*2.2. Methods*

Based on the above data, the aim of this study was to examine the distribution pattern of the railway network at the county level using a new railway network distribution index. In this study, the distribution pattern of the railway network was assessed by the railway network density; railway network proximity; the shortest travel time; train frequency, which reflect the railway frequency of each county; the population; GDP; the gross industrial value above designated size; and fixed asset investments, which provide an indication of the social economic index. These indicators comprehensively reflect the characteristics of railway network distribution from the aspects of regional support ability, external convenience degree, external accessibility, service ability, and coordination with the population and economy [20,26,27].

Based on these indicators, the railway network distribution index reflects the characteristics of railway network distribution. We have selected the analytic hierarchy process (AHP) method to assign the weight of each indicator. The main steps of the method procedure are: (1) the qualitative method is used firstly to select the indicators related to the railway network distribution, then the correlations between these indicators are analyzed, and finally eight indicators with less correlations are selected; (2) we construct the judgment matrix by Delphi expert investigation method, then solve for the weight of each indicator and judge the rationality of the weight vector. AHP provides a new, concise and practical decision-making method for the study of complex system which is composed of interrelated and mutually constrained factors [28].

This study mainly referred to the method proposed by Jin [4] with adaptations to local conditions. In his study, the concept of transportation superiority is presented from three aspects—quality, quantity, and field—to reflect the scale, technical level, and relative advantage of transport infrastructure in China. Then, the transport network density, degree of influence of the transport trunk line, and transport superiority degree of locations are used as basic indicators to express transport superiority by utilizing Geographical Information System (GIS) technology. Based on the concept of transportation superiority, we used eight indicators to indicate the railway network density, railway network proximity, the shortest travel time, train frequency, and also social-economic indicators, which are better able to describe the characteristics of railway network distribution.

The specific technical process is shown in Figure 1. The main research steps are as follows: (1) based on railway vector data, the railway network density was used to evaluate the supporting capacity of railway network facilities, and it is mainly applicable to linear transportation facilities [29,30]; (2) based on the location of the county center and railway station, the railway network proximity was used to reflect the convenience of the county with respect to other counties, the higher the proximity, the better the tra ffic conditions, the higher the support for regional development, and the greater the potential for external connection [31,32]; (3) based on the train frequency data, the shortest travel time was used to reflect the accessibility of the railway network, and the train frequency data also reflected the external service capacity of each railway station. In China, the railway is the main mode of transportation for medium—and long-distance passenger transportation, and the shortest travel time directly reflects the importance and connectivity of the county in the national railway network [33,34]; (4) the above indicators combine social and economic factors, such as population, GDP, gross industrial value above designated size, and fixed asset investment, to depict the pattern of the overall railway network distribution.

The specific methods for determining the distribution pattern of the railway network in China at the county level were as follows:

#### (1) Railway network density, *C*1*i*

The railway network density can be used to evaluate the supporting ability of railway infrastructure to regional development. Railway network density is a positive indicator: the larger its value, the denser the railway network and the better the regional railway conditions. Let the railway network density of county *i* be *C*1*i*, the length of the railway network of county *i* be *Li*, and the area of county *i* be *Ai*; then the county railway network density can be calculated as follows:

$$\mathbb{C}\_{1i} = \frac{L\_i}{A\_i} \quad i = \text{ (1,2,3,...,n)}.\tag{1}$$

**Figure 1.** Technical flow for describing the distribution pattern of the railway network.

#### (2) Railway network proximity, *C*2*i*

This reflects the ease of traveling from a county to other counties via the proximity of its railway network. In general, proximity is used to describe the reachability of two features in geographical space. In this study, proximity was used to characterize the impact of railways on regional transportation advantages. Based on the distance from the nearest railway station to the county center, qualitative indicators were quantified using expert scoring, and then a proximity value was assigned and classified. Table 2 presents the proximity of the railway network infrastructure in county *i*, and the distance from the nearest railway station to the center of county *i* is used to determine the weight; *i* represents a county in China. The expression of the proximity is as follows:

$$C\_{2i} = \sum P\_i \quad i = (1, 2, 3, \dots, n). \tag{2}$$


**Table 2.** Proximity of a railway network.

L is the distance from the nearest railway station to the county center. The weighted value refers to the provincial-level functional area division [35].

#### (3) The shortest travel time, *C*3*i*

The shortest travel time reflects the accessibility of the railway network in a region and is an important indicator to measure the external connections of a railway network. The shortest travel time is an hour, and the higher this value is, the worse the accessibility is.

This study used the train frequency data to calculate the overall shortest time in each county. *Tij* is the shortest railway time in county *i* and county *j*, and when there was no railway connection between the two counties, it was assumed that the shortest travel time was the maximum time for the railway service that can connect county *i* and county *j*. *N* is the number of counties in the region. The expression of the shortest travel time is as follows:

$$C\_{3i} = \sum\_{j=-1}^{n} T\_{\mathbb{H}} /\_{n} \quad i = \text{ (1,2,3,...,n)}.\tag{3}$$

(4) Train frequency, *C*4*i*

Train frequency data can reflect the railway's service to the county area. *i* represents the county of China, while *mi* is the number of train frequency through county *i*:

$$\mathbf{C}\_{4i} = m\_i \quad i = (1, 2, 3, \dots, n). \tag{4}$$

#### (5) Railway network distribution index, *C*

We integrated the railway network density *C*1*i*, railway network proximity *C*2*i*, the shortest travel time *C*3*i*, the train frequency *C*4*i*, socioeconomic indicators of the population *C*5*i*, the GDP *C*6*i*, the gross industrial value above designated size *C*7*i*, and fixed asset investment *C*8*i*, through the analytic hierarchy process (AHP) method and, combined with the existing research results [4], a railway network distribution index model was constructed. First, as an intermediate step, the eight indicators were standardized; as shown in Equation (5). *Cji* is the normalized value of index *j* in county *i*, *Eji* is the original value of index *j* in county *i*, *Max*(*Eji*) is the maximum value of index *j* in county *i*, and *Min*(*Eji*) is the minimum value of index *j* in county *i* (*j* = 1, 2, 3, 4, 5, 6, 7, 8, *i* ∈ (1, 2, 3, ... , *n*)).

$$C\_{ji} = \frac{E\_{ji} - \text{Min}(E\_{ji})}{\text{Max}(E\_{ji}) - \text{Min}(E\_{ji})}.\tag{5}$$

Secondly, the inverse indicator, such as the shortest travel time of the railway, was used to forward the reverse index. Thirdly, according to the degree of action of each indicator in the railway network distribution, the judgment matrix was constructed, and the weight of each indicator *aj* was calculated by AHP.

Hence, the railway network distribution index, *C*, of each county *i* can then be expressed as

$$\mathbb{C} = \sum\_{j=-1}^{8} a\_j \times \mathbb{C}\_{ji}(j=1,2,3) \tag{6}$$

where *aj* is the weight of each indicator, determined using the AHP.
