Zipf's law, Hierarchical Structure, and Shuffling-Cards Model for Urban Development

TL;DR

This paper explores the hierarchical structure of cities and their distribution following Zipf's law, proposing a shuffling-cards model to explain the self-organization of complex systems across various scientific fields.

Contribution

It introduces a hierarchical framework linked to Zipf's law and develops a card-shuffling model to interpret power-law distributions in complex systems.

Findings

Hierarchical city structures follow exponential and power-law distributions.

The card-shuffling model explains the emergence of Zipf's law in urban systems.

The approach can be generalized to other physical and social phenomena.

Abstract

A new angle of view is proposed to find the simple rules dominating complex systems and regular patterns behind random phenomena such as cities. Hierarchy of cities reflects the ubiquitous structure frequently observed in the natural world and social institutions. Where there is a hierarchy with cascade structure, there is a rank-size distribution following Zipf's law, and vice versa. The hierarchical structure can be described with a set of exponential functions that are identical in form to Horton-Strahler's laws on rivers and Gutenberg-Richter's laws on earthquake energy. From the exponential models, we can derive four power laws such as Zipf's law indicative of fractals and scaling symmetry. Research on the hierarchy is revealing for us to understand how complex systems are self-organized. A card-shuffling model is built to interpret the relation between Zipf's law and hierarchy of cities. This model can be expanded to explain the general empirical power-law distributions across the individual physical and social sciences, which are hard to be comprehended within the specific scientific domains.