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

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^<i>n</i>-rule to an <i>rⁿ</i>-rule on the size - number relationship of cities (<i>r</i> > 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.