Modeling fractal structure of city-size distributions using correlation function

TL;DR

This paper introduces a fractal-based model to explain city-size distributions and Zipf's law, revealing how internal and external city development effects influence the scaling exponent.

Contribution

It proposes a dual competition hypothesis and correlation functions to mathematically derive the range of the Zipf exponent and explain city evolution dynamics.

Findings

The Pareto exponent interval is derived as (0.5, 1].

The Zipf exponent interval is derived as [1, 2).

Equilibrium of effects leads to a Zipf exponent of 1.

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. Based on the idea from general fractals and scaling, this paper proposes a dual competition hypothesis of city develop 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. 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 Pareto effect indicating city number increase (external complexity), and the other 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.