Abstract
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 Zipf's rank-size distribution, andvice versa. However, we have no theory to explain the spatial dynamics associated with Zipf's law of cities. In this paper, a new angle of view is proposed to find the simple rules dominating complex systems and regular patterns behind random distribution of cities. 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 including Zipf's law indicative of fractals and scaling symmetry. A card-shuffling model is built to interpret the relation between Zipf's law and hierarchy of cities. This model can be expanded to illuminate the general empirical power-law distributions across the individual physical and social sciences, which are hard to be comprehended within the specific scientific domains. This research is useful for us to understand how complex systems such as networks of cities are self-organized.
Highlights
The well-known Zipf’s law is a very basic principle for city-size distributions, and empirically, the Zipf distribution is always associated with hierarchical structure of urban systems
Hierarchy is frequently observed within the natural world as well as in social institutions, and it is a form of organization of complex systems which depend on or produce a strong differentiation in power and size between the parts of the whole 1
Where mathematical models are concerned, a hierarchy of cities always bears an analogy to network of rivers 3, 4, while the latter has an analogy with earthquake energy distribution
Summary
The well-known Zipf’s law is a very basic principle for city-size distributions, and empirically, the Zipf distribution is always associated with hierarchical structure of urban systems. Hierarchy can provide a new angle of view for us to understand Zipf’s law and allometric scaling of cities, and vice versa Both Zipf’s law and allomtric growth law are related with fractals e.g., 6, 20–23 , and fractal theory is one of powerful tools for researching complexity and regularity of urban development. Zipf’s law, allometric scaling, and fractal relations will be integrated into the same framework based on hierarchy of cities, and, a model of playing cards will be proposed to explain the Zipf distribution and hierarchical scaling. From this framework, we can gain an insight into cities in the new perspective.
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