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(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.