Zipf's and Taylor's Laws

TL;DR

This paper demonstrates that Zipf's and Taylor's laws naturally emerge from simple stochastic models incorporating environmental variability or individual correlations, explaining their widespread empirical presence.

Contribution

It provides a unified theoretical framework showing how Zipf's and Taylor's laws can arise from basic stochastic processes without fine tuning.

Findings

Zipf's law and Taylor's law emerge from models with environmental variability or correlations.

The stationary distribution of the processes follows Zipf's law.

Conditional variance of population increments scales quadratically with population size.

Abstract

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 individuals 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.