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The mathematical relationship between Zipf’s law and the hierarchical scaling law

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  • Chen, Yanguang
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    Abstract

    The empirical studies of city-size distribution show that Zipf’s law and the hierarchical scaling law are linked in many ways. The rank-size scaling and hierarchical scaling seem to be two different sides of the same coin, but their relationship has never been revealed by strict mathematical proof. In this paper, the Zipf’s distribution of cities is abstracted as a q-sequence. Based on this sequence, a self-similar hierarchy consisting of many levels is defined and the numbers of cities in different levels form a geometric sequence. An exponential distribution of the average size of cities is derived from the hierarchy. Thus we have two exponential functions, from which follows a hierarchical scaling equation. The results can be statistically verified by simple mathematical experiments and observational data of cities. A theoretical foundation is then laid for the conversion from Zipf’s law to the hierarchical scaling law, and the latter can show more information about city development than the former. Moreover, the self-similar hierarchy provides a new perspective for studying networks of cities as complex systems. A series of mathematical rules applied to cities such as the allometric growth law, the 2n principle and Pareto’s law can be associated with one another by the hierarchical organization.

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    File URL: http://www.sciencedirect.com/science/article/pii/S0378437111009678
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    Bibliographic Info

    Article provided by Elsevier in its journal Physica A: Statistical Mechanics and its Applications.

    Volume (Year): 391 (2012)
    Issue (Month): 11 ()
    Pages: 3285-3299

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    Handle: RePEc:eee:phsmap:v:391:y:2012:i:11:p:3285-3299

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    Web page: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/

    Related research

    Keywords: Zipf’s law; Scaling law; Hierarchy; Cascade structure; Fractal; 1/fβ noise; Allometric growth; Rank-size distribution of cities;

    References

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    1. Yanguang Chen & Yixing Zhou, 2003. "The rank-size rule and fractal hierarchies of cities: mathematical models and empirical analyses," Environment and Planning B: Planning and Design, Pion Ltd, London, vol. 30(6), pages 799-818, November.
    2. Jia Shao & Plamen Ch. Ivanov & Boris Podobnik & H. Eugene Stanley, 2007. "Quantitative relations between corruption and economic factors," The European Physical Journal B - Condensed Matter and Complex Systems, Springer, vol. 56(2), pages 157-166, 03.
    3. Jia Shao & Plamen Ch. Ivanov & Boris Podobnik & H. Eugene Stanley, 2007. "Quantitative relations between corruption and economic factors," Papers 0705.0161, arXiv.org.
    4. Xavier Gabaix, 2008. "Power Laws in Economics and Finance," NBER Working Papers 14299, National Bureau of Economic Research, Inc.
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    6. Geoffrey B. West & James H. Brown & Brian J. Enquist, 1999. "The Fourth Dimension of Life: Fractal Geometry and Allometric Scaling of Organisms," Working Papers 99-07-047, Santa Fe Institute.
    7. Chen, Yanguang, 2012. "The rank-size scaling law and entropy-maximizing principle," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(3), pages 767-778.
    8. Xavier Gabaix & Yannis M. Ioannides, 2003. "The Evolution of City Size Distributions," Discussion Papers Series, Department of Economics, Tufts University 0310, Department of Economics, Tufts University.
    9. Xavier Gabaix, 1999. "Zipf'S Law For Cities: An Explanation," The Quarterly Journal of Economics, MIT Press, vol. 114(3), pages 739-767, August.
    10. Boris Podobnik & Davor Horvatic & Alexander M. Petersen & Branko Uro\v{s}evi\'c & H. Eugene Stanley, 2010. "Bankruptcy risk model and empirical tests," Papers 1011.2670, arXiv.org.
    11. Hern�n D. Rozenfeld & Diego Rybski & Xavier Gabaix & Hern�n A. Makse, 2011. "The Area and Population of Cities: New Insights from a Different Perspective on Cities," American Economic Review, American Economic Association, vol. 101(5), pages 2205-25, August.
    12. Stanley, Michael H. R. & Buldyrev, Sergey V. & Havlin, Shlomo & Mantegna, Rosario N. & Salinger, Michael A. & Eugene Stanley, H., 1995. "Zipf plots and the size distribution of firms," Economics Letters, Elsevier, vol. 49(4), pages 453-457, October.
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    Cited by:
    1. Chen, Yanguang & Wang, Jiejing, 2014. "Recursive subdivision of urban space and Zipf’s law," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 395(C), pages 392-404.

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