IDEAS home Printed from https://ideas.repec.org/a/eee/phsmap/v391y2012i3p767-778.html
   My bibliography  Save this article

The rank-size scaling law and entropy-maximizing principle

Author

Listed:
  • Chen, Yanguang

Abstract

The rank-size regularity known as Zipf’s law is one of the scaling laws and is frequently observed in the natural living world and social institutions. Many scientists have tried to derive the rank-size scaling relation through entropy-maximizing methods, but they have not been entirely successful. By introducing a pivotal constraint condition, I present here a set of new derivations based on the self-similar hierarchy of cities. First, I derive a pair of exponent laws by postulating local entropy maximizing. From the two exponential laws follows a general hierarchical scaling law, which implies the general form of Zipf’s law. Second, I derive a special hierarchical scaling law with the exponent equal to 1 by postulating global entropy maximizing, and this implies the pure form of Zipf’s law. The rank-size scaling law has proven to be one of the special cases of the hierarchical scaling law, and the derivation suggests a certain scaling range with the first or the last data point as an outlier. The entropy maximization of social systems differs from the notion of entropy increase in thermodynamics. For urban systems, entropy maximizing suggests the greatest equilibrium between equity for parts/individuals and efficiency of the whole.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:phsmap:v:391:y:2012:i:3:p:767-778
    DOI: 10.1016/j.physa.2011.07.010
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378437111005577
    Download Restriction: Full text for ScienceDirect subscribers only. Journal offers the option of making the article available online on Science direct for a fee of $3,000

    File URL: https://libkey.io/10.1016/j.physa.2011.07.010?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Hernán D. Rozenfeld & Diego Rybski & Xavier Gabaix & Hernán 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 2205-2225, August.
    2. repec:cai:popine:popu_p1998_10n1_0240 is not listed on IDEAS
    3. Xavier Gabaix, 2009. "Power Laws in Economics and Finance," Annual Review of Economics, Annual Reviews, vol. 1(1), pages 255-294, May.
    4. Vining, Daniel Jr., 1977. "The Rank-Size rule in the absence of growth," Journal of Urban Economics, Elsevier, vol. 4(1), pages 15-29, January.
    5. Gangopadhyay, Kausik & Basu, B., 2009. "City size distributions for India and China," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(13), pages 2682-2688.
    6. R. Bussière & F. Snickars, 1970. "Derivation of the Negative Exponential Model by an Entropy Maximising Method," Environment and Planning A, , vol. 2(3), pages 295-301, September.
    7. Ioannides, Yannis M. & Overman, Henry G., 2003. "Zipf's law for cities: an empirical examination," Regional Science and Urban Economics, Elsevier, vol. 33(2), pages 127-137, March.
    8. Jia Shao & Plamen Ch. Ivanov & Boris Podobnik & H. Eugene Stanley, 2007. "Quantitative relations between corruption and economic factors," Papers 0705.0161, arXiv.org.
    9. Chen, Yanguang & Zhou, Yixing, 2008. "Scaling laws and indications of self-organized criticality in urban systems," Chaos, Solitons & Fractals, Elsevier, vol. 35(1), pages 85-98.
    10. 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.
    11. Xavier Gabaix, 1999. "Zipf's Law for Cities: An Explanation," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 114(3), pages 739-767.
    12. Xavier Gabaix, 1999. "Zipf's Law and the Growth of Cities," American Economic Review, American Economic Association, vol. 89(2), pages 129-132, May.
    13. A Anastassiadis, 1986. "New Derivations of the Rank-Size Rule Using Entropy-Maximising Methods," Environment and Planning B, , vol. 13(3), pages 319-334, September.
    14. F. Semboloni, 2008. "Hierarchy, cities size distribution and Zipf's law," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 63(3), pages 295-301, June.
    15. Krugman, Paul, 1996. "Confronting the Mystery of Urban Hierarchy," Journal of the Japanese and International Economies, Elsevier, vol. 10(4), pages 399-418, December.
    16. Alexander M. Petersen & Boris Podobnik & Davor Horvatic & H. Eugene Stanley, 2010. "Scale invariant properties of public debt growth," Papers 1002.2491, arXiv.org.
    17. Chen, Yanguang, 2009. "Analogies between urban hierarchies and river networks: Fractals, symmetry, and self-organized criticality," Chaos, Solitons & Fractals, Elsevier, vol. 40(4), pages 1766-1778.
    18. Moura, Newton J. & Ribeiro, Marcelo B., 2006. "Zipf law for Brazilian cities," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 367(C), pages 441-448.
    19. Peng, Guohua, 2010. "Zipf’s law for Chinese cities: Rolling sample regressions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(18), pages 3804-3813.
    20. Boris Podobnik & Davor Horvatic & Alexander M. Petersen & Branko Urov{s}evi'c & H. Eugene Stanley, 2010. "Bankruptcy risk model and empirical tests," Papers 1011.2670, arXiv.org.
    21. S. Narayan, 2009. "India," Chapters, in: Peter Draper & Philip Alves & Razeen Sally (ed.), The Political Economy of Trade Reform in Emerging Markets, chapter 7, Edward Elgar Publishing.
    22. 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;EDP Sciences, vol. 56(2), pages 157-166, March.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Jiejing Wang & Yanguang Chen, 2021. "Economic Transition and the Evolution of City-Size Distribution of China’s Urban System," Sustainability, MDPI, vol. 13(6), pages 1-15, March.
    2. Nan Dong & Xiaohuan Yang & Hongyan Cai & Liming Wang, 2015. "A Novel Method for Simulating Urban Population Potential Based on Urban Patches: A Case Study in Jiangsu Province, China," Sustainability, MDPI, vol. 7(4), pages 1-20, April.
    3. Santos, M.R.F. & Gomes, M.A.F., 2020. "A heuristic model for the scaling linguistic diversity-area," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 555(C).
    4. Chen, Yanguang, 2012. "Zipf’s law, 1/f noise, and fractal hierarchy," Chaos, Solitons & Fractals, Elsevier, vol. 45(1), pages 63-73.
    5. Jian Feng & Yanguang Chen, 2021. "Modeling Urban Growth and Socio-Spatial Dynamics of Hangzhou, China: 1964–2010," Sustainability, MDPI, vol. 13(2), pages 1-25, January.
    6. 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.
    7. Karima Kourtit, 2021. "City intelligence for enhancing urban performance value: a conceptual study on data decomposition in smart cities," Asia-Pacific Journal of Regional Science, Springer, vol. 5(1), pages 191-222, February.
    8. Fernando Rubiera-Morollón & Ignacio del Rosal & Alberto Díaz-Dapena, 2015. "Can large cities explain the aggregate movements of economies? Testing the ‘granular hypothesis’ for US counties," Letters in Spatial and Resource Sciences, Springer, vol. 8(2), pages 109-118, July.
    9. Chen, Yanguang, 2012. "The mathematical relationship between Zipf’s law and the hierarchical scaling law," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(11), pages 3285-3299.
    10. Chen, Yanguang, 2015. "The distance-decay function of geographical gravity model: Power law or exponential law?," Chaos, Solitons & Fractals, Elsevier, vol. 77(C), pages 174-189.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Chen, Yanguang, 2012. "The mathematical relationship between Zipf’s law and the hierarchical scaling law," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(11), pages 3285-3299.
    2. 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.
    3. Arshad, Sidra & Hu, Shougeng & Ashraf, Badar Nadeem, 2019. "Zipf’s law, the coherence of the urban system and city size distribution: Evidence from Pakistan," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 513(C), pages 87-103.
    4. Valente J. Matlaba & Mark J. Holmes & Philip McCann & Jacques Poot, 2013. "A Century Of The Evolution Of The Urban System In Brazil," Review of Urban & Regional Development Studies, Wiley Blackwell, vol. 25(3), pages 129-151, November.
    5. Xinyue Ye & Yichun Xie, 2012. "Re-examination of Zipf’s law and urban dynamic in China: a regional approach," The Annals of Regional Science, Springer;Western Regional Science Association, vol. 49(1), pages 135-156, August.
    6. Sarabia, José María & Prieto, Faustino, 2009. "The Pareto-positive stable distribution: A new descriptive model for city size data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(19), pages 4179-4191.
    7. Duranton, Gilles & Puga, Diego, 2014. "The Growth of Cities," Handbook of Economic Growth, in: Philippe Aghion & Steven Durlauf (ed.), Handbook of Economic Growth, edition 1, volume 2, chapter 5, pages 781-853, Elsevier.
    8. Xavier Gabaix, 2009. "Power Laws in Economics and Finance," Annual Review of Economics, Annual Reviews, vol. 1(1), pages 255-294, May.
    9. Luckstead, Jeff & Devadoss, Stephen, 2014. "A nonparametric analysis of the growth process of Indian cities," Economics Letters, Elsevier, vol. 124(3), pages 516-519.
    10. Angelina Hackmann & Torben Klarl, 2020. "The evolution of Zipf's Law for U.S. cities," Papers in Regional Science, Wiley Blackwell, vol. 99(3), pages 841-852, June.
    11. Luckstead, Jeff & Devadoss, Stephen, 2014. "A comparison of city size distributions for China and India from 1950 to 2010," Economics Letters, Elsevier, vol. 124(2), pages 290-295.
    12. Rafael González-Val, 2019. "US city-size distribution and space," Spatial Economic Analysis, Taylor & Francis Journals, vol. 14(3), pages 283-300, July.
    13. Ronan Lyons & Elisa Maria Tirindelli, 2022. "The Rise & Fall of Urban Concentration in Britain: Zipf, Gibrat and Gini across two centuries," Trinity Economics Papers tep0522, Trinity College Dublin, Department of Economics.
    14. Chen, Yanguang, 2016. "The evolution of Zipf’s law indicative of city development," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 443(C), pages 555-567.
    15. Devadoss, Stephen & Luckstead, Jeff, 2016. "Size distribution of U.S. lower tail cities," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 444(C), pages 158-162.
    16. Devadoss, Stephen & Luckstead, Jeff & Danforth, Diana & Akhundjanov, Sherzod, 2016. "The power law distribution for lower tail cities in India," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 442(C), pages 193-196.
    17. Duranton, Gilles, 2002. "City size distributions as a consequence of the growth process," LSE Research Online Documents on Economics 20065, London School of Economics and Political Science, LSE Library.
    18. Sen, Hu & Chunxia, Yang & Xueshuai, Zhu & Zhilai, Zheng & Ya, Cao, 2015. "Distributions of region size and GDP and their relation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 430(C), pages 46-56.
    19. Rafael González‐Val, 2019. "Historical urban growth in Europe (1300–1800)," Papers in Regional Science, Wiley Blackwell, vol. 98(2), pages 1115-1136, April.
    20. Nikolay K. Vitanov & Marcel Ausloos, 2015. "Test of two hypotheses explaining the size of populations in a system of cities," Journal of Applied Statistics, Taylor & Francis Journals, vol. 42(12), pages 2686-2693, December.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:phsmap:v:391:y:2012:i:3:p:767-778. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.