The Rank-Size Rule and Fractal Hierarchies of Cities: Mathematical Models and Empirical Analyses

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 rⁿ-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.