Universality of Zipf’s law

Zipf’s law is the most common statistical distribution displaying scaling behavior. Cities, populations or firms are just examples of this seemingly universal law. Although many different models have been proposed, no general theoretical explanation has been shown to exist for its universality. Here, we show that Zipf’s law is, in fact, an inevitable outcome of a very general class of stochastic systems. Borrowing concepts from Algorithmic Information Theory, our derivation is based on the properties of the symbolic sequence obtained through successive observations over a system with an ubounded number of possible states. Specifically, we assume that the complexity of the description of the system provided by the sequence of observations is the one expected for a system evolving to a stable state between order and disorder. This result is obtained from a small set of mild, physically relevant assumptions. The general nature of our derivation and its model-free basis would explain the ubiquity of such a law in real systems.

Figure 1

(Color online) An example of the behavior of the normalized entropy for a multiplicative stochastic process exhibiting Zipf’s law. Here, we use the model described in [23] using a $80\times 80$ lattice where each node is described by a density of population $\rho \left(i,j\right)$. The rules of the model are very simple: i) At every time step, each node loses a fraction $\alpha$ of its contents, which is distributed among its four nearest neighbors. ii) At time $t+1$ the local population is multiplied, with probability $p$, by a factor $p^{-1}$. Furthermore, with probability $1-p$, the population of a node is set to zero. Additionally, at each step a random number $\eta$ is added to every node. In this way, we avoid falling into an absorbing state $\rho =0$. Here we use $0<\eta <0.01$, $\alpha =1/4$, and $p=3/4$. This is an extremely simplified (and yet successful) model of urban population dynamics. A snapshot (for $t=500$) is shown in (a) where we can appreciate the wide range of local densities, following Zipf’s law (b). If we plot the evolution of the normalized entropy $\mu$ over time (averaged over ${10}^2$ replicas) we observe a convergence toward a stationary value $\mu \approx 0.65$.

Figure 2

Normalized entropies of five power-law distributed systems of different size as functions of the exponent. The curves display 5 different sizes. $n=500000$ black circles, $n=10000$ white circles, $n=10000$ up triangles, $n=1000$ squares and $n=100$ down triangles, respectively. The most interesting feature of the numerical computations is the sharp decay of the normalized entropy when the values of the exponent are close to 1, which implies that a wide range of normalized entropies are obtained by tuning the exponent of the power-law distribution around unity. Furthermore, we observe that the decay is sharper as the size of the system grows, concentrating an increasing range of relative entropies near the exponent 1 (gray area).