Sustainable Transportation: Exploring the Node Importance Evolution of Rail Transit Networks during Peak Hours

Abstract

The scientific and rational assessment of the evolution of node importance in rail transit line networks is important for the sustainability of transportation systems. Based on the complex network theory, this study develops a weighted network model using the Space L method. It first considers the network topology, the mutual influence of neighboring nodes of the transportation system, and the land use intensity in the station influence domain to construct a comprehensive index evaluation system of node importance. It then uses the covariance-weighted principal component analysis algorithm to more comprehensively evaluate the node importance evolution mechanism and analyzes the similarity and difference of the sorting set by adopting three different methods. The interaction mechanism between the distribution of important nodes and the evolution of land use intensity is explored in detail based on the fractal dimension theory. The Xi’an rail transit network is considered an example of qualitative and quantitative analysis. The obtained results show that the importance of nodes varies at different times of the day and the complexity of the morning peak is more prominent. Over time, articulated fragments with significance values greater than 0.5 are formed around the station, which are aligned with the direction of urban development, creating a sustainable mechanism of interaction. As the network’s crucial nodes in the center of gravity increase and the southern network expands, along with the increased intensity of the city’s land utilization, the degree of alignment in evolution becomes increasingly substantial. Different strategies for attaching the network, organized based on the size of $S_i$ can lead to the rapid damage of the network (reducing it to 0.2). The identification of crucial nodes highlighted in this paper serves as an effective representation of the functional characteristics of the nodes in transportation networks. The results obtained can provide a reference for the operation and management of metro systems and further promote the sustainable development of transportation networks.

1. Introduction

The urban rail transit has been developing in China for more than fifty years. It has gradually become the first choice for travelers, and it has developed into the backbone of the urban public transportation system due to its high speed, punctuality, and reliability as well as its low delay [1]. In the past decades, the development of urban rail transit has been rapid, marked by the construction of numerous extension lines. Due to its spatial independence, urban rail transit is virtually immune to the influence of other modes of urban transportation [2]. However, recent emergencies, such as the urban flooding caused by the rainstorm in Zhengzhou in 2021, have posed significant challenges to the transportation system. The resulting damage to the stations or interval lines has potentially intensified the extent of the impact. Therefore, it is of great significance to further study the evolution laws of the important nodes of the urban rail transit line network in order to be able to perform emergency management and ensure its safe and efficient operation.

From a topological perspective, urban rail transit can be conceptualized as a complex network. The complex network theory has been widely used in public transport fields, including the railway [3,4], bus [5,6], and rail transit [7,8]. Many studies have also shown that the complex network theory is an effective tool for studying the urban traffic network. Ref. [9] designed a station importance ranking algorithm to explore key stations in the Shanghai metro network. Ref. [10] proposed a method to identify the importance of network stations using a combined subjective and objective weighted approach. Ref. [11] analyzed the centrality and robustness of nodes in topological URTN in several cities around the world, providing rail transit planners with ideas for highly reliable transit networks.

Some studies have been conducted on important stations of transportation networks. In addition, in the construction of rail transportation networks, most of the studies use full-day cross-sectional passenger flows as weights. For instance, Ref. [12] used full-day cross-sectional passenger flows for the weighted modeling of the Shenzhen metro network. They studied the evolution of important stations in different periods and analyzed the interaction mechanism between them and the built environment. Ref. [13] proposed a weighted WTOPSIS identification algorithm and verified its feasibility through examples in order to expose the evolution of important nodes in urban rail transit networks. Ref. [14] used full-day cross-sectional passenger flows, weighted the bus and metro networks to create a multi-modal network for public transport, and analyzed the relationship between performance changes in urban public transport networks and station capacity control parameters under cascading failure scenarios. Few studies have been conducted on the importance of rail transit network stations using unweighted networks. For instance, Ref. [15] used the Gaussian mixture model to divide the Xi’an Metro network into three stages and then combined multiple network centrality indicators into a comprehensive index of station importance using principal component analysis. It has been shown that dividing the development time sequence of urban rail transit and integrating multiple indicators of complex networks is efficient in studying and analyzing the importance of network stations. However, the weightless network only analyzes the importance of nodes from the perspective of topological space, which is far from real-world considerations. In addition, the weighted network approach using all-day passenger flow tends to ignore the tidal characteristics of urban traffic. The traffic tidal phenomenon is usually associated with the morning and evening peak travel times on weekdays [16], especially in complex urban transportation networks, where large-scale passenger flow often exists at important locations, time periods, and stations [17]. Therefore, it is necessary to study important stations under peak conditions of urban traffic.

In addition, the performance of the transit-oriented development (TOD) model in coping with rapid urbanization, population growth, and increased private car ownership largely depends on the land development intensity layout and related resource allocation in the service area [18,19]. Moreover, many studies discussed the definition and assessment methods of the TOD [20,21]. For instance, Ref. [22] proposed a POI-based visualization method for land use characteristics around urban rail transit stations, which verified that there is a strong correlation between the rail transit stations and the built environment around them. Based on the temporal distribution of smooth-level passenger traffic, a proxy method for land use types has been proposed [23]. In addition, a multiple regression model has been developed to explain the growth of the station-level passenger traffic with network expansion, and to respond to changes in network topology and surrounding land use. Therefore, further studies should be conducted on the quantitative analyses of the evolution of important nodes and land use intensity.

In summary, this paper uses the morning and evening peak cross-section passenger flow to develop a weighted network model of urban rail transit. By taking into consideration the network topology, passenger flow of the rail transportation system, mutual influence of adjacent nodes, and land use intensity in the site influence domain, a comprehensive evaluation system of the importance of the nodes, combined with the improved covariance-weighted principal component analysis (WCPCA), is constructed. This allows us to study the evolution mechanism of the important sites of the Xi’an Metro in the morning and evening peaks over the past ten years and explore the degree of matching of the land use intensity and its evolution using fractal and dimensional theories. Finally, various network attack methods are compared to verify the proposed theory. The contributions of this paper are summarized as follows:

(1) An importance contribution matrix of neighboring sites is constructed and a node location importance measure is proposed by considering the local and global importance of the site, the poor distribution of cross-section passenger flows, the number of people in the domain of influence of the site, and the intensity of land use.

(2) The CRITIC algorithm is used to determine the weights of the indicators. In addition, a comprehensive identification algorithm for important station degrees of urban rail transit approach solves the problem faced by the traditional principal component analysis. That is, the sample covariance matrix does not correctly reflect the correlation between variables. Moreover, it takes into account the topology and performance of the network.

(3) Based on the proposed identification algorithm, the weighted combined importance of all the nodes in different periods of the morning and evening peak is geographically visualized and quantitatively and qualitatively evaluated. It is deduced that the Xi’an Urban Rail Transit Line network (XURTN) exhibits some geographical heterogeneity. In addition, the complexity of the XURTN evolution under the weighted morning peak hour is more prominent than that of the evening peak hour. Moreover, when the structure of the network is improved, its center of gravity gradually expands to the east during the morning peak hour compared with the evening peak hour. The obtained results allow us to better understand the complex evolution of the urban rail transit network system under peak hours in different periods. This provides a reference for the metro operation management to strengthen targeted passenger flow control and evacuation and further improves the safety and resilience of the XURTN.

(4) The evaluation index of the matching degree between the land use and line network evolutions is constructed using the fractal dimension theory while considering the land use intensity characterization. The interactive matching mechanism between the importance distribution of the XURTN nodes and urban land use intensity is studied using the fractal dimensioning theory, which can promote the benign interaction between the urban rail transit line network and urban spatial development.

The remainder of this paper is organized as follows: Section 2 presents the methodological model used in this study, Section 3 describes the case study conducted using the methodology and its results, and Section 4 provides the conclusions. A summary of variables used in this paper is given in

Table 1. Nomenclature.

Notation	Description
G	An unweighted network
V	The nodes set
N	The number of nodes
$a_{ij}$	Connections between $i$ and j
A	Associated adjacency matrix
W	Weights set
$e$	Total number of the node’s connected edges
λ	Largest e of the matrix A
Ci	Closeness centrality of node $i$
$C_i^w$	Weighted closeness centrality of node $i$
Bi	Betweenness centrality of node $i$
Biw	Weighted betweenness centrality of node $i$
$d_{ij}$	Shortest path length between $i$ and j
G w	A weighted network
Vi	The $ith$ station in the network
E	The edges set
M	The number of edges
${pf}_{e_{ij}}(\Delta t)$	Weight between $i$ and j
$W_{ij}$	Weight between $i$ and j
Di	Unweighted degree of node $i$
$D_i^w$	Weighted degree of node $i$
$E_i^w$	Weighted eigenvector centrality of node $i$
$W_l^x$	Weight of the $xth$ edge of the $lth$ path from j to k
$W_l^{i,x}$	Weight of the $xth$ edge of the $lth$ path from j to k through $i$
M-Si w	Morning peak weighted comprehensive node importance values
E-Si w	Evening peak weighted comprehensive node importance values
Pi	Passenger flow values
Si	Unweighted comprehensive node importance values
$R_H$	Floor space owned per capita

.

2. Methodology

This paper aims to explore the evolution of important stations in the rail network with rail construction under urban morning and evening peak hours and study the corresponding heterogeneity. The different stages of this study are shown in Figure 1.

2.1. Development of the Weighted Network Model

The rail transit network model developed by the Space L method [24,25] emphasizes the connection relationship between the nodes in the network. Therefore, this paper adopts it to construct the network model and weights the adjacency matrix with the morning and evening peak segment passenger flows. Rail transit stations are abstracted as network nodes, and the lines between stations correspond to the network edges. The edges exist between two stations only when they are adjacent to each other back and forth. A traditional topological network [26] is denoted by $G=(V,E)$, where $V$ represents the set of network nodes and $E$ is the set of network edges. In the adjacency matrix of the network, $a_{ij}$ is used to describe the position relationship between two nodes. When $V_i$,$V_j$ are front and rear nodes (i.e., when the two nodes are connected), $a_{ij}=1$; otherwise, $a_{ij}=0$ and a weighted network is then used. The weighted network is denoted by $G=(V,E,W)$, where $W=\left\{{pf}_{e_{ij}}\left(\Delta t\right)\right\}$ represents the set of directed edge weights and ${pf}_{e_{ij}}\left(\Delta t\right)$ are the flow of passengers passing through the $i$-j section during the morning or evening peak periods.

Figure 2a,b, respectively, show the traditional topology model and weighted network model [27,28] that are adopted in this study. These two models are composed of twelve stations of three lines. In the topology model shown in Figure 2a, the weights of the edges between the stations are identical, and an undirected graph is considered since there is only one edge between the stations. In contrast to the results shown in Figure 2a, two different directional edges between the two stations are observed in Figure 2b. This consists of a directed graph where the weights are the morning (evening) cross-section passenger flows in the corresponding directions.

2.2. Node Importance Measurement

Node centrality measures: The use of accurate metrics proves to be the most efficient approach in evaluating the importance of nodes within the complex structure of a network. This study employs three network node metrics: the degree [29,30], betweenness [31], and closeness [32]. These metrics are initially considered unweighted ($D_i$,$B_i$,$C_i$). Weighted versions ($D_i^w$,$B_i^w$,$C_i^w$) are then incorporated to capture the time-varying effect of the passenger flow. The eigenvector index ($E_i^w$), which takes into consideration the passenger flow, is used to measure the impact of adjacent nodes. The eigenvector of a node denotes the value of its corresponding element in the eigenvector of the network partition matrix. In this study, the eigenvector element value corresponding to the largest eigenvalue pair is selected to characterize the centrality index. As previously mentioned, the node metric is an efficient tool for assessing the importance of a station in a complex network structure [33].

It is important to mention that the perception of passengers regarding travel convenience and reliability is also a crucial factor for determining the significance of a station in real-world scenarios. The efficiency reliability [34] ($I_i$) and site passenger flow loss ($Q_i$) metrics, which are based on the changes in passenger travel time resulting from site destruction, are used to capture this aspect. When a node in the network is destroyed due to disturbance, some of the passengers in the network are not able to complete their trips, and other passengers will have to make a detour in order to complete their initial commute. When the detour distance is greater than 0.5 times the initial commute distance, these passengers will leave the metro network. Therefore, in this study, the site passenger flow loss is defined as the flow of passengers who will not be able to complete their commute after the node is destroyed and the flow of passengers whose detour distance is greater than 0.5 times the initial distance. To evaluate the overall impact of each site on the network efficiency, this study systematically removes the underlying site from the network and then recalculates its efficiency. The specific calculations of the node centrality and reliability metrics are shown in

Table 2. The definition of node centrality indexes in the XURTN.

Index	$\mathbf{Formula}$	$\mathbf{Definition}$
Central indicators	$D_i=\sum _{j=1}^n{a}_{ij}$ $D_i^w=\sum _{j\in V_i}p{f}_{e_{ij}}(\Delta t)$	$D_i$ is the number of adjacent nodes connected to the node. $D_i^w$ is the total passenger flow of its connected edges [35].
	$$\begin{gathered} B_i={\displaystyle \frac{\sum _{o\ne d\ne i}{n}_{od}}{\sum _{o\ne d}{n}_{od}}} \\ {B}_i^w={\displaystyle \frac{\sum _{l=1}^{n^i}\sum _{x\in l}{w}_l^{i,x}}{\sum _{l=1}^n\sum _{x\in l}{w}_l^x}} \end{gathered}$$	$B_i$ and $B_i^w$ are the shortest number of paths through a station.
	$C_i=\left(N-1\right)\ast {\left[\sum _{j=1}^n{d}_{ij}\right]}^{-1}$ $C_i^w=\left(N-1\right)\ast {\left[\sum _{j=1}^n{d}_{ij}^w\right]}^{-1}$	$C_i$ and $C_i^w$ are used to measure the ability of a station to affect other stations through the network.
	$\mathrm{e}={[e_1,e_2,\dots e_n]}^T$ $\lambda E_i={\sum }_{j=1}^N{\alpha }_{ij}{w}_{ij}{e}_j$ $\lambda E_i^w={\sum }_{j=1}^N{\alpha }_{ij}{w}_{ij}{e}_j^w$	$E_i$ and $E_i^w$ can measure the influence of a node in the network.
Reliability indicators	$I\left(G\right)={\displaystyle \frac{1}{N(N-1)}}\sum _{i\ne j}{\displaystyle \frac{1}{d_{ij}}}$ $I_i=1-{\displaystyle \frac{R(G_i)}{R(G)}}$	Network efficiency ($I_i$) is an indicator characterizing the accessibility between stations.
	$Q_i={\displaystyle \frac{\acute{F}}{F}}$	The ratio of the passenger flow ($\acute{F}$) to the original passenger flow (F) in the removed network

.

Node location importance measure: By considering the local and global importance of the site, the distribution difference of cross-section passenger flow, the number of the population, and land use intensity in the influence domain of the site, this study constructs the importance contribution matrix of neighboring sites $R_C$, which is expressed as follows:

$$R_C=\left[\begin{array}{cccc}{Q}_1& {\displaystyle \frac{{\mathrm{D}}_2{Q}_2{V}_2{\delta }_{12}}{{\left(\overline{D_i^w}\right)}^2}}& \cdots & {\displaystyle \frac{D_n{Q}_n{V}_n{\delta }_{1n}}{{\left(\overline{D_i^w}\right)}^2}}\\ {\displaystyle \frac{D_1{Q}_1{V}_1{\delta }_{21}}{{\left(\overline{D_i^w}\right)}^2}}\hfill & Q_2& \cdots & \hfill {\displaystyle \frac{D_n{Q}_n{V}_n{\delta }_{2n}}{{\left(\overline{D_i^w}\right)}^2}} \\ ⋮& \hspace{1em}\hspace{1em}⋮\hfill & \cdots \hfill & ⋮\hfill \\ {\displaystyle \frac{D_1{Q}_1{V}_1{\delta }_{n1}}{{\left(\overline{D_i^w}\right)}^2}}\hfill & {\displaystyle \frac{D_2{Q}_2{V}_2{\delta }_{n2}}{{\left(\overline{D_i^w}\right)}^2}}& \cdots & Q_n\end{array}\right]$$

(1)

where the elements of the matrix indicate the importance contribution value of node $i$ to node $j$, ${\delta }_{ij}$ denotes the coefficient of the importance contribution of site $i$ to $j$, which is specified as a ratio of the cross-section passenger flow of the neighboring node to the weighted intensity of this site, and $V_1$, $V_2$, …, $V_n$ represent the values of land use intensity within the scope of the site influence domain (Equation (2)) calculated according to the actual situation in Xi’an city while considering 800 m as the threshold value of the scope of the site influence area as follows:

$$V_i={\displaystyle \frac{s_i^{\prime }}{S_i}}={\displaystyle \frac{N_i{R}_H}{S_i}} i=\mathrm{1,2},\dots ,n$$

(2)

where $N_i$ is the number of residents in the influence domain of site $i$, $R_H$ is the per capita owned floor space, and $S_i$ is the land area in the influence domain of site $i$.

The location importance value of each site $R_i$ is calculated according to the constructed neighboring site importance contribution matrix $R_C$:

$$R_i=D_i\times Q_i\times V_i\times {\sum }_{j=1,i\ne j}^n{D}_j{Q}_j{V}_j{\delta }_{ij}/{\left(\overline{D_i^w}\right)}^2$$

(3)

where $D_i$ and $Q_i$ are the degree and efficiency values of the $ith$ site, respectively.

2.3. Node Importance Ranking

WCPCA algorithm: Considering that each single indicator has its own strengths and limitations, this paper proposes an improved comprehensive evaluation algorithm of the weighted indicator system based on traditional principal component analysis. This algorithm is able to more accurately evaluate the data with two main components: determination of indicator weights and comprehensive importance ranking. The determination of indicator weights is a crucial step in the comprehensive evaluation and importance ranking. Commonly used methods for establishing indicator weights include the coefficient of variation method, the CRITIC method, and the entropy weight method.

This study adopts the CRITIC objective weighting method, which outperforms the entropy and standard deviation methods. It is based on the comparison strength of evaluation indicators and the conflict between them. It uses the objective properties of the data to comprehensively measure the objective weight of the indicators, which takes into account the size of their variability and the correlation between them. Therefore, a larger number does not indicate a higher importance, as shown in

Table 3. The detailed steps of the CRITIC method.

Step	Specific Contents
Step 1	Suppose there are n samples to be evaluated and m evaluation indicators, forming the original indicator data matrix X, where $X_{ij}$ denotes the value of the $jth$ evaluation indicator of the $ith$ sample.
Step 2	Dimensionless treatment: in this study, we use the positive treatment, and the larger the value of the index is, the better. $X_{ij}^{\prime }={\displaystyle \frac{X_j-X_{min}}{X_{max}-X_{min}}}$
Step 3	Calculate the indicator variability, $V_j$, which denotes the standard deviation of the $jth$ indicator. ${\overline{X}}_j={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{X}_{ij} V_j=\sqrt{{\displaystyle \frac{{\sum }_{i=1}^n{(X_{ij}-{\overline{X}}_j)}^2}{n-1}}}$
Step 4	Calculation of indicator conflict, $R_j$: The correlation coefficient is used to express the correlation between indicators, and the stronger the correlation with other indicators, the smaller the conflict between the indicator and other indicators, $r_{ij}.$ Then, evaluate the correlation coefficient between indicators $i$ and j: $R_j=\sum _{i=1}^p(1-r_{ij})$
Step 5	Calculate the amount of information, $G_j$: the larger $G_j$ is, the greater the role of the $jth$ evaluation index in the whole evaluation index system, and the more weight should be assigned to it. $G_j=V_j\sum _{i=1}^p(1-r_{ij})=V_j\times R_j$
Step 6	Calculate the objective weight, $W_j$: the objective weight of the $jth$ indicator $W_j$ is $W_j={\displaystyle \frac{G_j}{{\sum }_{j=1}^p{G}_j}}$

.

After the determination of the weights, the WCPCA algorithm is used to rank the combined importance of nodes. It uses the eigenvectors of the covariance matrix as the basis vectors to solve the problem faced by the traditional principal component analysis (i.e., the sample covariance matrix does not correctly reflect the correlation between variables). This allows us to obtain the comprehensive importance ranking results $S_i^w$ of the nodes under the weighted network. The detailed process is shown in

Table 4. Specific steps of the WCPCA.

Step	Data Processing and Calculation Steps
Input	The original index dataset, $X=\left\{x_{ij}\right\}$, $i=\mathrm{1,2},\dots ,n;j=\mathrm{1,2},\dots ,m$ The index weights set, $W_j=\{w_1,w_2,...w_m\}$
Process	1. Calculate the covariance matrix $C_{ij}={\displaystyle \frac{1}{n}}\sum _{i=1}^n{W}_i(X_i-\mu ){(X_j-\mu )}^T$, where μ is the mean value of X. 2. Compute the eigenvalues ${{\lambda }_1,{\lambda }_2,\dots ,\lambda }_n$ and eigenvectors $v_1,v_2,\dots ,v_n$ 3. Selecting principal components k: $(VCR>90\%)$: $VCR={\displaystyle \frac{{\sum }_{i=1}^k{\lambda }_i}{{\sum }_{i=1}^n{\lambda }_i}}$ 4. Calculate the principal component coefficients $b_{ij}$: $b_{ij}=\sum _{\{l=1\}}^{\left\{k\right\}}{v}_{\left\{jl\right\}}(X_{\left\{il\right\}}-u_l)$, where $b_{ij}$ is the coefficient of the $ith$ sample on the $jth$ principal component, $v\left\{jl\right\}$ is the value of the $jth$ eigenvector of the $lth$ principal component, $x\left\{il\right\}$ is the value of the $ith$ sample on the $lth$ variable, and ${\mu }_l$ is the mean value of X on the $lth$ variable. 5. Calculate the weighted ranking $S_i^w$: $S_i^w=\sum _{i=1}^m{b}_{ij}{v}_{ik}$ 6. Rank $S_i^w$ according to its value and $S_i^w$ ∈ [0, 1].
Output	Comprehensive weighted ranking set $S_i^w$.

.

2.4. Correlation Analysis

The built city is not static and its development is not balanced in all directions; instead, development is distributed differently in each direction. In this paper, the fractal theory is introduced to divide the city into several directions in the radial direction [36,37]. The distribution of cities and line networks is divided into $m$ dimensions and several circles, as shown in Figure 3. The combined weighted measure of population accumulation scale and land development intensity (i.e., the plot ratio) are chosen to characterize the land use intensity in the area and calculate the fractal dimension of land use distribution in each direction.

Theoretically, there is an anisotropic scaling relationship between an urban area and its population in different dimensions (i.e., the spatial anisotropy ratio) [38]. In this paper, the city is divided into four directions, the city center O is considered as the center of the circle, and the directions of the city are divided into several circles of radius r having different lengths. The sub-dimensional representation of land use distribution in each direction is given by ${ D}_{l1},D_{l2},D_{l3},...D_{lm}$. This reflects the changing trend of urban land use intensity from the city center to the peripheral areas. In addition, the weighted comprehensive distribution is expressed as ${ D}_{s1},D_{s2},D_{s3},...D_{sm}$, which reflects the development characteristics and trends of the urban rail transit network in different directions of the city in terms of spatial structure. The dimensional vector of land use distribution is given by $D_l=(D_{l1},D_{l2},D_{l3},...D_{lm})$, and the dimensional vector of weighted station importance distribution is expressed as $D_s=(D_{s1},D_{s2},D_{s3},...D_{sm})$. The specific calculations are shown in

Table 5. Matching degree-related calculation.

Index	Formula	Definition
Land usage ($D_l$)	Population distribution $P(r)\propto r^{D_P}$ $\rho (r)={\displaystyle \frac{dP(r)}{dA(r)}}\propto D_p{r}^{D_p-2}$	$P(r)$—The number of people in the urban ring with radius r. $D_p$—Fractal dimension of urban population distribution. $D_l$—Fractal dimension of the distribution of land use intensity in the line network. $A(r)$—Area of a circle with radius r.
	Plot ratio $R(r)={\displaystyle \frac{N{R}_H}{S}}\propto r^{D_R}$ $D_l=w_p{D}_p+w_R{D}_R$	$R\left(r\right)$—the ratio of the total floor area to the land area in the area. $w_p$,$w_R$—Values that are determined mainly using the linear weighting method.
Station ($D_S$)	$S^w(r)\propto r^{D_S}$ $S^w(r)=\sum _{i=1}^{n_r}{S}_i^w(r)$	$D_S$—Fractal dimension of the distribution of important stations in the line network.

.

The land use and rail network development is a dynamic and balanced reciprocal feedback relationship. Therefore, this paper considers the characteristic index of land use intensity based on the evolution mechanism of important nodes of rail transit. It constructs the evaluation index ${ M}_{I }$ for the matching degree between the land use evolution and the line network evolution (Equation (4)), which ranges between 0 and 1. The closer the value is to 1, the better the matching between the urban rail transit line network and urban land use intensity,

$$M_I={\displaystyle \frac{1}{1+d(D_s,D_l)}}$$

(4)

where $d(D_s,D_l)$ is the difference between the distribution of important stations in the urban rail transit network and the urban land use intensity distribution sub-dimension.

The specific calculation is shown in Equation (5).

$$\begin{gathered} d(D_s,D_l)={\displaystyle \frac{\sqrt{\sum _{i=1}^m{\Delta D}_i}}{m}}{(D_{li}-D_{si})}^2 \\ {\Delta D}_i=\left\{\begin{array}{c}{(D_{li}-2)}^2{(D_{si}-2)}^2+,(D_{li}-2)(D_{si}-2)<0\\ {(D_{li}-D_{si})}^2,(D_{li}-2)(D_{si}-2)\ge 0\end{array}\right. \end{gathered}$$

(5)

2.5. Validity Verification

The results of important node identification are first compared and validated using three different methods. Based on the traditional network resilience curve [39], a network resilience curve, which is consistent with the verification of the proposed research method, is proposed (Figure 4). $B_i^w$,$D_i$, random, $S_i^w$, S_i, and $E_i^w$ strategies are adopted to attack the network in order to verify the applicability of the proposed methodology by comparing the rates of degradation of network performance. In the latter, the complex network is perturbed at ${t=t}_s$, which results in decreasing the network performance $p(t)$. This holds until the $t_a$ network disturbance ends, and the network performance decreases by F. More precisely, two indicators are defined to describe the network. The different characteristics of the stable phase and the post-attack phase include fragility (F) and disruption rapidity (D), which characterize the ability of the network to resist external disturbances.

F describes the sensitivity of the system to disruptions that may degrade the serviceability of the network. It is one of the representative indicators of the network outage phase, which is computed as follows:

$$F\equiv P(t_a)-P(t_s)$$

(6)

D identifies the absorption capacity of the system in the case of disruption [40]. It is approximated by the average slope of the performance curve during the attack period. It can be expressed as follows:

$$D\equiv {\displaystyle \frac{{\int }_{t_s}^{t_a}[P(t)-P(t_s)]dt}{t_a-t_s}}$$

(7)

3. Case Analysis and Research

3.1. Data Collection and Analysis

The Xi’an Metro is considered an example in this study. The aforementioned method is applied to the evolution of important stations. Xi’an is located in the middle of the Guanzhong Plain in China, bordering the Wei River in the north and the Qinling Mountains in the south. The Xi’an Metro has 153 stations and eight operating lines: Line 1, Line 2, Line 3, Line 4, Line 5, Line 6, Line 9, and Line 14. These stations and operating lines have been functioning for almost 10 years (2011–2022). The core area of the XURTN is less affected by the geographical environment. The XURTN is divided into six periods according to the opening sequence of each line. Figure 5 shows the network opening times, station increments, and other time expansion and spatial distribution information.

In this study, the existing historical passenger flow data were used to identify the peak hours in the study city by statistical analysis through a Gaussian clustering algorithm for clustering the full-day operation hours. The dataset used in this case is provided by the AFC system and partial voting data of urban rail transit in Xi’an for the time range of 6:00–24:00. PostgreSQL cleaning and denoising were performed on the dataset. Based on a previous study, this study mainly uses the morning and evening peak data from 2011 to 2021 (morning: 8:00–9:00; 18:00–19:00) for network modeling and analysis.

The evolution of the XUTRN is divided into six stages according to the opening sequence of subway lines.

Table 6. Time expansion and spatial distribution of the XURTN opening times and station increments.

Period	Lines Operated	Date	Sections	M	A-PF ×104	M-PF ×104	E-PF ×104
I (The single-line operation)	The first period of L-2	September 2011	1–17 (BKZ-HZZ)	16	14.76	1.56	1.54
Ⅱ (The “cross” operation)	The first period of L-1; The second period of L-2	September 2013 June 2014	18–21 (HWZ-FZC) 22–39 (SY-WQN)	38	93.91	13.91	14.60
Ⅲ (The basic skeleton of the XURTN is built)	L-3	November 2016	40–63 (YHZ-BSQ)	63	160.81	14.42	14.74
Ⅳ (The parallel line is opened)	L-4	December 2018	64–88 (HTX-BGC)	90	253.70	22.44	18.08
Ⅴ (The airport line is opened)	The second period of L-1; L-14	September 2019	64–88 (FHS-FDZ) 93–100 (JCX-BGC)	102	262.40	23.92	20.56
Ⅵ (The number of stations increased by more than 50%)	L-5 L-6 L-9	December 2020	101–128 (MTK-CXG) 129–139 (GJY-XGD) 140–153 (FZC-QLX)	158	427.06	29.66	23.74

and Figure 6 illustrate the period division, number of stations (N), number of edges (M), annual average daily passenger flow (D-PF), annual average morning passenger flow (M-PF), and evening passenger flow (E-PF) distribution in the network from 2011 to 2021.

In order to develop a network model that considers the passenger flow as a weight, the 153 stations opened by Xi’an Metro are coded in the order of opening. It is important to mention that the transfer station continues to use the code of the previous period. For example, THM is coded as 34 in Period II and skipped in Period III, thus modeling the topological network of the XURTN. Moreover, based on the previous study, the linear relationship between the number of stations N and the number of edges M is determined as M = 1.047N − 2.437 and R² = 0.9999. The relationship between N and the average morning peak passenger flow volume is verified (y = −11.272x² + 3851.6x − 30,638, and R² = 0.9961). The relationship between N and the average evening peak passenger flow volume is also verified (y = −11.946x² + 3508.8x − 23,510, and R² = 0.9932).

3.2. Node Importance Evolution Analysis

This paper incorporates the feature vector centrality to measure the quality of connections between nodes. Using the proposed ranking algorithm, the ideal $S_i^w$ value is obtained through the node importance based on the complex network centrality index and reliability index. The higher its value, the higher the importance. Finally, the ranking set of node importance under different methods ($S_i^w,S_i,P_i$) is obtained. In general, the identification results obtained by the different methods have mainly the same trend, as shown in Figure 7a. In other words, when the ratio increases, the number and proportion of hub nodes significantly decrease from Period I to Period II and then become stable.

It can be seen from Figure 7b that the proportion of important stations in the morning and evening peaks is at a high level (almost 60%) in the first period of single-line operation. From the second phase onwards, it is maintained between 8% and 17%. In contrast to the continuous growth of the number of hub nodes, the proportion of hub nodes in the whole network is relatively stable. It can be seen that in the early stage when the network structure is relatively simple, more important nodes can be identified using the site passenger flow loss and traditional unweighted topological network for important node identification. However, from the fourth period onwards, the proposed algorithm gradually outperforms the other methods. Thus, it can be deduced that, when the complexity of the network further increases, the weighted comprehensive important node identification method will further lead to better results.

To clearly describe the evolution of the importance distribution of nodes, the spatial distribution of the results obtained in the latter three periods is visualized while considering the morning peak passenger flow weighting as the analysis object, as shown in Figure 8, Figure 9 and Figure 10. It can be seen that in general, the weighted comprehensive method can identify a wider range of important stations. For Period VI, the red area on the east side in Figure 8 is deeper and has a larger surrounding diffusion area compared with the result shown in Figure 9. This denotes a significant increase in the important nodes compared with the result shown in Figure 10, which is due to the fact that it takes into account the topology of the network and highlights the importance of passenger flow weighting rather than simply adding them.

Table 7. Stations with the top 10 node importance values and total passenger flow volume.

(a) Top 10 of $M$-$S_i^w$, $S_i^w$ and Pi values of the morning peak in Period IV
Number	Weighted				Unweighted				Passenger flow
	ID	M-Siw	Station		ID	Si	Station		ID	Pi	Station
1	10	1.000	BDJ		10	1.000	BDJ		15	1.000	XZ
2	31	0.930	WLK		34	0.952	THM		10	0.707	BDJ
3	34	0.908	THM		31	0.874	WLK		45	0.561	JXC
4	15	0.876	XZ		15	0.868	XZ		34	0.550	THM
5	46	0.871	DYT		46	0.842	DYT		44	0.516	TBNL
6	33	0.659	KFL		4	0.624	XZZX		16	0.456	WYJ
7	32	0.599	CYM		52	0.534	HJM		43	0.437	KJL
8	45	0.501	JXC		30	0.508	SJQ		17	0.424	HZZX
9	9	0.493	AYM		32	0.505	CYM		31	0.373	WLK
10	30	0.486	SJQ		53	0.482	SJJ		46	0.356	DYT
(b) Top 10 of $M$-$S_i^w$,$S_i^w$ and Pi values of the morning peak in Period V
Number	Weighted				Unweighted				Passenger flow
	ID	M-Siw	Station		ID	Si	Station		ID	Pi	Station
1	10	1.000	BDJ		10	1.000	BDJ		15	1.000	XZ
2	31	0.922	WLK		34	0.858	THM		10	0.727	BDJ
3	15	0.883	XZ		31	0.808	WLK		34	0.612	THM
4	34	0.823	THM		15	0.784	XZ		45	0.585	JXC
5	46	0.757	DYT		46	0.743	DYT		44	0.533	TBNL
6	4	0.607	XZZX		4	0.737	XZZX		43	0.451	KJL
7	32	0.545	CYM		30	0.534	SJQ		16	0.439	WYJ
8	30	0.535	SJQ		9	0.489	AYM		31	0.416	WLK
9	9	0.530	AYM		29	0.485	YXM		46	0.415	DYT
10	33	0.527	KFL		52	0.474	HJM		17	0.408	HZZX
(c) Top 10 of $M$-$S_i^w$, $S_i^w$ and Pi values of the morning peak in Period VI
Number	Weighted				Unweighted				Passenger flow
	ID	M-Siw	Station		ID	Si	Station		ID	Pi	Station
1	34	1.000	THM		10	1.000	BDJ		10	1.000	BDJ
2	10	0.987	BDJ		34	0.949	THM		34	0.944	THM
3	31	0.913	WLK		31	0.912	WLK		43	0.933	KJL
4	13	0.839	NSM		13	0.827	NSM		15	0.847	XZ
5	15	0.821	XZ		15	0.799	XZ		16	0.629	WYJ
6	43	0.724	KJL		46	0.733	DYT		17	0.581	HZZX
7	46	0.695	DYT		74	0.696	JZ·L		4	0.577	XZZX
8	74	0.659	JZ·L		43	0.693	KJL		31	0.540	WLK
9	48	0.607	QLS		109	0.656	XBGY		30	0.514	SJQ
10	109	0.607	XBGY		48	0.650	QLS		29	0.499	YXM

presents the top 10 important stations identified in the last three periods under the early peak moment, which allows us to further compare the evolutionary characteristics of different methods. It can be clearly seen from Table 7 and Figure 9 that in the unweighted network, the important stations in Period IV are mainly concentrated in the BDJ, THM, and WLK stations, where the BDJ station with the highest $S_i$ value is the first important station in the network. After L-14 is opened for operation, the importance area with a high combined value in Period V starts to extend toward the airport. Its evolution shows that the opening of new lines leads to important changes in the topological characteristics of the network, and its transfer stations often have a greater degree of influence compared with the standard stations. In addition, considering the passenger flow and POI points, the distribution of the important nodes shown in Figure 8 will significantly change, and stations having high passenger flow are usually commercial and entertainment centers (XZ), transportation hubs (BDJ and THM), and tourist attractions (DYT). In period IV, the XZ station, which ranks first in passenger flow, ranks fourth in weighted importance due to its low topological importance, while WLK, which has a low passenger flow, ranks in the top 2. In summary, the ranking results significantly vary in the same period and under different circumstances. However, this study considers the topology of the network while also incorporating its morning and evening passenger flow.

To further explore the evolution of hub nodes during the morning and evening peak hours, the importance scores of the nodes at each stage were mapped to the whole urban space, as shown in Figure 11 and Figure 12. It can be observed that the evolutions have many similarities. In particular, the hub nodes in Period I are highly consistent with the urban development core. As the network reaches Period II, the current distribution of hub nodes rapidly narrows to the intersections of the network, which is highly consistent with the role assumed by the transfer stations in the topological network structure. In the later periods of network expansion, the red portion appears mainly in the red areas of the network intersecting parts. After the network is developed, the number of important nodes increases and the importance of the surrounding nodes also increases, as shown in Periods V and VI. When the number of red dots increases, the yellow–green area around the red dots also gradually expands. Finally, these important nodes, having a score greater than 0.5, are clustered into fragments, and the area also overlaps with the center of gravity of the core development zone of the city. This indicates that the evolution of the network nodes and the development direction of the city promote and complement each other.

It can also be seen from Table 6 that the top ten stations basically belong to commercial and entertainment center complexes, important transportation hubs, and typical residential-type stations. In addition, their important stations have only slight differences between the morning and evening peaks, except for the overall situation of individual nodes with complex configurations and requirements, whose importance evolution should be further analyzed according to specific circumstances. A detailed analysis example of the KFL and TBNL stations will be presented in a future study.

The KFL station is a typical important station combining commercial and financial land with medical institutions. Its spatial mapping distribution is shown in Figure 11. Period VI, Direction 1, No. 33 ranks 11th in the morning peak $S_i^w$ = {0.425} and 19th in the evening peak $S_i^w$ = {0.515}, which is mainly due to the influence of large hospitals. In fact, when they start to receive medical treatment in the morning, more patients, medical staff, and family members will flock to the metro station, creating a passenger flow peak. Although there are major periods of visiting and accompanying patients in its vicinity in the evening, its morning peak node is more important due to several factors such as the work hours.

The TBNL station is a typical combination of residential and working land. It is directly connected to an undergraduate university. The mapping of its spatial distribution is shown in Figure 12. Period VI, Direction 3, No. 44 ranks 22nd in the morning peak $S_i^w$ = {0.386} and 14th in the evening peak $S_i^w$ = {0.425}. This is due to the fact that the surrounding buildings are mainly apartments and family homes of various units, and university students basically live on campus. The distance to and from work, which is close to the morning peak travel, is relatively unfocused. However, affected by the topology of the neighboring node (JXC) and commercial entertainment complex (XZ), a large number of students and off-duty people choose to shop, dine, and entertain in this area, which leads to the greater importance of the evening peak node.

It can also be seen from Figure 7a that the range of important stations is larger in the morning peak compared with the evening peak. In particular, by comparing the results of Figure 11 and Figure 12, it can be deduced that there is a darker color in the middle of the network near the eastern branch. This is due to the concentrated travel of passengers during the morning peak and the wide distribution of residential land, while the purpose of travel during the evening peak is more abundant than that of the morning peak, and the passenger flow is dispersed in different directions. In summary, it can be concluded that the XURTN exhibits geographical variability at different times of the day, and its complexity is more prominent in the morning peak hours than in the evening peak hours.

3.3. Correlation Evolutionary Analysis

Based on the relevant analysis approach of the proposed methodology, the (108.954094, 34.259568) location is used as the center of the circle to divide the city into four directions, and the radius length difference between adjacent circles is set to 1000 m in order to divide Xi’an into multiple circles at different development stages, as shown in Figure 13.

The land use intensity of each quadrant, obtained using the combined weighted measure of $P(r)$ and $R(r)$ from Figure 14, shows that the development status of the city in each direction is uneven. Therefore, it is necessary to divide the city into multiple orientations, count the relationship between the radius of different circles in each quadrant and the land use intensity and important nodes in the circles, and conduct linear regression in a double logarithmic coordinate system. Figure 15 shows the results of the $S^w(r)$ function of r in Period VI.

The distribution of land use intensity and important nodes within each area is also separately calculated for the last three periods of the evening peak weighting. The obtained results are shown in

Table 8. Matching degree values of important node evolution and land use evolution of the XURTN.

Period	${\mathit{D}}_{\mathit{l}1}$	${\mathit{D}}_{\mathit{l}2}$	${\mathit{D}}_{\mathit{l}3}$	${\mathit{D}}_{\mathit{l}4}$	${\mathit{D}}_{\mathit{s}1}$	${\mathit{D}}_{\mathit{s}2}$	${\mathit{D}}_{\mathit{s}3}$	${\mathit{D}}_{\mathit{s}4}$	${\mathit{M}}_{\mathit{I}}$
IV	1.071	1.865	1.101	1.219	0.483	0.465	0.214	0.078	0.656
V	1.039	0.660	1.069	1.188	0.498	0.508	0.199	0.065	0.724
VI	0.974	0.660	0.906	1.105	0.528	0.353	0.348	0.375	0.790

. It can be seen that $D_{li}$ is smaller than 2 and gradually decreases in all the periods. This indicates that the evolution of urban land use intensity in Xi’an gradually decreases from the city center to the periphery in the area of orientation $i$. $D_{si}$ is also less than 2. The degree of distribution of the important nodes of the rail transit line network then decreases from the city center to the periphery in the area of urban orientation $i$. The smaller its value, the faster the decrease. However, in Period VI, when Lines 5 and 6 are opened, the center of gravity and the number of important nodes in the XURTN evolve southward, as $D_{s3} \mathrm{a}\mathrm{n}\mathrm{d}$ $D_{s4}$ show a significant increase. In addition, all the $M_I$ are less than 1 and gradually increasing, which indicates that the match between the evolution of the important nodes of the XURTN and urban land use intensity is increasingly fitting with the increase in the stage over time.

In summary, the evolution of node importance in complex rail transit networks is affected by various social or political factors such as the changes in topological complexity, dynamic spatial and temporal characteristics of passenger flows, and urban planning of rail transit. Some stations pose a challenge in determining their importance solely through topology or passenger flow analysis. A comprehensive qualitative and quantitative assessment becomes necessary, taking into account various factors to perform a well-rounded evaluation. In addition, there is a beneficial and reciprocal feeder relationship between the urban rail transit line network and land use. The purpose of travel, population distribution, and land use attributes are key components for determining the normal operation of a city, especially in the morning and evening peak hours, which has a strong constraint on the nodal importance and land use intensity evolution of urban rail transit networks.

3.4. Evolutionary Scheme Verification

An attack strategy suitable for the presented study is derived from the resilience verification curve principle introduced in the methodology. In the network disruption scenario, six network node attack strategies are used: node weighted betweenness, node degree, comprehensive weighted importance, comprehensive unweighted importance, node eigenvector centrality rankings, and random attack, respectively. And network performance is characterized by the largest relative subgraph size (LRS). The maximum connectivity subgraph in this study is the largest connected portion of the network, i.e., the largest set of nodes in the graph after the network has been disrupted, where any two nodes can be connected to each other by a path.

Figure 16 shows the results obtained by taking the evening peak as an example. It can be seen that in most scenarios, the strategy of the comprehensive weighted importance ranking method ($E$-$S_i^w)$ results in the fastest degradation of network performance, which proves that the hub nodes obtained by the methodology described in this paper have a profound impact on network performance.

In addition, the simulated node attack shows that the XURTN is robust to random attacks, and even if many nodes are destroyed, it does not necessarily lead to the paralysis and collapse of the network. However, if many key nodes are attacked, the whole network will quickly collapse. This analysis further shows that the XURTN exhibits clear scale-free and heterogeneous properties. This emphasizes the significance of studying URTN to identify important nodes within the network.

4. Conclusions

This study first reviewed the development history of the XURTN since the beginning of its operation. Based on the study of full-day passenger flow, the morning and evening peaks were then used as the weights for the development of the passenger flow weighting network. The centrality and reliability indexes of each node within the network, including $D_i$ and $D_i^w$; $E_i^w$, $C_i$ and $C_i^w$; $B_i$ and $B_i^w$; and $I_i$ and $Q_i$ were calculated. The constructed node location importance index ($R_i$) was also calculated. The weighted comprehensive ranking algorithm, which combines the CRITIC and WCPCA methods, was incorporated into the unweighted comprehensive importance ranking set, the weighted comprehensive importance ranking set, and the station traffic loss ranking set that were calculated and analyzed. Afterward, the similarity and dissimilarity of the results under the three different node importance identification methods were evaluated. From the results of the comparison of the three methods, it can be seen that the weighted comprehensive method proposed in this study can identify important sites that have been overlooked by previous research methods. The evolution mechanism of important nodes during the morning and evening peak hours was qualitatively and quantitatively studied. The importance and heterogeneity of using morning and evening peak passenger flows as the weighting network weights were further illustrated. On this basis, the correlation between the distribution of important nodes and the evolution of urban land use intensity of the XURTN was detailed. Moreover, a corresponding network attack verification strategy was developed based on the principle of the resilience verification curve for the proposed comprehensive method of important node identification. Although the case study investigated in this study is based on the metro network of Xi’an, the methodology used takes into account the network structure, station locations, and dynamic passenger transfer effects, which are generic for other cities.

Furthermore, in the development of the rail network, the evolution of the importance of different stations is affected by changes in the network topology and passenger flow to different degrees. In other words, the node importance changes are more likely to be affected by the topology compared with the passenger flow, while the opposite holds for others. It is necessary to conduct a specific analysis in the context of the actual situation. As for the network, the XURTN continues to evolve and grow with increasing connectivity and reliability. The network in the urban core is dense, it is extended to the peripheral areas, and it becomes more complete as the number of lines continues to increase. This paper provides a reference for studying the evolution mechanism of node importance in the development of the XURTN. The obtained results have important theoretical and practical significance for the stable operation and future development of the XURTN.

The future research plan of this study is as follows: (1) Consider introducing the evaluation factors of the convenience of the station’s public transportation connection when establishing the comprehensive node importance index system, and then carry out an all-round assessment combined with other transportation modes to comprehensively analyze the evolution mechanism and law of the important stations along with the development of the urban transportation network. (2) Subsequently, we will refine the calculation of the intensity of the land use, collect more data around the metro stations, and categorize the different types of stations to study the evolution law of the importance of different types of stations. We will also collect more data about the metro stations, classify the stations, and study the evolution of the importance of different types of stations.