The Mathematical Relationship between Zipf's Law and the Hierarchical Scaling Law

TL;DR

This paper establishes a rigorous mathematical link between Zipf's law and the hierarchical scaling law in city-size distributions, revealing their underlying relationship and implications for urban systems analysis.

Contribution

It provides a formal mathematical proof connecting Zipf's law with hierarchical scaling, offering a new theoretical foundation for understanding city development patterns.

Findings

Derived exponential distribution of city sizes from hierarchy

Established hierarchical scaling equation from Zipf's distribution

Validated results with mathematical experiments and city data

Abstract

The empirical studies of city-size distribution show that Zipf's law and the hierarchical scaling law are linked in many ways. The rank-size scaling and hierarchical scaling seem to be two different sides of the same coin, but their relationship has never been revealed by strict mathematical proof. In this paper, the Zipf's distribution of cities is abstracted as a q-sequence. Based on this sequence, a self-similar hierarchy consisting of many levels is defined and the numbers of cities in different levels form a geometric sequence. An exponential distribution of the average size of cities is derived from the hierarchy. Thus we have two exponential functions, from which follows a hierarchical scaling equation. The results can be statistically verified by simple mathematical experiments and observational data of cities. A theoretical foundation is then laid for the conversion from Zipf's law to the hierarchical scaling law, and the latter can show more information about city development than the former. Moreover, the self-similar hierarchy provides a new perspective for studying networks of cities as complex systems. A series of mathematical rules applied to cities such as the allometric growth law, the 2^n principle and Pareto's law can be associated with one another by the hierarchical organization.