Truncated lognormal distributions and scaling in the size of naturally defined population clusters

Using population data of high spatial resolution for a region in the south of Europe, we define cities by aggregating individuals to form connected clusters. The resulting cluster-population distributions show a smooth decreasing behavior covering six orders of magnitude. We perform a detailed study of the distributions, using state-of-the-art statistical tools. By means of scaling analysis we rule out the existence of a power-law regime in the low-population range. The logarithmic-coefficient-of-variation test allows us to establish that the power-law tail for high population, characteristic of Zipf's law, has a rather limited range of applicability. Instead, lognormal fits describe the population distributions in a range covering from a few dozen individuals to more than $1\times {10}^6$ (which corresponds to the population of the largest cluster).

Figure 1

(a) Whole data set: Latitude and longitude (dots) of the 7 586 888 inhabitants of Catalonia on January 1, 2013. (b) Zoom of the data around the Barcelona zone. Notice that we are representing the coordinates of the residence place of each individual, so the high resolution of the data becomes apparent in this plot. The contour line delimits the municipality of Barcelona.

Figure 2

(a) Empirical probability mass functions $f\left(h\right)$ of number of inhabitants $h$ per cell, for several values of cell width $\ell$, in the grid approach, using the grid-in-km procedure. (b) Corresponding empirical probability densities of population density per cell. Observe the enormous variability, from less than 0.1 inhabitant per ${\mathrm{km}}^2$ to more than ${10}^5$. One can conclude that the concept of average density has a rather limited value. Unpopulated cells are not considered.

Figure 3

Part of the cluster associated to the Barcelona urban area (in red diamonds), for cell width $\ell =0.{002}^{\circ }$. The cluster, which is the largest one for this $\ell$, includes part of other municipalities in addition to Barcelona, but there are parts of the Barcelona municipality not included in the cluster. Part of the second-largest cluster (in green crosses) is also shown. Smaller clusters are not highlighted.

Figure 4

Top ten clusters following the ball approach. The maps show the result of the percolation on the position of the portals (households), at different distance thresholds. At 100 m the red cluster shows a high overlap with Barcelona.

Figure 5

Empirical probability mass functions $f\left(s\right)$ of cluster size $s$ (in number of inhabitants) for several values of cell width $\ell$, using square grid in longitude-latitude (grid-in-degrees, bottom curves), square grid in distance (grid-in-km, $y\mathrm{axis}$ multiplied by a factor 10, for clarity sake), and ball approach ($y$ axis multiplied by a factor 100). Results for the usual approach based on municipalities are also included, for the sake of comparison. Continuous lines are truncated lognormal fits.

Figure 6

(a) Empirical probability mass functions $f\left(s\right)$ rescaled by their mean value $\langle s\rangle$ for some values of $\ell$ in the grid-in-degrees, grid-in-km, and ball approaches. The curve is the fit corresponding to the ball approach, as shown in the table. (b) Same probability mass functions using the nontrivial rescaling given by Eq. (2) (see also Ref. [40]). The scalings in (a) and (b) can be considered as visually equivalent, which rules out the existence of a power law with $\alpha >1$ for small $s$. This could be made more quantitative using the collapse algorithm of Ref. [41].

Figure 7

Empirical value of the “residual logarithmic coefficient of variation” $c_v$ of the cluster population as a function of the number of residual points in the tail of the distribution, $n_{cv}$. Several examples are shown, corresponding to our three approaches. The solid lines correspond to the simulated 5th and 95th percentiles of the distribution of $c_v$ under the power-law null hypothesis. In all cases the first crossing below the fifth percentile takes place for $n_{cv}$ around 100, which corresponds to $n_{\text{PL}}$ in Table 1. Note that the horizontal axis is reversed, to display the tail on the right side.

Figure 8

Empirical probability density of rescaled cluster sizes ($s/\langle s\rangle$) aggregated for diverse values of $\ell$ (100, 200, 400, and 800 m), in the ball approach. The axes have been transformed to evaluate the goodness of the lognormal fit (Table 1, continuous line). Note that both axes are doubly logarithmic. Data are the same as those in Fig. 6 for the ball approach.

Figure 9

Comparison of the original distribution of cluster sizes $f\left(s\right)$ with the distribution of cluster sizes obtained when the values $h$ of the population of each cell are randomized between occupied cells. The less interesting case in which the values of $h$ are randomized between all cells (occupied and unoccupied) is also shown. The example shown corresponds to the grid-in-km approach, with $\ell =222$ m. Data in the $y$ axis multiplied by integer powers of 10, for clarity sake.

Figure 10

Same results as in Fig. 5, represented in terms of the rank-size relation. The size (vertical axis) has been multiplied by integer powers of 10, for the sake of clarity. Continuous lines are the truncated lognormal fits translated into the rank-size representation (the mathematical expression is not lognormal anymore but involves the inverse of the error function).