Figure 1.
Annual number of homicides in cities of Colombia, Mexico and Brazil versus population size (2007).
Large cities are defined in terms of metropolitan areas which are aggregations of municipalities (red circles) while non-metropolitan municipalities are shown separately (green squares). The solid blue line fits only the scaling of homicides for metropolitan areas. Large variations, especially among the smaller population units, and the fact that many municipalities have (not shown) prevent a direct scaling analysis. However, it is possible to analyze the data consistently through the estimation of conditional probabilities.
Figure 2.
Cumulative normalized distributions of homicides in Colombia, Mexico and Brazil (2007) are well described by power-law distributions.
Here we plot not the density function but the complementary cumulative distribution to attenuate the tail fluctuations and ease visual interpretation. Best fits (dashed red line) of the form were estimated using the procedure in [17] to the density function (see Methods section). Standard errors are reported in parenthesis. The solid blue line shows the minimum value of
for which a power-law fit holds. While the distribution of total homicides is scale invariant, this is the result of tracing more predictable conditional distributions for each city over a broad distribution of city sizes (see text).
Figure 3.
Normalized frequency histograms of the logarithm of city population for varying number of observed homicides Y.
Each column corresponds to a different country and each row, from top to bottom, corresponds to the values homicides per year. A log-normal distribution (notice the x-axis is expressed in terms of
) is shown as a solid red line, with parameters obtained via maximum likelihood estimation.
Figure 4.
Collapsed histograms of P(N | Y) across values of Y in 2007.
Log-normal probability density functions for the three nations are shown as solid red lines. This shows that power-law distributions describing total homicides in the urban systems have in fact more predictable statistics when conditioned on city population size.
Figure 5.
Estimates of (via maximum likelihood) for different values of
, for Colombia, Mexico, and Brazil.
A different curve was constructed for every year of the analysis (see Methods). The plots show the average over several years. Error bars represent one standard deviation intervals (67% confidence level). The plots show no clear systematic -dependence of
. This suggests, in turn, that each country has a characteristic variance of its indicators conditioned on other urban quantities. In this respect, it is interesting to note the similarities between Colombia and Brazil.
Figure 6.
Estimates of (via maximum likelihood) for different values of
, for Colombia, Mexico, and Brazil.
A different curve was constructed for every year of the analysis, and the points plotted are the averages over several years. The error bars represent one standard deviation intervals about the mean. Plots show a logarithmic dependence on , from which a scaling relationship emerges in terms of expectation values (see text). Best fits were obtained using a Levenberg-Marquardt algorithm, weighting every point by its error, see Methods.
Figure 7.
Q-Q plot of the standardized log-variables of the populations of the cities for several values of Y.
This shows that a log-normal distribution is an excellent description of , for the three nations, notwithstanding a number of small exceptions at the extremes (a perfect straight line in the dots would correspond to an exact normal distribution of log-populations).
Figure 8.
Cumulative normalized distributions of city populations in Colombia, Mexico and Brazil (2007) fitted with pure-power-law distributions.
Best fits (dashed red line) of the form were estimated using the procedure in [17] to the density function. Not disregarding the long-held debate about the city-size distribution, we believe the fit to a power-law distribution stands as a first approximation consistent with our proposed statistical framework.