Zipf’s law for atlas models

2020 ◽  
Vol 57 (4) ◽  
pp. 1276-1297
Author(s):  
Ricardo T. Fernholz ◽  
Robert Fernholz
Keyword(s):  
Pareto Distribution ◽  
Household Wealth ◽  
City Size ◽  
Zipf’S Law ◽  
Mathematical Explanation ◽  
Zipf's Law ◽  
First Order ◽  
Pareto Distributions ◽  
Straight Line ◽  
Atlas Model

AbstractA set of data with positive values follows a Pareto distribution if the log–log plot of value versus rank is approximately a straight line. A Pareto distribution satisfies Zipf’s law if the log–log plot has a slope of $-1$. Since many types of ranked data follow Zipf’s law, it is considered a form of universality. We propose a mathematical explanation for this phenomenon based on Atlas models and first-order models, systems of strictly positive continuous semimartingales with parameters that depend only on rank. We show that the stationary distribution of an Atlas model will follow Zipf’s law if and only if two natural conditions, conservation and completeness, are satisfied. Since Atlas models and first-order models can be constructed to approximate systems of time-dependent rank-based data, our results can explain the universality of Zipf’s law for such systems. However, ranked data generated by other means may follow non-Zipfian Pareto distributions. Hence, our results explain why Zipf’s law holds for word frequency, firm size, household wealth, and city size, while it does not hold for earthquake magnitude, cumulative book sales, and the intensity of wars, all of which follow non-Zipfian Pareto distributions.

A Dynamic Generalization of Zipf's Rank-Size Rule

1982 ◽  
Vol 14 (11) ◽  
pp. 1449-1467 ◽  
Author(s):  
B Roehner ◽  
K E Wiese
Keyword(s):  
Stochastic Model ◽  
Size Distribution ◽  
Urban Growth ◽  
Simple Form ◽  
Deterministic Model ◽  
City Size ◽  
Zipf’S Law ◽  
Zipf's Law ◽  
City Size Distribution ◽  
Zero Growth

A dynamic deterministic model of urban growth is proposed, which in its most simple form yields Zipf's law for city-size distribution, and in its general form may account for distributions that deviate strongly from Zipf's law. The qualitative consequences of the model are examined, and a corresponding stochastic model is introduced, which permits, in particular, the study of zero-growth situations.

scholarly journals Scaling in words on Twitter

2019 ◽  
Vol 6 (10) ◽  
pp. 190027 ◽  
Author(s):  
Eszter Bokányi ◽  
Dániel Kondor ◽  
Gábor Vattay
Keyword(s):  
The United States ◽  
City Size ◽  
Zipf’S Law ◽  
Zipf's Law ◽  
Scaling Properties ◽  
Scaling Relationship ◽  
City Population ◽  
Urban Scaling ◽  
Core Vocabulary ◽  
Relationship Of

Scaling properties of language are a useful tool for understanding generative processes in texts. We investigate the scaling relations in citywise Twitter corpora coming from the metropolitan and micropolitan statistical areas of the United States. We observe a slightly superlinear urban scaling with the city population for the total volume of the tweets and words created in a city. We then find that a certain core vocabulary follows the scaling relationship of that of the bulk text, but most words are sensitive to city size, exhibiting a super- or a sublinear urban scaling. For both regimes, we can offer a plausible explanation based on the meaning of the words. We also show that the parameters for Zipf’s Law and Heaps' Law differ on Twitter from that of other texts, and that the exponent of Zipf’s Law changes with city size.

scholarly journals Demography and the emergence of universal patterns in urban systems

2020 ◽  
Vol 11 (1) ◽  
Author(s):  
Luís M. A. Bettencourt ◽  
Daniel Zünd
Keyword(s):  
Urban Areas ◽  
Small Towns ◽  
City Size ◽  
Zipf’S Law ◽  
Urban Systems ◽  
Size Distributions ◽  
Vital Rates ◽  
Zipf's Law ◽  
Large Cities ◽  
Population Sizes

Abstract Urban areas exist in a wide variety of population sizes, from small towns to huge megacities. No proposed form for the statistical distribution of city sizes has received more attention than Zipf’s law, a Pareto distribution with power law exponent equal to one. However, this distribution is typically violated by empirical evidence for small and large cities. Moreover, no theory presently exists to derive city size distributions from fundamental demographic choices while also explaining consistent variations. Here we develop a comprehensive framework based on demography to show how the structure of migration flows between cities, together with the differential magnitude of their vital rates, determine a variety of city size distributions. This approach provides a powerful mathematical methodology for deriving Zipf’s law as well as other size distributions under specific conditions, and to resolve puzzles associated with their deviations in terms of concepts of choice, symmetry, information, and selection.

scholarly journals Investigation of urban regularities for Croatia in the period from 1857 to 2011

2020 ◽  
Vol 71 (4) ◽  
pp. 307-330
Author(s):  
Hrvoje Jošić ◽  
Berislav Žmuk
Keyword(s):  
Size Distribution ◽  
Unit Root ◽  
Urban Economics ◽  
Unit Roots ◽  
City Size ◽  
Zipf’S Law ◽  
Urban Hierarchy ◽  
Zipf's Law ◽  
Gibrat’S Law ◽  
Gibrat's Law

Two main regularities in the field of urban economics are Zipf’s law and Gibrat’s law. Zipf’s law states that distribution of largest cities should obey the Pareto rank-size distribution while Gibrat’s law states that proportionate growth of cities is independent of its size. These two laws are interconnected and therefore are often considered together. The objective of this paper is the investigation of urban regularities for Croatia in the period from 1857 to 2011. In order to estimate and evaluate the structure of Croatian urban hierarchy, Pareto or Zipf’s coefficients are calculated. The results have shown that the coefficient values for the largest settlements in different years are close to one, indicating that the Croatian urban hierarchy system follows the rank-size distribution and therefore obeys Zipf's law. The independence of city growth regarding the city size is tested using penal unit roots. Results for Gibrat's law testing using panel unit root tests have shown that there is a presence of unit root in growth of settlements therefore leading to the acceptance of Gibrat’s law.

scholarly journals Does Zipf’s law hold for Polish cities?

2016 ◽  
Vol 20 (4) ◽  
pp. 5-10 ◽  
Author(s):  
Andrzej Cieślik ◽  
Jan Teresiński
Keyword(s):  
Size Distribution ◽  
Empirical Analysis ◽  
City Size ◽  
Zipf’S Law ◽  
Empirical Literature ◽  
Zipf's Law ◽  
First Approximation ◽  
City Size Distribution ◽  
The City ◽  
Equal Population

Abstract In this paper we study Zipf’s law, which postulates that the product of a city’s population and its rank (the number of cities with a larger or equal population) is constant for every city in a given region. We show that the empirical literature indicates that the law may not always hold, although its general form, the rank-size rule, could be a good first approximation of city size distribution. We perform our own empirical analysis of the distribution of the population of Polish cities on the largest possible sample to find that Zipf’s law is rejected for Poland as the city sizes are less evenly distributed than it predicts.

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