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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.

Suggested Citation

  • Yanguang Chen & Yixing Zhou, 2003. "The Rank-Size Rule and Fractal Hierarchies of Cities: Mathematical Models and Empirical Analyses," Environment and Planning B, , vol. 30(6), pages 799-818, December.
    • Handle: RePEc:sae:envirb:v:30:y:2003:i:6:p:799-818
    DOI: 10.1068/b2948
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    References listed on IDEAS

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    1. R White & G Engelen, 1993. "Cellular Automata and Fractal Urban Form: A Cellular Modelling Approach to the Evolution of Urban Land-Use Patterns," Environment and Planning A, , vol. 25(8), pages 1175-1199, August.
    2. Lucien Benguigui & Daniel Czamanski & Maria Marinov & Yuval Portugali, 2000. "When and Where is a City Fractal?," Environment and Planning B, , vol. 27(4), pages 507-519, August.
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    Cited by:

    1. Lang, Wei & Long, Ying & Chen, Tingting & Li, Xun, 2019. "Reinvestigating China’s urbanization through the lens of allometric scaling," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 525(C), pages 1429-1439.

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