Scaling Analysis of the Cascade Structure of the Hierarchy of Cities

Abstract

The scaling relations indicating fractal nature can be employed to associate a hierarchy and a network of cities. The cascade structure of an urban hierarchy follows the 2n rule of size class, which can be formulated as a set of exponential functions. From a pair of exponential laws, we can derive a power law indicating the scaling relation between city number and city size of different classes. The scaling exponent is the fractal dimension of the city-size distribution. Owing to the relationship of mathematical transformation between a hierarchy and a network, the scaling analysis of hierarchical structure can be used to enrich geographical spatial analysis such as spatial interaction studies. The interaction among cities from different classes has locality property, i.e., in theory, cities of one class act merely on cities of the immediate adjacent classes. Not all cities have significant impact on other cities where hierarchy is concerned. Only the interaction between adjacent cities, both in space and size, is strong enough to have a significant effect on the structure of urban systems.