Figure 1

Top: quantiles from 0.1 (red) to 0.9 (blue) (each of 0.1) for the distribution of the scaled growth rates $p_{\Xi }\left(\xi \right)$ (see text) as a function of the population $x$ (the median is shown in black). Bottom: cumulative distribution of the scaled growth rates $P_{\Xi }\left(\xi \right)$ for each of Spain's provinces.Reuse & Permissions

Figure 2

Variance $s^2/x$ vs population $x$ (see text), each dot representing one municipality. Red lines: quantiles from 0.1 to 0.9 (each of 0.1). Solid black line: fit to the median value following Eq. (13). Dashed black lines: extrapolation of the finite-size term [Eq. (14)], and extrapolation of the multiplicative term [Eq. (15)].Reuse & Permissions

Figure 3

$\dot{x}/x$ vs population (see text) for 12 Spanish provinces (each dot represents one municipality per year). The mean $\langle \dot{x}/x\rangle$, and the limits of the standard deviation as functions of the population, are drawn with black lines. The fit to an expression of the type represented by Eq. (17) is displayed using red lines. The results correspond, respectively, to: Asturias, Almería, Cáceres, Cuenca, Baleares, Lleida, Badajoz, Ávila, Guipúzcoa, Castellón, Valladolid, and Guadalajara.Reuse & Permissions

Figure 4

Top: cumulative distribution $P\left(x\right)$ at three different times, pertaining to a simulated $q$-metric diffusion process with $q=1/2$. We start with the green-dotted distribution. Things evolve and one reaches the red-dashed distribution. Finally, one arrives to the blue-solid configuration. This last distribution is compared with the cumulative distribution of Salamanca's towns' population (blue dots). Inset: variance of the relative growth (in years${}^{-1}$) vs log population (dots), confirming the $\sqrt{x}$ dependence for $q=1/2$ dynamics (red line) (see text). Bottom: geometric diffusion $q=1$ compared with the population distribution of Florida (US). Inset: same as top panel's inset, confirming this time the size independence of our results and thus the proportional nature of the underlying dynamics (see text).Reuse & Permissions

Figure 5

Equilibrium rank distributions for $q$-metric diffusion processes with maximum size constraints for (from left to right), $q=0.5$, $1,$ $1.5,$ and $n=10000$. Insets: evolution of the density distribution of the diffusion process, starting with a $\delta$ configuration in $x$ and passing through the red and green configurations, until equilibrium is attained for the black distribution (see text).Reuse & Permissions

Figure 6

Evolution of the density distribution from an initial $\delta$ state (gray solid bar), passing through intermediate nonequilibrium configurations red-green-blue, until reaching equilibrium (solid black line). We deal with $N$-constrained, $q$-metric diffusion processes for (from top to bottom), $q=0$, 1, and 2. The smooth black lines follow Eq. (73) in $u$ space, for each $q$ value.Reuse & Permissions

Figure 7

An example of population restriction arising out of geographical reasons, as a mountainous or island landscape. Left: d'Aosta valley (Italy) [41]. Right: Marshall islands [42].Reuse & Permissions

Figure 8

SFIG-in-a-box examples, from top to bottom: rank distribution of Huelva province, Spain (2010), D'Aosta Valley, Italy (2000 in red and 2010 in blue), and Marshall Islands (2010), compared with distribution (68), plotted as a continuous curve. In the insets, the relative growth $\dot{u}$ in years${}^{-1}$ vs the log population $u$.Reuse & Permissions

Figure 9

Empirical rank distribution of SFIGs under total population constraint. These examples refer, respectively, to: Navarra, Lleida, Zaragoza, Girona, Granada, Almería, Alicante, and Vizcaya (dots) compared with Eq. (75) (continuous lines). Insets: relative growth $\dot{x}/x$ in years${}^{-1}$ vs log population $\ln \left(x\right)$, fitted to Eq. (17) with the same value of $q$ used for the rank distribution.Reuse & Permissions