Research on the Influence of a High-Speed Railway on the Spatial Structure of the Western Urban Agglomeration Based on Fractal Theory—Taking the Chengdu–Chongqing Urban Agglomeration as an Example

By shortening the transportation time between cities, high-speed rail shortens the spatial distance between cities and exerts a far-reaching influence on urban agglomerations’ spatial structures. In order to explore the influence of high-speed rail on the spatial reconstruction of an urban agglomeration in western China, this paper employs fractal theory to compare and analyze the spatial structure evolution of the Chengdu–Chongqing urban agglomeration in western China before and after the opening of a high-speed railway. The results show that after the completion of the high-speed railway, the intercity accessibility is improved. The Chengdu–Chongqing urban agglomeration’s spatial distribution shows a decreasing density from the central city to the surrounding areas. Furthermore, the urban system presents a trend of an agglomeration distribution. Therefore, strengthening the construction of high-speed rail channels between primary and medium-sized cities, as well as accelerating the construction of intercity railway networks and rapid transportation systems based on high-speed rail cities, would help develop urban agglomerations in western China.


Introduction
In recent years, infrastructure construction mainly based on the high-speed railway has led to urban form transformation. In the great transformation from the ordinary railway era to the high-speed railway era, the urban form has undergone important changes both in the urban agglomeration and in the urban interior [1]. According to the space-time compression theory, the change of high-speed railway speed will affect the spatial-temporal relationship of the urban network, which has an important influence on the regional spatial reconstruction. According to this theory, in Europe in the second half of the 20th century, the EU invested a lot to promote the construction of transnational and trans-regional high-speed rail. Studies have confirmed that high-speed railways can improve different nodes' accessibility, reducing the time and space distances between Asia and Europe [2]. Scholars believe that high-speed rail links many cities together to create a new area or corridor with higher inter-regional accessibility [3]. Besides, studies have also focused on the negative impact of high-speed rail construction. While absolute accessibility and mobility have improved, the relative accessibility of some regions and cities has decreased. Studies have shown that the primary beneficiaries of impact on the urban spatial structure [30][31][32]. This study studies the fractal characteristics of the urban agglomeration spatial structure from the perspective of spatial and temporal differences caused by different traffic modes before and after the completion of high-speed rail. Due to the reduction in traffic time after the opening of high-speed rail, the connection between cities will be strengthened. Therefore, it is more scientific to study the urban agglomeration spatial structure's fractal characteristics using traffic time distance instead of spatial distance [33].
This article aims to explore the impact of high-speed rail on the spatial reconstruction of western urban agglomerations. First, Zipf's law is combined with fractal theory to analyze the rank-size structure of the Chengdu-Chongqing urban agglomeration, the Guanzhong Plain urban agglomeration, the Lanzhou-Xining(Lan-Xi) urban agglomeration, and the Hohhot-Baotou-Ordos-Yulin (HBOY) urban agglomeration. Then, based on these, we take the Chengdu-Chongqing urban agglomeration as an example, and analyze the spatial cohesion and spatial correlation of the urban agglomeration before and after the opening of the high-speed rail using the aggregation dimension and correlation dimension in the fractal, to reveal the changes in the spatial structure of the Chengdu-Chongqing urban agglomeration. The research results will provide scientific reference for the western region to formulate a reasonable high-speed rail construction plan and promote the urban agglomeration scale and structure optimization.

Methods and Data
In order to test the evolution of the spatial structure of urban agglomerations in western China during the period of the rapid development of the western railway, four urban agglomerations in western China were selected as the research areas: they are the Chengdu-Chongqing urban agglomeration, the Guanzhong Plain urban agglomeration, the Lanzhou-Xining (Lan-Xi) urban agglomeration, and the Hohhot-Baotou-Ordos-Yulin (HBOY) urban agglomeration. In February 2010, the Zhengzhou-Xi'an high-speed railway, the first high-speed rail with a maximum speed of 350 km per hour in the central and western regions, was completed and put into operation, marking the beginning of the era of high-speed rail in the western region. Therefore, when studying the western urban agglomeration's rank-size structure, 2010-2017 was selected as the research time limit. To further explore the impact of high-speed rail on the spatial structure of urban agglomerations, the Chengdu-Chongqing urban agglomeration, which has the fastest high-speed rail network construction and the most mature development among the western urban agglomerations, was taken as an example. Fractal theory was used to study the spatial cohesion and spatial correlation of the urban agglomeration after completing the Chengdu-Chongqing high-speed rail.

Fractal Model of Urban Scale Sequence Structure
The rank-size structure of the urban system has fractal characteristics. The most basic method to determine the urban system's rank-size structure is the Hausdorff fractal dimension [22]. In previous studies, many scholars combined the Hausdorff dimension with Zipf's law and selected the urban population as an indicator, making the data collection and calculation process easier [34]. The Hausdorff dimension's basic idea is to give an object's measure its volume through a set of measurement scales, r, and the measurement result must be a value related to r, denoted as N(r). Obviously, the smaller the scale of the r, the larger the measurement result (N(r)); the larger the scale of the r, the smaller the measurement result (N(r)). Generally, the relationship between N(r) and r is expressed as [35]: Among them, the constant D is usually called the Hausdorff fractal dimension, and C is a constant. Taking the logarithm of both ends of the above formula, the following can be obtained: A series of point pairs (r, N(r)) can be formed by the scale r and the corresponding measurement result N(r), and the linear regression obtains the fractal dimension D.
The formula of Zipf's law is which can be transformed into Among them, r represents the city sequence; P(r) represents the population of the j-th city; P 1 represents the population of the primary city; and q represents the Zidf dimension. Chen (1999) proved Formula (3) is equivalent to Formula (1), and D =1/q [35]. D in different intervals represents different meanings. If D > 1, it means that the urban population is evenly distributed in the city, the number of medium-sized cities is more, and the city size is centralized distribution. If D < 1, it indicates that the urban population distribution is uneven, the primary city is in an obvious monopoly position, and the city size is distributed in a decentralized way. The case of D = 1 is too ideal and will not be discussed here.

Correlation Model of Urban Spatial Structure
The correlation dimension can simulate the correlation characteristics among the important nodes in an urban agglomeration. The specific model is as follows: In order to facilitate the actual calculation, Formula (5) is often changed into If the relationship between C(r) and r is satisfied, then The fractal characteristics of the system is Among them, A is a constant; H is the Heaviside function; r is the spatial scale; N is the number of cities in the urban agglomeration; and d ij refers to the distance between the node cities. When d ij represents the Euclidean distance, D reflects the spatial distribution characteristics of the urban system. When d ij represents the actual traffic mileage, D reflects the connectivity characteristics of the transportation network between the inner cities in the urban system. If the correlation fractal dimension D value is between 0 and 2, the smaller the value, the higher the urban space distribution concentration, and the stronger the city spatial correlation. D tending towards 0 means the cities' spatial distribution is highly concentrated in the primary city; D tending towards 2 means that the cities are evenly distributed in the urban agglomeration space [36].
Suppose the fractal dimension D calculated based on actual traffic mileage is closer to the fractal dimension D calculated based on the Euclidean distance. In that case, it indicates that the connectivity between the cities within the urban agglomeration is better [37]; that is, the closer the network accessibility ρ is to 1, the better the accessibility between the cities. The calculation formula of the network accessibility is

Agglomeration Model of Urban Spatial Structure
It is assumed that the urban agglomerations surrounding cities gather around the central city in a cohesive state and follow some self-similar law [38]. Since the value of r will affect the fractal dimension, the average radius is used instead of r. There are Among them, R s is the average radius; r i is the distance from the i th city to the central city; S represents the number of cities in the circle with r as the radius; and D is the fractal dimension of the agglomeration. If D > 2, it indicates that the distribution of the urban agglomeration elements is funneled discrete, and the density decreases from the periphery to the city center, which is abnormal. If D = 2, it indicates that the urban agglomeration elements change uniformly along the urban radius. If D < 2, it indicates that the urban agglomeration elements present an increasing trend from the surrounding cities to the center, and the distribution of the urban agglomeration elements is in a concentrated state [39].

Data
In this paper, the urban population data are taken as samples to calculate the rank-size structure of the urban agglomeration. The data mainly come from (1) the statistical yearbooks of the relevant provinces, cities, and autonomous regions in Western China; (2) China's urban statistical yearbook; and (3) the statistical annual report of national economic and social development.
When calculating the correlation dimension and the aggregation dimension of the urban spatial structure, it is introduced to use the traffic time distance to replace the traditional geographic space measurement distance [34]. The main modes of transportation in the Chengdu-Chongqing urban agglomeration are expressways, ordinary railways, high-speed railways, and aviation. As aviation is mostly used between long-distance cities, it is not considered. The specific calculation formula for the traffic time distance with or without high-speed rail is Among them, T 1 and T 2 respectively represent the traffic time distance between the cities before and after the completion of the high-speed rail; b, p, and g respectively represent the traffic time between the cities using expressways, ordinary railways, and high-speed railways. The traffic time data of ordinary railways and high-speed railways between the cities are obtained from the China Railway Customer Service Center website's train schedule. When two cities have direct high-speed railways or ordinary railways, the shortest time is used as the traffic distance. In the absence of direct railways, first calculate the time required to use ordinary railways or high-speed railways to reach the transit city closest to the destination, plus the highway traffic distance from the transit city to the destination city.

The Rank-Size Structure of Urban Agglomerations in the Western Region
We used Formula (3) to nonlinearly regress the population data of each urban agglomeration in Western China from 2010 to 2017, and summarized the Hausdorff dimension of each urban agglomeration in Figure 1. The trend in the Hausdorff dimension of each urban agglomeration from 2010 to 2017 in the figure is displayed with different-colored broken lines. It is important to note that the reference axis of the line graph of the HBOY urban agglomeration in the picture is the secondary axis (the secondary axis is indicated in yellow). The figure shows that the Hausdorff dimension of Chengdu-Chongqing urban agglomeration and HBOY urban agglomeration constantly changes during the period, and is always greater than the equilibrium value 1, showing a typical "rank-size" spatial structure. Besides, the Chengdu-Chongqing high-speed rail was completed and put into operation at the end of 2015. It can be seen from the figure that after one year of operation, the Hausdorff dimension of the Chengdu-Chongqing urban agglomeration has dropped significantly and tends to the ideal value 1. Although the fractal dimension of the spatial structure of Lan-Xi urban agglomeration and Guanzhong Plain Urban Agglomeration showed a gradual increase trend from 2010 to 2017, it was still less than the equilibrium value of 1, showing a "primacy" spatial structure.
To clarify the rank-size structure changes of the four urban agglomerations in the western region from 2010 to 2017, the average value of the fractal dimensions of the four urban agglomerations was obtained to depict the changing trend, shown in Figure 2. Huasdorff dimension The trend in the Hausdorff dimension of each urban agglomeration from 2010 to 2017 in the figure is displayed with different-colored broken lines. It is important to note that the reference axis of the line graph of the HBOY urban agglomeration in the picture is the secondary axis (the secondary axis is indicated in yellow). The figure shows that the Hausdorff dimension of Chengdu-Chongqing urban agglomeration and HBOY urban agglomeration constantly changes during the period, and is always greater than the equilibrium value 1, showing a typical "rank-size" spatial structure. Besides, the Chengdu-Chongqing high-speed rail was completed and put into operation at the end of 2015. It can be seen from the figure that after one year of operation, the Hausdorff dimension of the Chengdu-Chongqing urban agglomeration has dropped significantly and tends to the ideal value 1. Although the fractal dimension of the spatial structure of Lan-Xi urban agglomeration and Guanzhong Plain Urban Agglomeration showed a gradual increase trend from 2010 to 2017, it was still less than the equilibrium value of 1, showing a "primacy" spatial structure.
To clarify the rank-size structure changes of the four urban agglomerations in the western region from 2010 to 2017, the average value of the fractal dimensions of the four urban agglomerations was obtained to depict the changing trend, shown in Figure 2.
the fractal dimension of the spatial structure of Lan-Xi urban agglomeration and Guanzhong Plain Urban Agglomeration showed a gradual increase trend from 2010 to 2017, it was still less than the equilibrium value of 1, showing a "primacy" spatial structure.
To clarify the rank-size structure changes of the four urban agglomerations in the western region from 2010 to 2017, the average value of the fractal dimensions of the four urban agglomerations was obtained to depict the changing trend, shown in Figure 2.  As shown in the figure, the mean value of the fractal dimension increased significantly from 2010 to 2011, and then showed a stable upward trend from 2011 to 2017. Generally speaking, the fractal dimension gradually increased from 2010 to 2017. That means the overall spatial structure of the western urban agglomeration tends to be agglomerative. The population growth rate of the small and medium-sized cities is far greater than that of the primary city.
The urban agglomeration space has an obvious scale-free area, so the spatial structure shows obvious fractal characteristics [16]. The superposition of different traffic network levels not only changes the fractal characteristics of the urban traffic networks but also affects the urban spatial structure at the global level.

The Spatial Structure of the Chengdu-Chongqing Urban Agglomeration
The Chengdu-Chongqing urban agglomeration is the area with the best economic foundation and the strongest economic strength in the western region. The urban system is relatively sound, with 113 towns per 10000 square kilometers, much higher than other areas in the west, and has huge potential for future development. Moreover, its high-speed rail network construction is highly valued and actively supported by the state, and it has an absolute advantage in the western urban agglomeration. Therefore, we further study its spatial structure by using fractal theory.

The Spatial Structure Correlation of the Chengdu-Chongqing Urban Agglomeration
According to the development plan of the Chengdu-Chongqing urban agglomeration, 15 important cities were selected to study. To clearly demonstrate the current high-speed rail traffic between these cities, we have drawn a schematic diagram of the high-speed rail links between them, as shown in Figure 3.
According to the development plan of the Chengdu-Chongqing urban agglomeration, 15 important cities were selected to study. To clearly demonstrate the current high-speed rail traffic between these cities, we have drawn a schematic diagram of the high-speed rail links between them, as shown in Figure 3. To compare the transformation of spatial structure correlation of the Chengdu-Chongqing urban agglomeration with or without high-speed rail, we used Formula (13) to calculate the traffic time distance of these cities in these two cases and to integrate the results into Table 1. In order to save space, the table integrates the non-zero elements of the traffic time distance matrix in the two To compare the transformation of spatial structure correlation of the Chengdu-Chongqing urban agglomeration with or without high-speed rail, we used Formula (13) to calculate the traffic time distance of these cities in these two cases and to integrate the results into Table 1. In order to save space, the table integrates the non-zero elements of the traffic time distance matrix in the two types of situations; that is, the upper triangle of the matrix in the table is the traffic time distance without high-speed rail, and the lower triangle of the matrix is the traffic time distance with high-speed rail. In the calculation, the zero elements of the two matrices should be supplemented. The data processing methods of the two cases are the same. Take the high-speed railway distance as an example. The total distance points, in this case, are N 2 = 15 × 15, to build a 15 × 15 matrix, and take the step size ∆r = 20 as the scale r, and get r = 420, 400, . . . , 20. Then the number of points C(r) of the cities within the range of r will change with r. The composition of point column (r, C(r)) is shown in Table 2.
Based on the data in Table 2, a double logarithmic coordinate map (only the selected scale-free section images) is obtained. It can be seen that the urban spatial correlation dimension D 1 is 1.1371 with high-speed rail. Similarly, the urban spatial correlation dimension D 2 is 1.22 without high-speed rail.
It can be seen from Figure 4 that D 1 < D 2 . In view of the correlation dimension's nature, the completion of high-speed rail will improve the spatial distribution of the Chengdu-Chongqing urban agglomeration to the primary city. It can be seen from Figure 5 that the fractal dimension under the Euclidean distance of the Chengdu-Chongqing urban agglomeration is 1.1517. Using Formula (10), it can be calculated that the network accessibility after the completion of the high-speed railway is 0.9873 and that before the completion of the high-speed railway it is 1.0593. After the completion of high-speed rail, the network accessibility is closer to 1, which indicates that the accessibility of the Chengdu-Chongqing urban agglomeration has improved.  It can be seen from Figure 4 that D1 ＜ D2. In view of the correlation dimension's nature, the completion of high-speed rail will improve the spatial distribution of the Chengdu-Chongqing urban agglomeration to the primary city. It can be seen from Figure 5 that the fractal dimension under the Euclidean distance of the Chengdu-Chongqing urban agglomeration is 1.1517. Using Formula (10), it can be calculated that the network accessibility after the completion of the high-speed railway is 0.9873 and that before the completion of the high-speed railway it is 1.0593. After the completion of high-speed rail, the network accessibility is closer to 1, which indicates that the accessibility of the Chengdu-Chongqing urban agglomeration has improved.

The Agglomeration of the Spatial Structure of the Chengdu-Chongqing Urban Agglomeration
In this section, we take Chengdu as the central city to study the spatial structure agglomeration of the Chengdu-Chongqing urban agglomeration. According to the agglomeration dimension model, the results are shown in Table 3.  It can be seen from Figure 4 that D1 ＜ D2. In view of the correlation dimension's nature, the completion of high-speed rail will improve the spatial distribution of the Chengdu-Chongqing urban agglomeration to the primary city. It can be seen from Figure 5 that the fractal dimension under the Euclidean distance of the Chengdu-Chongqing urban agglomeration is 1.1517. Using Formula (10), it can be calculated that the network accessibility after the completion of the high-speed railway is 0.9873 and that before the completion of the high-speed railway it is 1.0593. After the completion of high-speed rail, the network accessibility is closer to 1, which indicates that the accessibility of the Chengdu-Chongqing urban agglomeration has improved.

The Agglomeration of the Spatial Structure of the Chengdu-Chongqing Urban Agglomeration
In this section, we take Chengdu as the central city to study the spatial structure agglomeration of the Chengdu-Chongqing urban agglomeration. According to the agglomeration dimension model, the results are shown in Table 3.

The Agglomeration of the Spatial Structure of the Chengdu-Chongqing Urban Agglomeration
In this section, we take Chengdu as the central city to study the spatial structure agglomeration of the Chengdu-Chongqing urban agglomeration. According to the agglomeration dimension model, the results are shown in Table 3.
It can be seen from the figure that the spatial agglomeration dimension of the Chengdu-Chongqing urban agglomeration with high-speed rail is 1.2061, and the agglomeration dimension without high-speed rail is 1.2558. After the opening of the high-speed railway, the spatial agglomeration dimension is smaller. This indicates that the radiation capacity of the core cities is enhanced after the opening of high-speed rail. The spatial distribution density of the Chengdu-Chongqing urban agglomeration along the central city decreases gradually, and the degree of agglomeration gradually increases. According to Table 3, the double logarithmic coordinate diagram is obtained as follows in Figure 6: According to Table 3, the double logarithmic coordinate diagram is obtained as follows in Figure  6: It can be seen from the figure that the spatial agglomeration dimension of the Chengdu-Chongqing urban agglomeration with high-speed rail is 1.2061, and the agglomeration dimension without high-speed rail is 1.2558. After the opening of the high-speed railway, the spatial agglomeration dimension is smaller. This indicates that the radiation capacity of the core cities is enhanced after the opening of high-speed rail. The spatial distribution density of the Chengdu-Chongqing urban agglomeration along the central city decreases gradually, and the degree of agglomeration gradually increases.

Conclusion and Discussion
Many scholars in the world have studied the impact of high-speed rail construction on urban structure. They found that high-speed rail in Europe improves the accessibility between cities, and the population migrates from the core area of the urban agglomeration to its peripheral areas. This article studies the impact of high-speed rail on the Chengdu-Chongqing urban agglomeration based on fractal theory. The research results confirm the findings that high-speed rail construction does help improve cities' accessibility. We also found that the opening of high-speed rail has enhanced the

Conclusions and Discussion
Many scholars in the world have studied the impact of high-speed rail construction on urban structure. They found that high-speed rail in Europe improves the accessibility between cities, and the population migrates from the core area of the urban agglomeration to its peripheral areas. This article studies the impact of high-speed rail on the Chengdu-Chongqing urban agglomeration based on fractal theory. The research results confirm the findings that high-speed rail construction does help improve cities' accessibility. We also found that the opening of high-speed rail has enhanced the degree of spatial agglomeration of the urban agglomerations through a comparative analysis. Populations and resources flowed from the core cities to surrounding cities through high-speed rail, which promoted the spatial structure of the Chengdu-Chongqing urban agglomeration to develop toward the "rank-size" type. However, not all the western urban agglomerations' spatial structures have reached the "rank-size" type. Therefore, this article combines the urban agglomeration development plan and proposes the following suggestions for the optimization of the spatial structure of the western urban agglomeration from the perspective of high-speed rail network construction.
(1) Through the research on the spatial structure of the western urban agglomeration, the results show that the Hausdorff dimension of the Lan-Xi urban agglomeration and Guanzhong Plain urban agglomeration is less than 1. The primary city is in a prominent monopoly position, and the spatial structure is an unbalanced "primacy" type. For such urban agglomerations, it is suggested that the leading role of the primary city should be actively used in the construction of high-speed rail by means of "point-line-surface". Specifically, strengthening the construction of high-speed rail channels between the primary and medium-sized cities promotes the flow of the industry, population, and resources of the primary city to the medium-sized city, and then cultivate the regional secondary central city. Finally, it promotes the spatial structure of the urban agglomeration to develop toward the "rank-size" type. (2) By comparing the traffic distance matrix between the cities and the average radius of the cities with or without high-speed rail, it was found that the traffic time distance and average radius show an apparent downward trend after the opening of high-speed rail. This indicates that the opening of high-speed rail is indeed conducive to improve the internal correlation and agglomeration degree of urban agglomerations. Therefore, to promote the economic restructuring and spatial pattern reconstruction of an urban agglomeration, we should take the Chengdu-Chongqing urban agglomeration as a reference. Specifically, speeding up the construction of intercity railway networks and rapid transportation systems based on high-speed rail cities. With the convenience of high-speed rail networks, we can promote the exchange and cooperation of excellent industries between high-speed rail cities and non-high-speed rail cities, further accelerating the formation of an urban agglomeration pattern with high-speed rail as the link. (3) It was found from the development planning documents of the four urban agglomerations that different areas of the same province are divided into different urban agglomerations. This means that the linkage development among the western urban agglomerations needs to strengthen the cooperation between the urban agglomerations. The Chengdu-Chongqing high-speed railway and Xi'an-Chengdu high-speed railway connect the Chengdu-Chongqing urban agglomeration, which is the key to the National Southwest open channel strategy, and to the Guanzhong Plain Urban Agglomeration, which is the strategic hub of the Northwest Open Channel. It connects Xi'an, Chongqing, and Chengdu, which are the fastest developing cities in western China. Furthermore, it promotes the in-depth exchange of human resources, information, resources, science, and technology between Sichuan, Chongqing, and northwest China. Therefore, it is urgent to speed up the construction of a high-speed rail network in western China to promote urban agglomeration in western China.

Conflicts of Interest:
The authors declare that there are no conflicts of interest regarding the publication of this paper.