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The rank-size scaling law and entropy-maximizing principle

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  • 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
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    References listed on IDEAS

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    Cited by:

    1. Marcel Ausloos & Roy Cerqueti, 2016. "Religion-based urbanization process in Italy: statistical evidence from demographic and economic data," Quality & Quantity: International Journal of Methodology, Springer, vol. 50(4), pages 1539-1565, July.
    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, Open Access Journal, vol. 7(4), pages 1-20, April.
    3. 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.
    4. 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.
    5. 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.

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