Abstract
Zipf's law is one the most conspicuous empirical facts for cities, however, there is no convincing explanation for the scaling relation between rank and size and its scaling exponent. Using the idea from general fractals and scaling, I propose a dual competition hypothesis of city development to explain the value intervals and the special value, 1, of the power exponent. Zipf's law and Pareto's law can be mathematically transformed into one another, but represent different processes of urban evolution, respectively. Based on the Pareto distribution, a frequency correlation function can be constructed. By scaling analysis and multifractals spectrum, the parameter interval of Pareto exponent is derived as (0.5, 1]; Based on the Zipf distribution, a size correlation function can be built, and it is opposite to the first one. By the second correlation function and multifractals notion, the Pareto exponent interval is derived as [1, 2). Thus the process of urban evolution falls into two effects: one is the Pareto effect indicating city number increase (external complexity), and the other the Zipf effect indicating city size growth (internal complexity). Because of struggle of the two effects, the scaling exponent varies from 0.5 to 2; but if the two effects reach equilibrium with each other, the scaling exponent approaches 1. A series of mathematical experiments on hierarchical correlation are employed to verify the models and a conclusion can be drawn that if cities in a given region follow Zipf's law, the frequency and size correlations will follow the scaling law. This theory can be generalized to interpret the inverse power-law distributions in various fields of physical and social sciences.
Highlights
If a region is large enough to encompass a great number of cities, the size distribution of the cities usually follow Zipf’s law [1]
It is easy to prove that the density function of the Pareto distribution is a special density correlation function—a kind of hierarchical correlation functions (Figure S1)
The hierarchical correlation models are built for understanding the city rank-size distribution and its scaling exponent
Summary
If a region is large enough to encompass a great number of cities, the size distribution of the cities usually follow Zipf’s law [1]. Zipf’s law for cities is one of the most conspicuous empirical facts in the social sciences [2,3]. In urban geography, this empirical regularity is known as the rank-size rule [4,5,6,7,8,9]. Few social science problems have generated more researches than the urban rank-size distribution of cities, and numerous models have been proposed to account for variations in rank-size regularity. There is no convincing explanation for the rank-size rule and the scaling exponent value of city rank-size distribution, despite the frequency with which it has been observed [11]. The pending problem requires further theoretical study before it will lead us to the underlying rationale of the empirical rule
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