Zipf's law in city size from a resource utilization model

We study a resource utilization scenario characterized by intrinsic fitness. To describe the growth and organization of different cities, we consider a model for resource utilization where many restaurants compete, as in a game, to attract customers using an iterative learning process. Results for the case of restaurants with uniform fitness are reported. When fitness is uniformly distributed, it gives rise to a Zipf law for the number of customers. We perform an exact calculation for the utilization fraction for the case when choices are made independent of fitness. A variant of the model is also introduced where the fitness can be treated as an ability to stay in the business. When a restaurant loses customers, its fitness is replaced by a random fitness. The steady state fitness distribution is characterized by a power law, while the distribution of the number of customers still follows the Zipf law, implying the robustness of the model. Our model serves as a paradigm for the emergence of Zipf law in city size distribution.

Figure 1

The probability density $P\left(s\right)$ for a fraction of restaurants with $s$ agents. The data are shown for different system sizes $N=2^8,2^9,2^{10},2^{11},2^{12},2^{13},2^{14},2^{15}$. The power law exponent is compared with $s^{-2}$.

Figure 2

The probability density $P\left(s\right)$ for a fraction of restaurants with $s$ agents, for different distributions $\Pi \left(p\right)=(1+\delta ){(1-p)}^{\delta }$, with $\delta =-0.5,0,1.0$. The power law exponents agree with $\nu =2+\delta$. The data are shown for $N=2^{13}$.

Figure 3

Utilization fraction $f$ in the model (a) with fixed fitness $p$. The data are shown for $N=1024$. The dotted line gives the analytical estimate. (b) $f$ for the case of uniformly distributed $p$, i.e., for $\Pi \left(p\right)=1$: $p_i=1-i/N$ (red squares) and uniformly random (blue circles) for various system sizes $N$. The inset shows the variation of $f$ with $1/lnN$. Error bars are also shown.

Figure 4

(a) The distributions $D\left(p\right)$ of $p$ for cases I (without cutoff) and II (with cutoff). The inset shows $D\left(p\right)$ vs $1-p$, and the solid line is a guide to $\zeta =0.5$. The data are shown for $N={10}^4$. (b) The distributions $P\left(s\right)$ of $s$ for case I (without cutoff) at different times $t={10}^7,2\times {10}^7,4\times {10}^7$ for $N={10}^4$. The solid line is a guide to $\nu =2$. (c) The distributions $P\left(s\right)$ of $s$ for case II (with cutoff) at different times $t={10}^7,2\times {10}^7,4\times {10}^7$ for $N={10}^4$. The solid line is a guide to $\nu =2$.

Figure 5

Plot of size $s$ with rank $k$ of a city for different countries. The two straight lines are, respectively, $s\propto k^{-1}$ and $s\propto k^{-2/3}$, guides to the approximate extremes of the power law behavior of the data sets. The actual exponents of the power law fits are given in Table. 1.