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The rank-size rule and fractal hierarchies of cities: mathematical models and empirical analyses

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  • Yanguang Chen
  • Yixing Zhou

Abstract

This paper contributes to the demonstration that the self-similar city hierarchies with cascade structure can be modeled with a pair of scaling laws reflecting the recursive process of urban systems. First we transform the Beckmann's model on city hierarchies and generalize Davis's 2 n -rule to an r n -rule on the size - number relationship of cities ( r > 1), and then reduce both Beckmann's and Davis's models to a pair of scaling laws taking the form of exponentials. Then we derive an exact three-parameter Zipf-type model from the scaling laws to revise the commonly used two-parameter Zipf model. By doing so, we reveal the fractal essence of central place hierarchies and link the rank - size rule to central place model logically. The new mathematical frameworks are applied to the class counts of the 1950 - 70 world city hierarchy presented by Davis in 1978, and several alternative approaches are illustrated to estimate the fractal dimension.

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  • 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.
  • Handle: RePEc:pio:envirb:v:30:y:2003:i:6:p:799-818
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    Cited by:

    1. 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.
    2. Yanguang Chen, 2009. "Urban chaos and perplexing dynamics of urbanization," Letters in Spatial and Resource Sciences, Springer, vol. 2(2), pages 85-95, October.
    3. 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.
    4. 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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