Discrete Dynamics in Nature and Society

Volume 2017, Article ID 5243287, 15 pages

https://doi.org/10.1155/2017/5243287

## A Hierarchical Allometric Scaling Analysis of Chinese Cities: 1991–2014

Department of Geography, College of Urban and Environmental Sciences, Peking University, Beijing 100871, China

Correspondence should be addressed to Jian Feng; nc.ude.ukp@naijgnef

Received 1 January 2017; Revised 22 March 2017; Accepted 9 April 2017; Published 22 May 2017

Academic Editor: Charalampos Skokos

Copyright © 2017 Yanguang Chen and Jian Feng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The law of allometric scaling based on Zipf distributions can be employed to research hierarchies of cities in a geographical region. However, the allometric patterns are easily influenced by random disturbance from the noises in observational data. In theory, both the allometric growth law and Zipf’s law are related to the hierarchical scaling laws associated with fractal structure. In this paper, the scaling laws of hierarchies with cascade structure are used to study Chinese cities, and the method of analysis is applied to analyzing the change trend of the allometric scaling exponents. The results show that the hierarchical scaling relations of Chinese cities became clearer and clearer from 1991 to 2014 year; the global allometric scaling exponent values fluctuated around 0.85, and the local scaling exponent approached 0.85. The Hurst exponent of the allometric parameter change is greater than 0.5, indicating persistence and a long-term memory of urban evolution. The main conclusions can be reached as follows: the allometric scaling law of cities represents an evolutionary order rather than an invariable rule, which emerges from self-organized process of urbanization, and the ideas from allometry and fractals can be combined to optimize spatial and hierarchical structure of urban systems in future city planning.

#### 1. Introduction

Cities as systems (individuals) and systems of cities (groups) are scale-free complex systems, which cannot be effectively described by the traditional mathematical methods based on characteristic scales in many respects. The ideas from scaling can be used to research urban systems (see, e.g., [1–7]). Two correlated scaling laws are often employed to analyze a hierarchy of cities: one is Zipf’s law, and the other is the law of allometric growth. Zipf’s law, allometric growth law, and distance-decay law compose three basic laws of urban geography. Zipf’s law indicates the rank-size pattern of cities in a geographical region [8–12], and the allometric growth law describes the relationship between size and shape in the growth of human settlements [13–17]. In fact, Zipf’s law reflects urban growth [1, 9], and the allometric scaling law can be derived from dual Zipf’s models of rank-size distributions of urban population and area [18]. This suggests that Zipf’s law and the allometric scaling law represent different sides of the same coin. Zipf’s law proved to be equivalent to a hierarchical scaling law, and the rank-size allometric scaling can be replaced by the hierarchical allometric scaling [19]. Hierarchy mirrors a universal structure in natural and social systems [20]. Zipf’s law is a signature of hierarchical structure. Based on hierarchical scaling, we can develop a new approach to studying urban systems.

China bears a large set of cities with a long history. Studies on Chinese cities will help us understand the hidden order of complex systems. There is a dispute about whether the size distribution of Chinese cities follows Zipf’s law [22–27]. In fact, Zipf’s law is a rule of evolution rather than that of existence. The rank-size pattern emerges from the edge of chaos [28]. A new discovery is that Chinese city-size distribution can be described by the three-parameter Zipf’s model instead of the two-parameter Zipf’s model [29]. The three-parameter Zipf’s law suggests an incomplete hierarchy with cascade structure. This implies that the hierarchical scaling law can be used to research the allometry and rank-size pattern of Chinese cities.

This paper is devoted to making a hierarchical allometric analysis of Chinese systems of cities. The aim of this study is as follows. First, we try to reveal the evolutional process and characteristics of the allometric scaling in Chinese cities. Second, we attempt to bring to light the causality behind the hierarchical allometry of Chinese cities. Third, we will sum up a general framework of hierarchical allometric analysis of cities. The trait of this case study rests with large samples, continuous time series, and new angle of view. By this work, we can obtain useful geographical information about spatiotemporal evolution of Chinese cities and new knowledge about scaling in cities. Moreover, the study lends further support to the suggestions that the geographical laws are evolutional laws and there are inherent relationships between Zipf’s law and the law of allometric growth of cities. The remaining parts of the paper are organized as below. In Section 2, the basic mathematical models of hierarchical structure are presented and explained; in Section 3, empirical analyses of hierarchical allometric scaling in Chinese cities are made by means of two algorithms, to show the evolutional regularity of Chinese cities; in Section 4, several questions are discussed, and a general process of allometric analysis is proposed for urban studies. Finally, in Section 5, the article will be concluded by summarizing the mains of this study.

#### 2. Models

##### 2.1. Hierarchical Scaling Law of Cascade Structure

The mathematical models of self-similar hierarchies of cities can be expressed as a set of exponential functions and power functions. Using these models, we can make allometric scaling analysis based on city-size distribution. Suppose that the cities in a geographical region are grouped into* M* classes in a top-down order according to the generalized 2^{n} principle [19, 30, 31]. The cascade structure of the urban system can be modeled by three exponential equations as follows:where* m* refers to the top-down ordinal number of city level (), denotes the number of cities of order ; correspondingly, and represent the mean population size and urban area at the th level. The meaning of the parameters is as below: refers to the number of the top-level cities, and are the mean population size and urban area of the top-level cities, / is the interclass* number ratio* of cities, is the population* size ratio*, and is the urban* area ratio*. Equations (1), (2), and (3) compose the mathematical expressions of the generalized 2^{n} rule [19, 21], which is based on Beckmann-Davis models [30, 32]. In theory, if as given, then it will follow that and vice versa. Here the arrow denotes “approach” or “be close to.” If , the generalized 2^{n} rule will return to the normal 2^{n} rule presented [31].

##### 2.2. Hierarchical Allometric Rescaling

The cascade structure of a hierarchy of cities suggests allometry, fractal, and scaling. A set of power-law relations including the three-parameter Zipf’s law can be derived from the above exponential laws [21, 29]. The power-law models are as below:where , /, , , , and . Equation (4) is termed the hierarchical* size-number scaling relation* of cities, which is equivalent to the Pareto law of city-size distribution, and* D* is the fractal dimension of urban hierarchies measured with urban population. Equation (5) is termed the hierarchical* area-number scaling relation* of cities, which is equivalent to the Pareto law of city-area distribution, and* d* can be treated as the fractal dimension of urban hierarchies measured with urban area. According to Chen [19], both* D* and* d* are actually paradimension rather than real fractal dimension. Equation (6) is termed the hierarchical allometric scaling relation between urban area and population, and* b* is the allometric scaling exponent of an urban hierarchy. The inverse functions of (4) and (5) are equivalent to Zipf’s models of urban population size and urban area distributions. The allometric scaling exponent is actually the ratio of the fractal dimension of urban area size distribution to the fractal dimension of urban population size distribution; that is, . Generally speaking, , and is close to 1. Thus, value comes between 0 and 1.

The exponential laws and the power laws reflect two relations of a hierarchy, respectively: longitudinal relations and latitudinal relations. The longitudinal relations are the associations across different classes, while the latitudinal relations are the correspondences between different measures such as city population size and urbanized area [21]. Formally, these hierarchical and measurement relations can be illustrated by dual hierarchies (Figure 1). If cities in a region follow Zipf’s law, they can be organized into a hierarchy with cascade structure. On the other hand, if an urban system bears cascade structure, the cities in the system follow Zipf’s law or can be described with a three-parameter Zipf’s model. The Zipf distribution is a signature of self-similar hierarchies associated with fractal patterns and self-organized processes.