Dynamical approach to Zipf's law

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 present 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.