The Evolution of urban agglomerations in China and how it deviates from Zipf’s law

ABSTRACT

Urban Agglomeration (UA) is regarded as an emerging complex urban system in China. The development of UA demands a reasonable scale structure, which can be investigated by Zipf’s law. However, few studies have been conducted to quantify the optimal scale of UA and how its development deviates from the optimal scale. With the continuous urban expansion, the problem of UAs’ scale structure has received increasing attention. In this study, we propose a method based on Zipf’s law for estimating the theoretical optimal scale of UAs in China and assessing the deviation rate from their optimal scales. Twelve typical UAs in China are selected, and their development is assessed via urban impervious surface data from 2000 to 2018. The results show that the average deviation rate of the investigated UAs decreased from 3.40% in 2000 to 2.32% in 2018, demonstrating that these UAs are on a positive evolution trajectory. Furthermore, according to the development stage, we make recommendations on “large cities vs. medium/small-sized cities and promoting vs. restraining” to each UA based on its size. The conceptual and analytical knowledge, as well as the results from this study, are expected to offer valuable insights and new references for regulating and managing UAs’ development in China.

KEYWORDS:

• Urban agglomeration
• Zipf’s law
• scale structure
• deviation
• urban impervious surface

1. Introduction

Rapid urbanization is one of the most significant changes occurring in contemporary human society (Huang, Liu, and Li 2021). The process of urbanization has promoted linkages between cities as well as their surroundings, and the dependence between neighboring cities is increasing (Fang, Li, and Song 2017). Urban Agglomeration (UA) is regarded as an emerging complex urban system, which is the highest spatial organization form in the mature stage of urban development (Fang 2019; He et al. 2019). In China, the process of urbanization has been accelerated since the reform and opening up in 1978. The policies are a series of plans guided by government decision-making and focused on the sustainable use of land resources. They include policies concerning the economy, culture, public service layout and land regional division and can determine future regional plan activities by optimizing the land function structure and supporting the accomplishment of established development goals and sustainable development (Yang and Shi 2017; Li et al. 2022).

With the constant expansion of UAs in China, the Urban Impervious Surface (UIS), a foundational element of urban growth that characterizes human activities and socioeconomic development, has increased dramatically. UIS is a general term for built-up land, which includes land use types of buildings, roads, squares, parking lots, etc. (Shao et al. 2019; Tang et al. 2021). In 2015, the UIS area in China reached 48,800 km² (Kuang 2020). However, there exists a notable contradiction between UAs’ development, land protection and environmental sustainability in China (Fang et al. 2019; Trinder and Liu 2020; Wang et al. 2021). In the early 1990s, China began to improve land use management by safeguarding arable land services and limiting the growth of built-up areas (Yu, Zhou, and Yang 2019). The Chinese government has implemented several policy measures, such as the policy of “Increasing vs. decreasing the balance of urban-rural built-up land” (Chen et al. 2016). Nonetheless, the conflict between regional land sources and economic growth is becoming increasingly serious. Therefore, determining effective land management strategies for UAs’ sustainable development is much needed.Rank-size distribution is one of the basic laws in interurban geography (Chen and Zhou 2006). Under the concept of Rank-size distribution, Zipf’s law, an efficient way to understand the static organization, represents a Self-Organized Critical (SOC) state of complex spatial systems (Zhang, Yu, and Fan 2012; Bee, Riccaboni, and Schiavo 2019). SOC provides inspiration to investigate the urbanization process and to discuss the coordinated evolution and development of reasonable city sizes. In recent years, Zipf’s law is often used to explore the land structure of the urban system, including evaluating the evolution of urban structure on the national scale (Gangopadhyay and Basu 2010) and quantifying the complexity and diversity of urban internal forms (Blumenfeld-Lieberthal and Portugali 2010; Rezayan et al. 2010). Ghosh and Basu (2019) argued that a complete urban system should have a reasonable self-sustaining hierarchical structure. Although Zipf’s law is regarded as the universal law of the system, we have to acknowledge that policy and legal intervention in urban systems at all scales in China can affect the self-organized development of the system. There are numerous studies that focused on sustainable land use strategies and that proposed few frameworks to evaluate the implementation of land plans and policies in China, which are important for macro-regional socioeconomic development. Wan et al. (2020) confirmed the opposing impacts of government interventions and market factors on the deviations of city sizes from Zipf’s law. Liu, Wang, and Guo (2018) explored the adjustment and strategies of land use under the background of “new normal”. However, few studies paid attention to the issue of the effects of land policies and deviations from Zipf’s law in a quantitative manner (González-Val 2010; Wan et al. 2020). Finance and Swerts (2020) raised the issue of self-organization degradation or alienation, implying that if urban management interventions are counterproductive, the fractal structure and self-organized distribution could be disturbed. It is unclear whether the land evolution of UAs is insufficient or surplus under the land policy. This paper attempts to quantify the deviations between the actual scale and optimum scale based on Zipf’s law and provide valuable references for UAs’ future adjustment and development.The main objective of this study is to explore the land structure evolution of UAs in China and assess the deviations from Zipf’s law. More specifically, using UIS remote sensing datasets of 12 typical UAs from 2000 to 2018 in China, we make the following contributions. First, most studies on Zipf’s law are based on data at the national scale, whereas we concentrate on the UAs, emerging complex urban systems in China. We systematically investigated the land expansion and structural evolution of the UAs. Second, we propose a model for quantifying the UAs’ optimal scale from Zipf’s law based on continuous expression instead of the discrete expression and estimate the deviation rates of UAs between the actual scale and optimal scale. Third, according to the scale structure and deviation rate from our findings, we give specific development suggestions for each UAs’ land regulation. The results of this study offer valuable insights and new references for the rational urban growth model and sustainable goals of UAs in regulation and management.

2. Study area and data

2.1. Study area

China’s New Urbanization strategy released in 2014 highlights the progressive development of UAs in China (NDRC 2014). Considering the data availability and the development status of UAs, a total of 12 typical UAs in China were chosen as study areas to be evaluated, as shown in Table 1.. Three UAs that include Beijing-Tianjin-Hebei (BTH-UA), Yangtze River Delta (YRD-UA), and Pearl River Delta (PRD-UA) are expected to grow into world-class UAs, representing the most developed areas of the country with a strong economic foundation, which have the most potential to attract growth (Cai et al. 2021; Guan et al. 2018; Liu et al. 2020). Middle Reaches of Yangtze River (MRYR-UA), Chongqing-Chengdu (CC-UA), Shandong Peninsula (SP-UA), and West Coast of Taiwan Straits (WCTS-UA) are planned to become large nation-class UAs. Nation-class UAs represent rapidly growing areas that can mimic those of world-class UAs. The developing process of region-class UAs that include Harbin-Changchun (HC-UA), Central Plain (CP-UA), Beibu Gulf (BG-UA), Central and Southern Liaoning (CSL-UA), Guanzhong Plain (GP-UA) is expected to be accelerated. These UAs generally lag behind world-class and national class UAs. More details regarding the definition of UAs can be found (Fang and Yu 2017; Guo et al. 2007). The selected UAs consist of a total of 158 cities, mainly in the east and middle of China (Figure 1).

Figure 1. The distribution of 12 selected urban agglomerations in China.

Table 1. Status among the selected 12 urban agglomerations in 2018.

Urban Agglomeration	Area (10^4km^2)	Numbers of cities	Current core cities	Total Population (10^7)
YRD-UA	21.17	26	Shanghai	225.00
BTH-UA	21.58	13	Beijing, Tianjin	118.62
PRD-UA	5.43	9	Shenzhen, Guangzhou	57.20
MRYR-UA	35.01	31	Wuhan	125.00
CC-UA	23.89	17	Chengdu, Chongqing	95.00
SP-UA	7.32	8	Jinan, Qingdao	47.36
WCTS-UA	27.06	19	Fuzhou, Xiamen, Wenzhou	94.14
CSL-UA	9.57	9	Shenyang, Dalian	31.06
CP-UA	16.31	30	Zhengzhou, Luoyang	163.53
HC-UA	32.30	11	Harbin, Changchun	46.32
GP-UA	16.22	10	Xi’an	38.65
BG-UA	10.67	10	Nanning, Zhanjiang	41.41

2.2. Data acquisition and processing

The rapid development of earth observation technologies has supplied valuable scientific information for monitoring and investigating the urbanization process as well as dynamic patterns from a spatial perspective (Ding et al. 2021; Shao, Wu, and Li 2021). Two types of remote sensing datasets are used to describe urban boundaries and impervious surface expansion in this study. UIS data is derived to characterize urban built-up land by clipping impervious surface data with urban boundaries data, as shown in Figure 2. The investigated temporal period covers the years 2000, 2005, 2010, 2015, and 2018.

Figure 2. An example of urban impervious surfaces expansion of BTH-UA during the investigated period.

Impervious surface data is extracted from the Global Artificial Impervious Area (GAIA) dataset, an annual product in raster format with a 30 × 30 m resolution between 1985 and 2018, with an average accuracy of more than 90% for multiple years (Gong et al. 2020). Furthermore, Urban Boundary (UB) data is retrieved from the Global Urban Boundaries (GUB) dataset by Li et al. (2020), which is a multi-temporal vector dataset from 2000 to 2018 based on GAIA data. The GUB dataset has great consistency with other products and can better capture the irregular shapes of cities around urban fringes.

3. Methodology

The designed model contains three major steps: 1) determining the structure of UA; 2) diagnosing the scale deviations of UA; and 3) providing suggestions for the sustainable development of UA. The following sub-sessions detail these steps.

3.1. Determining the structure of UAs

Zipf’s law provides an opportunity to determine which scale class dominates the urban system and how this distribution changes over time (Farrell and Nijkamp 2019). The common form of Zipf’s law of rank-size distribution (Zipf 1949) is defined as:

(1) $\mathit{Y}(\mathit{r})=\mathit{\alpha }{r}^{-\mathit{q}}$(1)

where $\mathit{r}$ denotes the rank of the city in UA, $\mathit{Y}(\mathit{r})$ is the UIS scale of the cities, $\mathit{\alpha }$ is a constant and $\mathit{q}$ can be termed Zipf’s scaling exponent that represents the scale structure of UAs. Chen and Zhou (2006) have proved that the fractal dimension, $1/\mathit{f}$ noise and Zipf’s dimension are mathematically expressed in relation to each other, which can illustrate three different self-organizing critical states of the system. According to the value of $\mathit{q}$, the scale structure of UA is categorized into three types (Gan, Li, and Song 2006). When $\mathit{q}$ approaches 1, the UA scale distribution satisfies Zipf’s criterion, suggesting the optimal distribution in the natural state; when $\mathit{q}<1$, the scale distribution of cities is relatively scattered, as large cities in the UA are not prominent while small and medium-sized cities are developed; when $\mathit{q}>1$, the scale of cities is relatively concentrated, as the UA is dominated by large cities while small and medium-sized cities are insufficiently developed. When $\mathit{q}=1$, all cities are developed in an equal manner. A $\mathit{q}$ that approaches infinite suggests that this UA has only one city (Zhang et al. 2019).

3.2. Deviation rate from Zipf’s law

Zipf’s law is used to reflect the scale of cities and their relationship with the entire system in terms of rank. It has been used as a quantitative criterion for optimizing the scale of the urban system (Fang and Wang 2015). The optimal scale of UA based on Zipf’s law by discrete expression is defined as:

(2) $\widehat{S_d}=\sum _{\mathit{k}=1}^N\mathit{P}(\mathit{r})=\mathit{\alpha }\sum _{\mathit{k}=1}^N{r}^{-\mathit{q}}$(2)

where $\widehat{S_d}$ refers to the optimal area of UA, $\mathit{P}(\mathit{r})$ is the optimal UIS scale of the cities in UAs, $\mathit{\alpha }$ and $\mathit{q}$ are Zipf’s scaling exponents determined by Equation (1)(1) $\mathit{Y}(\mathit{r})=\mathit{\alpha }{r}^{-\mathit{q}}$(1) .

Zipf’s distribution can be abstracted as a simple problem of $\mathit{p}$-series whose $\mathit{p}$ is exactly the $\mathit{q}$ of Zipf’s distribution. Zipf’s model of rank-size distributions is a well-known two-parameter model that is proved to be equivalent to Pareto’s cumulative distribution function. For pure Zipf’s distribution, the exponent $\mathit{q}$ equals 1. In this case, the two-parameter model is reduced to a one-parameter model, i.e. $\mathit{p}=\mathit{q}=1$, which describes the inverse relationship between rank and scale. If $\mathit{p}=\mathit{q}=1$, the p-series changes to the harmonic sequence. The changing regularity of a sequence can be judged through generalized integral (Chen 2021). For a scaling function, both its differential and integral results obey the scaling law (Finance and Swerts 2020). Equation (2)(2) $\widehat{S_d}=\sum _{\mathit{k}=1}^N\mathit{P}(\mathit{r})=\mathit{\alpha }\sum _{\mathit{k}=1}^N{r}^{-\mathit{q}}$(2) reaches theoretical optimal when the number of cities tends to infinity. However, in reality, the number of cities in UA is limited. In theoretical studies, the analytical process based on continuous expression is superior to that based on discrete sequences in the condition of a limited number of cities. Therefore, the optimal scale of UA based on Zipf’s law with continuous expression is proposed as:

(3) $\widehat{S_c}={\int }_{r=1}^N\mathit{P}(\mathit{r})\mathrm{dr}+\mathit{\epsilon }=\mathit{\alpha }{\int }_{r=1}^N{r}^{-\mathit{q}}\mathrm{dr}+\mathit{\epsilon }$(3)

(4) $\mathit{\epsilon }=\mathit{\alpha }\sum _{\mathit{r}=1}^N\frac{1}{r}-\mathit{\alpha }{\int }_{r=1}^N\frac{1}{r}\mathrm{dr}=\mathit{\alpha C}$(4)

where $\widehat{S_c}$ is the optimal scale of UA under continuous variables; $\mathit{\epsilon }$ is considered as the modification value between discrete sequences and the integration of continuous variables; $\mathit{C}=0.577,216\text{break}\dots$ denotes one of Euler’s constant for the sum of harmonic series (Alexanderson 2005).We can further diagnose UA’s scale by measuring the deviation between the current scale and the optimal scale. The deviation rate can help eliminate the scale effect and facilitate the horizontal comparison among UAs due to their different scales. When the current scale exceeds the optimal scale (the deviation rate is positive), it indicates that the UA scale is overextended; when the current scale is smaller than the optimal scale (the deviation rate is negative), it indicates that the UA scale is underdeveloped; The closer the current scale and the optimal scale are, the healthier the UA’s scale is. The deviation rate is defined as follows:

(5) $\mathit{DR}=\frac{S-\widehat{S_c}}{\mathit{S}}\times 100\mathrm{\%}$(5)

where $\mathit{DR}$ denotes deviations rate, and $\mathit{S}$ is the current UIS scale of UA.

3.3. Model for the development suggestion of urban agglomeration

We propose a model for UA’s development suggestion based on the structure index ($\mathit{q}$) by Equation (1)(1) $\mathit{Y}(\mathit{r})=\mathit{\alpha }{r}^{-\mathit{q}}$(1) and the Deviation Rate ($\mathit{DR}$) by Equation (4)(4) $\mathit{\epsilon }=\mathit{\alpha }\sum _{\mathit{r}=1}^N\frac{1}{r}-\mathit{\alpha }{\int }_{r=1}^N\frac{1}{r}\mathrm{dr}=\mathit{\alpha C}$(4) . $\mathit{DR}$ represents the process of decision-making, where a positive value of $\mathit{DR}$ means “promotion” and a negative value of $\mathit{DR}$ means “restraining”. It is assumed that the major domination in UA is “large cities” when $\mathit{q}>1$ and “small-medium sized cities” when $\mathit{q}<1$. We argue that UA’s regulation needs to approach the origin of the coordinate system where $\mathit{q}=1$ and $\mathit{DR}=0$. Thus, the development suggestions of UA are categorized into four types: 1) restraining medium-small-sized cities (L-H), 2) promoting large cities (L-L), 3) promoting medium-small-sized cities (H-L), and 4) restraining large cities (H-H). Figure 3 shows the concept of the four-quadrant graph of our proposed suggestion model.

Figure 3. The four-quadrant graph of suggestion model for the UA’s development.

4. Results

4.1. The expansion of urban agglomerations

Figure 4 shows the magnitude of UA’s expansion. Among the selected 12 UAs, the YRD-UA, the UA with the most UIS growth, expanded by a total of 20,476 km² from 2000 to 2018. YRD-UA and BTH-UA are UAs with the top two fastest rates of 1717 km²/yr and 1656 km²/yr during 2015–2018, respectively. While the expansion scales of different levels of UAs differed over the past two decades, most UAs had experienced substantial expansion during the investigated period. The UIS and UB scales of various UAs experienced rapid and synchronous development, particularly in the period from 2015 to 2018, when the accelerated trends occurred.

Figure 4. Expansion scales of 12 urban agglomerations from 2000 to 2018.

We also measured the proportion of UIS within the UB, denoted as PUIS, which was shown in Figure 5. PUIS variations are mainly used to represent the changes in human activity intensities and local environmental conditions (Kuang 2020). The results showed that the average PUIS is 64.27%±13.86%. A notable exception is CY-UA, whose PUIS was less than 55%. We observed that PUIS had a marked decline from 2015 to 2018, owing to the fact that UB expanded faster than UIS. Our findings coincide with those of Kuang (2020) and Sun and Zhao (2018), who also documented similar trends.

Figure 5. PUIS of 12 urban agglomerations from 2000 to 2018.

4.2. Scale structure of urban agglomerations

Table 2 presents Zipf’s scaling exponent, which aids in identifying the structural types of UAs intuitively. The coefficients of determination were greater than 0.95, and measurement results passed testing by 1%, indicating that the fitted values and real values of regression formulas corresponded well. This shows that the land scale structure of UAs accords with Zipf’s law. As of 2018, two UAs (i.e. YRD-UA and PRD-UA) were in the optimal distribution ($0.9<\mathit{q}<1$), suggesting optimal rank-size structures that exhibit well-fitting distributions. Six UAs were in scattered distribution ($\mathit{q}<0.9$) whose medium/small-sized cities were relatively developed. These UAs include MRYR-UA, BTH-UA, HC-UA, CP-UA, SP-UA, BG- UA, and WCTS-UA, whose scale distributions only differed slightly, and the urban scale systems were more balanced. What’s more, the scale structure of four UAs (i.e. CC-UA, GP-UA, LZ-UA, and CSL-UA) was unbalanced ($\mathit{q}>1$). It also means that big cities were over-developed or medium/small-sized cities were under-developed. We observed that the scale difference between cities was considerable, as few hub cities had greater aggregating potential and magnetic forces to attract socioeconomic variables or flows, potentially resulting in the cumulative Matthew effect (Liu, Wang, and Guo 2018).Furthermore, we found that the scaling index of UAs fluctuated from 2000 to 2018. Theoretically, a reasonable self-organized system tends to gradually approach the direction of optimized scale structure ($\mathit{q}=1$). However, several UAs, in particular, SP-UA and MRYR-UA, gradually progressed in the direction of deviation. Such fluctuation of the scale structure of UA reflected the uncertainty introduced by the external impact of land policies.

Table 2. Zipf’s scaling exponent of the selected 12 urban agglomerations in China.

	Urban Agglomeration	2000	2005	2010	2015	2018
World-class UA	YRD-UA	1.036	1.110	1.075	1.079	0.993
	BTH-UA	0.832	0.907	0.919	0.914	0.881
	PRD-UA	1.037	1.074	1.006	0.984	0.938
National-class UA	MRYR-UA	0.776	0.801	0.795	0.736	0.729
	CC-UA	1.777	1.849	1.843	1.709	1.710
	SP-UA	0.652	0.628	0.636	0.613	0.564
	WCTS-UA	0.937	0.959	0.983	0.984	0.867
Regional-class UA	CSL-UA	1.015	1.031	1.045	1.040	1.133
	CP-UA	0.658	0.639	0.611	0.608	0.632
	HC-UA	0.818	0.837	0.862	0.883	0.877
	GP-UA	1.380	1.414	1.389	1.326	1.279
	BG-UA	1.003	1.017	1.000	0.943	0.891

4.3. Scale deviations of urban agglomerations and suggestions

Figure 6 illustrates the deviation rates against the scales of the 12 UAs based on Equation (5(5) $\mathit{DR}=\frac{S-\widehat{S_c}}{\mathit{S}}\times 100\mathrm{\%}$(5) ). The UAs can be classified into two different categories based on the deviation rate: exceeding scale ($\mathit{DR}>0$) and inadequate scale ($\mathit{DR}<0$). As of 2018, five UAs that include PRD-UA, SP-UA, CSL-UA, HC-UA, and BG-UA, were at exceeding scales. The remaining seven UAs were at inadequate scales. From 2000 to 2018, the majority of the deviation from Zipf’s law displayed a declining pattern, with the exception of HC-UA, whose deviation rates were increasing. The average value of the absolute deviation rate of 12 UAs decreased from 3.40% in 2000 to 2.32% in 2018, which demonstrates that the UAs are on a positive trajectory.

Figure 6. Deviation rates from Zipf’s law of the selected 12 urban agglomerations.

Furthermore, we generated four-quadrant graphs to intuitively reach the development suggestions, as shown in Figure 7. Give the situation in the year 2018, H-L includes CC-UA and GP-UA; L-H includes CP-UA, MRYR-UA, BTH-UA, WCTS-UA, and YRD-UA; H-H includes only CSL-UA; L-L includes SP-UA, HC-UA, PRD-UA, BG-UA. Note that there existed quadrant changes among UAs. For example, YRD-UA was H-L during 2000–2015, but it changed to L-H in 2018. Such quadrant changes suggest that land policies should not be fixed but adjustable in a dynamic manner. In addition, BG-UA and CSL-UA with increased distances from the origin deviated from expectations during the investigated period. The above results are expected to provide quantitative and intuitive references for the systematic policy recommendations.

Figure 7. The four-quadrant graphs for the 12 selected urban agglomerations.

5. Discussion and conclusion

In this study, we investigate the evolution of UA under rapid land urbanization. Specifically, we focus on the scale, structure, and deviation of UAs with a novel framework. Our study areas are located in China, an ideal country to analyze urbanization due to its fast urbanization rate and policies implemented by governments at all levels in regulating land urbanization as well as the scale distributions of UAs. Various land policies and macro-control strategies are the propellers of urban system evolutions. On the basis of Zipf’s law, we target the structure of twelve selected UAs and propose a developed model to quantify the deviation rate of UAs’ scale. According to the four-quadrant graphs shown in Figure 7, the land development suggestions of UAs are intuitively divided into four modes: “large cities vs. medium/small-sized cities and promoting vs. restraining”. The results from this study facilitate the identification of the structure and deviations of UAs and provide targeted development suggestions for various UAs, benefiting the realization of sustainable urban growth.

Zipf’s law is an observational regulation that has been a common product of spontaneous or natural phenomena. In China, government interventions are expected to increase while market forces are expected to decrease, resulting in deviations of UA development from Zipf’s law (Wan et al. 2020). Although the average deviation rate of the selected twelve UAs demonstrates that the UAs are on a positive trajectory, our findings suggest that the scale and deviation of UAs present great diversity. In 1988, the “Paid Transfer of Land Use Rights” scheme was implemented, aiming to promote the growth of the local land scale (Wu, Guo, and Hewings 2019). In the early 2000s, the land use program was implemented as a major part of macro-control policies of the national economy (Wu, Guo, and Hewings 2019). Since then, local governments started to assist the expansion of built-up areas by increasing land supply to facilitate investment and foster manufacturing as well as real estate. However, researchers and policymakers have been concerned over the land urbanization bubble, a rapid and catastrophic rise and fall in urban development fueled by a combination of misplaced exuberance and speculation (Jiao and Liu 2012; Zhang and Li 2020). Restriction on growth has been a major trend of local governance under the supervision of the central government. The evolution of Chinese land policies experienced a transformation from the stringent management of cultivated land quotas to the intensive use of existing land, with a shift from an emphasis on quantity to quality (Wang et al. 2018; Zhuang et al. 2021). However, indicated by the homogeneity of scale expansion and the chaotic volatility of the structure index, certain UAs gradually lost their self-organized characteristics, with policies, to a certain degree, failing to diminish entropy.

Fortunately, UA planning has recently been acknowledged as a critical component of China’s territorial spatial planning system. Cities are no longer regarded as isolated entities. Instead, they form complex urban systems connected by broad economic and social structures that evolve in a continuous manner. The scale structure management of UA has been highlighted in the planning material of “Regulations for Compilation of Urban Land Spatial Planning in 2021”. We compare the historical UAs’ development planning with the suggestions shown in Figure 7. Taking YRD-UA and HC-UA as examples, the YRD-UA highlighted the development vitality of small and medium-sized cities, as well as the leading role of large cities such as Shanghai, which is consistent with our findings (Liu 2012; Xu et al. 2021). However, contradicting the “restraining medium/small-sized cities”, HC-UA has been encouraging the growth of large cities such as Harbin, leading to increased deviation from its optimal scale indicated by Zipf’s law. Furthermore, a rising number of UAs are following “promoting large cities” and “restraining medium/small-sized cities” policies, potentially resulting in large cities with an insufficient land scale and medium/small-sized cities with excessive expansion. By investigating the evolution of selected UAs in China, the results from this study provide valuable strategies for UAs’ optimization and development, preventing the entropy increase in the evolution process and allowing for healthy evolution in the context of sustainable self-organization.

We also need to acknowledge the limitation of this study. First, we assume that the number of cities in UAs is constant during the investigated period. In China, however, we have seen a changed number of cities in certain UAs in response to the policies regulation. This limitation, however, is expected to slightly alter the analytical results and does not harm the validity of our conclusions. For future studies, a more detailed survey of UAs is recommended to determine the rank-size rule or the primate-city rule in Zipf’s law. Furthermore, we quantitatively evaluated the deviation of the UAs’ development from Zipf’s law in this study, while the city-level deviation within UAs remains underexploited. Future studies are encouraged to shed light on the evolution of the UAs’ microstructures.

Acknowledgments

The authors are sincerely grateful to the editors as well as the anonymous reviewers for their valuable suggestions and comments that helped us improve this paper significantly.

Data availability statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported in part by the National Natural Science Foundation of China [Grants number 42090012]; 03 Special Research and 5G Project of Jiangxi Province in China [Grants number 20212ABC03A09]; Zhuhai Industry University Research Cooperation Project of China [Grants number ZH22017001210098PWC].

Notes on contributors

Bowen Cai

Bowen Cai received the BS degree in remote sensing science and technology from Wuhan University in 2018. He is currently pursuing his PhD degree at the School of Remote Sensing and Information Engineering, Wuhan University. His research interests include urban remote sensing, land use policy and remote sensing applications.

Zhenfeng Shao

Zhenfeng Shao received his PhD degree in photogrammetry and remote sensing from Wuhan University in 2004. Since 2009, he has been a Full Professor with the State Key Laboratory of Information Engineering in Surveying, Mapping and Remote Sensing (LIESMARS), Wuhan University. He has authored or coauthored over 70 peer-reviewed articles in international journals. His research interests include urban remote sensing, pattern recognition, and urban remote sensing applications.

Shenghui Fang

Shenghui Fang is an associate professor in the School of Remote Sensing and Information Engineering, Wuhan University. His research interests include quantitative remote sensing and remote sensing applications.

Xiao Huang

Xiao Huang received his PhD degree in Geography from the University of South Carolina in 2020. He is currently an Assistant Professor in the Department of Geosciences at the University of Arkansas. His research interests include remote sensing, big data and social sensing.

Yun Tang

Yun Tang received her MS degree in Cartography and Geographic Information System, Fuzhou University in 2018. She currently pursuing her PhD degree at the School of Remote Sensing and Information Engineering, Wuhan University. Her research interests include urban remote sensing, and geography.

Muchen Zheng

Muchen Zheng is currently pursuing her PhD degree at the School of Resource and Environmental Sciences, Wuhan University. Her research interests include map generalization, geography and remote sensing applications.

Hao Zhang

Hao Zhang is currently pursuing his PhD degree at the School of Remote Sensing and Information Engineering, Wuhan University. His research interests include digital photogrammetry and remote sensing.