Zipf's and Taylor's laws

Zipf's law states that the frequency of an observation with a given value is inversely proportional to the square of that value; Taylor's law, instead, describes the scaling between fluctuations in the size of a population and its mean. Empirical evidence of the validity of these laws has been found in many and diverse domains. Despite the numerous models proposed to explain the presence of Zipf's law, there is no consensus on how it originates from a microscopic process of individual dynamics without fine-tuning. Here we show that Zipf's law and Taylor's law can emerge from a general class of stochastic processes at the individual level, which incorporate one of two features: environmental variability, i.e., fluctuations of parameters, or correlations, i.e., dependence between individuals. Under these assumptions, we show numerically and with theoretical arguments that the conditional variance of the population increments scales as the square of the population, and that the corresponding stationary distribution of the processes follows Zipf's law.

Figure 1

(a) Zipf's law, data. Probability density function $P\left(n\right)$ ($y$ axis) vs size $n$ ($x$ axis) for cities (circles) and counties (triangles) within the United States. The solid lines are guides for the eye corresponding to $P\left(n\right)\propto n^{-2}$. Data on the population of cities are obtained from the geonames dataset [4]. County-level data are obtained from the U.S. Census Bureau [5]. The distribution for U.S. cities has been shifted by a factor of 10 along the $y$ axis for clarity. (b) Taylor's law, data. The variance of population in year $t+1$ conditioned to the population in year $t$ ($y$ axis) vs the average population in year $t+1$ conditioned to the population in year $t$ ($x$ axis) for cities (circles) and counties (triangles) in the United States during the period 1970–2010. The vertical dashed line denotes the crossover city size $n_c$ at which Taylor's exponent defined in Eq. (2) transitions from $\alpha =1/2$ to 1 (the best-fit exponents in the regime $n>n_c$ are $\alpha =0.92\pm 0.01$ for cities and $\alpha =0.97\pm 0.03$ for counties). As shown in (a), $n_c$ corresponds to the crossover city size at which the distributions start following Zipf's law. The data for U.S. cities have been shifted by a factor of 10 along the $y$ axis for clarity.

Figure 2

(a) Zipf's law, models. Stationary distributions of group sizes $P\left(n\right)$ for modified branching processes with environmental variability (circles) and correlated individuals (triangles). See the main text for details on the numerical simulations and the parameter values. The distributions have been shifted by a factor of 10 along the $y$ axis for clarity. The vertical dashed line denotes the crossover city size $n_c$ at which the distributions start following Zipf's law: $n_c={10}^4$ for both models. (b) Taylor's law, models. The variance of population change in the time interval $[t,t+1]$ conditioned to the population at time step $t$ ($y$ axis) vs the average population change in the time interval $[t,t+1]$ conditioned to the population at time step $t$ ($x$ axis) for branching processes with environmental variability (circles) and correlated variables (triangles). Simulations and parameter values are the same as in (a). We observe a transition of Taylor's exponent from $\alpha =1/2$ for $n<n_c$ to $\alpha =1$ for $n>n_c$. The dashed lines correspond to the analytical results of Eqs. (3) and (5). Curves have been shifted by a factor of 10 along the $y$ axis for clarity.

Figure 3

Numerical simulations support the ansatz of Eq. (4) for processes with environmental variability [panels (a) and (b)] and with correlated individuals [panels (c) and (d)]. (a) Distribution of the fluctuations of the population increments in consecutive time steps, $\mathrm{\Delta }n=n_{t+1}-n_t$, for a process with environmental variability where $P\left(x\right|\lambda )=\text{Poiss}(\lambda )$ and $G\left(\lambda \right)=\mathrm{\Gamma }\left(\lambda \right|\kappa ,\nu )$ with $\kappa =19.8,\nu =0.05$. Different curves correspond to different values of populations $n_t=n$. (b) The curves in panel (a) collapse on the same distribution, a standard normal distribution (dashed line), when the population increments are shifted by removing the mean and rescaled by the square root of the variance derived in Eq. (3). (c) Distribution of the fluctuations of the population increments in consecutive time steps for a process with correlated individuals, where the $n$ Poisson random variables have covariance matrix $\text{Cov}(x_i,x_j)={\delta }_{ij}\lambda +(1-{\delta }_{ij})\rho \lambda$ with $\lambda =0.999,\rho =0.1$. (d) The curves in panel (c) collapse on the standard normal distribution (dashed line) when the population increments are shifted by removing the mean and rescaled by the square root of the variance derived in Eq. (5).