Dynamical approach to Zipf's law

TL;DR

This paper introduces a dynamical framework explaining how systems evolve towards or temporarily exhibit Zipf's law, emphasizing the importance of parameter coherence and applying it to various phenomena like earthquakes and urban populations.

Contribution

It provides a novel dynamical approach to understanding Zipf's law, identifying coherence conditions, and applying the framework to diverse systems for predictive insights.

Findings

Earthquakes can only evolve incoherently, leading to spurious Zipf's law.

Coherent Zipfian dynamics are non-additive, explaining differences between US and world cities.

The framework helps estimate maximum urban populations and assess earthquake magnitudes.

Abstract

The rank-size plots of a large number of different physical and socio-economic systems are usually said to follow Zipf's law, but a unique framework for the comprehension of this ubiquitous scaling law is still lacking. Here we show that a dynamical approach is crucial: during their evolution, some systems are attracted towards Zipf's law, while others presents Zipf's law only temporarily and, therefore, spuriously. A truly Zipfian dynamics is characterized by a dynamical constraint, or coherence, among the parameters of the generating PDF, and the number of elements in the system. A clear-cut example of such coherence is natural language. Our framework allows us to derive some quantitative results that go well beyond the usual Zipf's law: i) earthquakes can evolve only incoherently and thus show Zipf's law spuriously; this allows an assessment of the largest possible magnitude of an earthquake occurring in a geographical region. ii) We prove that Zipfian dynamics are not additive, explaining analytically why US cities evolve coherently, while world cities do not. iii) Our concept of coherence can be used for model selection, for example, the Yule-Simon process can describe the dynamics of world countries' GDP. iv) World cities present spurious Zipf's law and we use this property for estimating the maximal population of an urban agglomeration.