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.
Highlights
Zipf’s law [1,2] is an empirical scaling relation that connects the sizes of a set of objects with their ranking when sorted according to the size itself
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 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
We argue that Zipfian dynamics may emerge as a consequence of the effect of correlations induced by grammar and semantic rules, as can be seen considering two different systems in which such rules are adopted to a different extent [43,44]
Summary
Zipf’s law [1,2] is an empirical scaling relation that connects the sizes of a set of objects with their ranking when sorted according to the size itself. There are currently numerous approaches to explain Zipf’s law based on different mechanisms including multiplicative processes [9,10], adjacent possible framework [11,12], sample space reducing processes [13], and information theory arguments [14,15]. While all these models give insights on the upset of Zipf’s scaling, they fail in providing a general explanation of the phenomenon.
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