYi(t) when grouping the cities whose urban indicator was (A) above and (B) below the allometric law in the year t = 1991. The colorful bars are empirical data (red for t = 1991, blue for t = 2000 and green for t = 2010) and gray bars represent the predictions obtained from the linear model of Eq 6 (see discussion in main text). The error bars are 95% bootstrap confidence intervals for the average values. Note that the average DYi(t) displays a statistically significant decreasing tendency for cities whose urban indicator was initially above the allometric law and an increasing tendency is observed for cities whose urban indicator was initially below the allometric law. The only exception to this pattern is the case of illiteracy, where the average DYi(t) has increased for cities initially above the allometric law and it is almost a constant for those cities initially below. We further observe that the predicted values (gray bars) keep this main tendency.
Fig 4.
Memory effects in the evolution of the scale-adjusted metric DYi(t).
The purple dots show the values of DYi(2010) versus DYi(2000) for each city (see S3 Fig for DYi(2000) versus DYi(1991)). Despite the different scattering degrees (see the value of Pearson correlation coefficient ρ in the plots), we observe a linear correspondence between DYi(2010) and DYi(2000). The dashed lines are fits of the linear model DYi(2010) = Ai + αi DYi(2000) (Eq 3) obtained via ordinary least-square regression. The values of αi and their standard errors are shown in the plots and also summarized in Table 2. We note that αi < 1 for almost all indicators, a fact that is in agreement with the approach to allometric laws observed for the averages value of DYi(t) (Fig 3). Illiteracy is the only exception where αi ≳ 1. This result also agrees with the departing from the allometric law of the average DYi(t) in the case of illiteracy.
Table 2.
Linear coefficients αi of the memory relationships between the scale-adjusted metrics in different years.
Values of the parameters αi obtained via least square fitting the model of the Eq 3 for the relationships between the scale-adjusted metrics: DYi(2000) versus DYi(1991) and DYi(2010) versus DYi(2000). We have omitted the values of the parameters Ai because they are very small (∼ 10−6). The values inside the brackets are the standard errors (SE) in the last decimal of the estimated parameter. The last column shows the values of the Pearson linear correlation coefficients ρ for each relationship.
Fig 5.
Geographic visualization of the predicted changes in the scale-adjusted metrics DYi(t) between the years t = 2010 and t = 2020.
Each circle represents a city and the radius of the circle is proportional to ∣DYi(2020) − DYi(2010)∣. We color the circles according to the difference between DYi(2020) and DYi(2010): shades of azure indicate that we expect a decrease in the values of DYi(2020), whereas shades of red show the cities where we expect an increase in the values of DYi(2020). The labeled cities are the capitals of the twenty-seven Federal Units of Brazil (the Brazilian states and the federal district). The forecast for the values of DYi(2020) were obtained through the linear model of Eq 6, where the linear coefficient Ck were averaged over the two combinations of years 2000—1991 and 2010—2000. We note that the changes in DYi(t) appear spatially clustered for most indicators, forming regions where most cities are expected to increase or decrease the value of the scale-adjusted metric.