Hierarchical Scaling in Systems of Natural Cities.

Yanguang Chen,Bin Jiang

doi:10.3390/e20060432

Yanguang Chen, Bin Jiang

Open Access

https://doi.org/10.3390/e20060432

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Abstract

Hierarchies can be modeled by a set of exponential functions, from which we can derive a set of power laws indicative of scaling. The solution to a scaling relation equation is always a power law. The scaling laws are followed by many natural and social phenomena such as cities, earthquakes, and rivers. This paper reveals the power law behaviors in systems of natural cities by reconstructing the urban hierarchy with cascade structure. Cities of the U.S.A., Britain, France, and Germany are taken as examples to perform empirical analyses. The hierarchical scaling relations can be well fitted to the data points within the scaling ranges of the number, size and area of the natural cities. The size-number and area-number scaling exponents are close to 1, and the size-area allometric scaling exponent is slightly less than 1. The results show that natural cities follow hierarchical scaling laws very well. The principle of entropy maximization of urban evolution is then employed to explain the hierarchical scaling laws, and differences entropy maximizing processes are used to interpret the scaling exponents. This study is helpful for scientists to understand the power law behavior in the development of cities and systems of cities.

Highlights

Hierarchy is one of the basic characters of complex systems such as cities and networks of cities
In light of the generalized 2n principle expressed by Equations (1) and (2), we can organize the cities in each country into a hierarchy with cascade structure
We investigate the systems of natural cities in three European and one North

Summary

Introduction

Hierarchy is one of the basic characters of complex systems such as cities and networks of cities. A fractal object is a self-similar hierarchy [2,3]. A finding is that a self-similar hierarchy can be described with two or three exponential functions, from which it follows a set of power functions indicative of scaling [6,7]. This suggests that the scaling in cities can be described with power laws, it can be understood through exponential laws. If the transformed result based on any scale factor bear the same structure with the original object, we will say that the process obeys scaling law [3,8]. Exponential laws can be derived by using the Entropy 2018, 20, 432; doi:10.3390/e20060432 www.mdpi.com/journal/entropy

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