An Analysis Method for Air-High-Speed ​​Rail Intermodal Transport Network

Abstract

This invention discloses a three-step method for analyzing air-high-speed rail intermodal transport networks: topology analysis, centrality measurement, and cluster analysis. First, a directed weighted network is constructed to compare the complex characteristics of the intermodal network and its sub-networks. Results show that, in addition to small-world characteristics, the network exhibits wide-area scale characteristics, with a rapidly decreasing degree distribution growth rate. Due to hub capacity limitations, the network deviates from a scale-free network. Centrality analysis reveals the increasing importance of secondary hubs, which are strategic transit points connecting major cities and surrounding areas. Cluster analysis reveals a dual clustering structure in the complex network: the air network supports national connectivity, while the high-speed rail network supports regional integration along specific corridors. This invention, employing the aforementioned air-high-speed rail intermodal transport network analysis method, combines the centralized structure of the entire air network with the decentralized, regionally centralized structure of the high-speed rail network, which is of significant importance for transportation planning and regional development.

Description

Technical Field

[0001] This invention relates to the field of air-rail intermodal transport technology, and in particular to a method for analyzing air-high-speed rail intermodal transport networks. Background Technology

[0002] Air-high-speed rail intermodal transport is a strategic integration of the air and high-speed rail systems, leveraging the advantages of both modes of transport to provide passengers and cargo with a seamless, efficient, and cost-effective transportation solution. This mode of transport fully utilizes the advantages of air transport—its speed and suitability for long-distance and high-value transshipment—while also taking advantage of the wide coverage and high carrying capacity of high-speed rail, especially on short- and medium-distance routes. By implementing air-high-speed rail intermodal transport, passengers can enjoy convenient and smooth transfers between airports and train stations, significantly saving time and costs. This model not only improves the market coverage and customer retention rates of airlines and railway companies but also alleviates the intense competition between the two modes of transport, promoting a shift from competition to cooperation.

[0003] In recent years, China has made significant progress in promoting air-high-speed rail intermodal transport. As of February 2024, the 12306 air-high-speed rail intermodal service has integrated the ticketing systems of major airlines such as Air China, China Eastern Airlines, and China Southern Airlines. Currently, the service covers 78 cities and more than 2,000 routes nationwide, marking a significant milestone in the development of intermodal transport. Despite these advancements in practice, there is still an urgent need to understand the basic structural characteristics and evolution patterns of China's air-high-speed rail intermodal transport network. Previous studies have mainly focused on analyzing air and high-speed rail networks separately, often neglecting the complex intermodal connections and their impact on network topology and regional connectivity. Summary of the Invention

[0004] The purpose of this invention is to provide a method for analyzing air-high-speed rail intermodal transport networks. It comprehensively analyzes these networks using complex network theory, examining the structure of subnetworks and the overall air-high-speed rail network to ensure consistency in data processing and analysis methods. By comparing the topological characteristics of the air-high-speed rail network and its subnetworks in the same spatiotemporal dimensions, the latest topological features and trends are discovered, including network topological characteristics, evolution patterns, centrality metrics, and cluster analysis. This provides policy recommendations for future infrastructure development and network optimization, offering insights to improve network connectivity, efficiency, and resilience.

[0005] To achieve the above objectives, this invention provides a method for analyzing air-high-speed rail intermodal transport networks, comprising the following steps:

[0006] Step S1: Construct a directed weighted network;

[0007] Step S2: Select network topology indicators;

[0008] Step S3: Analyze the network topology characteristics and verify whether the network has small-world properties;

[0009] Step S4: Calculate the network centrality index and perform various centrality analyses;

[0010] Step S5: Cluster analysis.

[0011] Preferably, in step S1, constructing the directed weighted network specifically involves:

[0012] In the high-speed rail sub-network, the city where the station is located is taken as the node. When the same high-speed train stops at two stations, it indicates that there is a connection between the two cities.

[0013] In the aviation subnetwork, the city where the airport is located is used as a node. When there are flights between two nodes, it indicates that there is a connection between the two cities.

[0014] In a directed weighted network, the edges from city A to city B, and from city B to city A, are set as two independent directed connections.

[0015] Preferably, in step S2, nodes, edges, average degree, average weighted degree, network diameter, modularity, average clustering coefficient, and average path length are selected as network topology indicators to describe the network structure and topological characteristics of the directed weighted network.

[0016] Preferably, Gephi software is used to calculate network topology metrics, specifically:

[0017] A node represents a single entity or point connected by an edge. A node is determined as follows: if a city has both a high-speed rail station and an airport, or if a city has more than one airport or high-speed rail station, then the city itself is considered a node.

[0018] An edge represents the relationship or interaction between nodes, and the number of edges is the total number of direct connections between nodes in a directed weighted network.

[0019] Average degree is used to measure the average number of connections of each node in a directed weighted network. It is calculated by adding the degrees of all nodes and then dividing by the total number of nodes. The degree of a node refers to the number of edges connected to that node.

[0020] The average weighted degree is calculated by adding the weighted degrees of all nodes and then dividing by the number of nodes. The weighted degree of a node refers to the sum of the weights of the edges connected to that node.

[0021] The network diameter represents the longest and shortest path between any two nodes in the network. It is calculated by determining the shortest path between all pairs of nodes and the maximum length of these paths.

[0022] The modularity level is measured by the degree to which the network is divided into modules or communities. A highly modular network has dense connections between nodes within modules or communities, while connections between nodes in different modules or communities are sparse.

[0023] The average clustering coefficient is calculated as follows: it is calculated from the average of the clustering coefficients of all nodes. The clustering coefficient of a single node is the ratio of the number of connections between its neighboring nodes to the maximum possible number of connections. The clustering coefficient of a node is used to measure the degree of connection between its neighboring nodes and to describe the overall trend of nodes clustering together in the network.

[0024] The average path length refers to the average number of steps taken by all possible node pairs along the shortest path in the network. It is calculated by adding the shortest path lengths between all node pairs and then dividing by the number of such node pairs.

[0025] Preferably, in step S3, the network topology characteristics of the aviation sub-network, high-speed rail sub-network, and aviation-high-speed rail intermodal network are analyzed based on network topology indicators to verify whether each network has small-world properties. This specifically includes the following steps:

[0026] S31. Draw a degree distribution map of all nodes;

[0027] S32. Draw a cumulative degree double logarithmic distribution plot based on the node degree distribution;

[0028] S33. Use Python and NetworkX to generate a random network with the same number of nodes, directed edges, and average weights as the existing network, and calculate the average path length and clustering coefficient of the random network.

[0029] Preferably, in step S4, Gephi software is used to calculate the network centrality index of the aviation subnetwork, the high-speed rail subnetwork, and the aviation-high-speed rail intermodal network to evaluate the relative importance of nodes. The network centrality index includes degree centrality, weighted degree centrality, betweenness centrality, harmonic proximity degree centrality, and eigenvector centrality.

[0030] Preferably, in step S4, the degree centrality is calculated based on the number of direct connections of a node, and the calculation formula is as follows:

[0031] C D (v) = in-deg(v) + out-deg(v);

[0032] Where in-deg(v) is the number of edges entering node v, and out-deg(v) is the number of edges leaving node v;

[0033] The formula for calculating weighted degree centrality is:

[0034] Cw i)=Σ j∈N(i) w ij ;

[0035] Where N(i) is the set of nodes i, and W ij It is the weight of the edge between nodes i and j;

[0036] The formula for calculating betweenness centrality is:

[0037]

[0038] Where, σ st σ is the total number of shortest paths from node s to node t. st (v) is the number of all shortest paths, the sum of which is taken from the sum of all node pairs s and t, excluding node v itself;

[0039] The formula for calculating harmonic proximity centrality is:

[0040]

[0041] Where d(u,v) is the shortest path distance between nodes u and v; if u and v are not connected, then d(u,v) is considered infinite. It is zero;

[0042] The formula for calculating the eigenvector centrality is:

[0043]

[0044] Among them, C E (v) is the eigenvector centrality of node v, λ is a constant, and A is the largest eigenvalue of the adjacency matrix. uv This represents the connection between nodes u and v.

[0045] Preferably, in step S4, the K-means clustering method in the XLSTAT statistical software is used, and cluster analysis is performed on the nodes in the network based on the calculated centrality index.

[0046] Preferably, in step S5, the silhouette coefficient score is used to evaluate the clustering quality. The silhouette coefficient score is used to evaluate the quality of clustering results in data analysis. The evaluation includes the similarity of an object to other clusters and to its own cluster. The higher the score, the clearer the clustering.

[0047] Therefore, the present invention employs the above-mentioned air-high-speed rail intermodal transport network analysis method, and the beneficial effects are as follows:

[0048] (1) This invention, through topological analysis, demonstrates that the air-high-speed rail intermodal network and its subnetworks exhibit small-world characteristics, characterized by high clustering coefficients and short average path lengths. Furthermore, these networks also exhibit wide-area network characteristics, where the degree distribution growth rate rapidly decreases after specific boundary points. This finding indicates that, due to capacity constraints at major hubs, the network deviates from the traditional scale-free model. Unlike purely scale-free networks with unrestricted growth, the capacity constraints of large hubs in the integrated network limit further expansion, resulting in a more balanced network structure. Notably, the connectivity and overall efficiency of the integrated network surpass those of the individual air and high-speed rail subnetworks, demonstrating the advantage of intermodal transport methods in achieving robust connectivity between countries and regions.

[0049] (2) This invention uses a set of five centrality indicators to measure hub cities in a network. This method is different from the traditional indicators used in previous literature and provides a more suitable method for evaluating hub cities and their hierarchical status in the network.

[0050] (3) This invention identifies different regional clusters in the air-high-speed rail intermodal transport network through cluster analysis, which further enriches the understanding of the network structure. The analysis highlights the advantages of major cities and intermodal hubs, and reveals the dual cluster mode and hybrid module structure. Among them, air hubs serve as important nodes in the national network connection, while high-speed rail nodes support regional integration along economic corridors. The identification of these clusters and the roles played by different cities provide key positions for the hierarchical and regional structure of the integrated network.

[0051] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0052] Figure 1 This is an overall flowchart of an embodiment of the air-high-speed rail intermodal transport network analysis method of the present invention;

[0053] Figure 2 This is a topology diagram of the aviation sub-network of an embodiment of the air-high-speed rail intermodal transport network analysis method of the present invention;

[0054] Figure 3 This is a degree distribution diagram of the aviation sub-network of an embodiment of the air-high-speed rail intermodal transport network analysis method of the present invention;

[0055] Figure 4 This is a log-cumulative degree distribution diagram of the aviation sub-network of an embodiment of the air-high-speed rail intermodal transport network analysis method of the present invention;

[0056] Figure 5 This is a topology diagram of the high-speed rail sub-network of the aviation sub-network, which is an embodiment of the air-high-speed rail intermodal transport network analysis method of the present invention.

[0057] Figure 6 This is a degree distribution diagram of the high-speed rail sub-network, which is an embodiment of the air-high-speed rail intermodal transport network analysis method of the present invention.

[0058] Figure 7 This is a log-cumulative degree distribution diagram of the high-speed rail sub-network, which is an embodiment of the air-high-speed rail intermodal transport network analysis method of the present invention.

[0059] Figure 8 This is a topology diagram of an air-high-speed rail intermodal transport network according to an embodiment of the air-high-speed rail intermodal transport network analysis method of the present invention;

[0060] Figure 9 This is a degree distribution map of an air-high-speed rail intermodal transport network, according to an embodiment of the air-high-speed rail intermodal transport network analysis method of the present invention.

[0061] Figure 10 This is a log-cumulative distribution diagram of an air-high-speed rail intermodal transport network, according to an embodiment of the air-high-speed rail intermodal transport network analysis method of the present invention.

[0062] Figure 11 This refers to the silhouette coefficient score in the cluster analysis of the aviation sub-network of an embodiment of the air-high-speed rail intermodal transport network analysis method of the present invention.

[0063] Figure 12 This refers to the silhouette coefficient score in the high-speed rail sub-network clustering analysis of an embodiment of the air-high-speed rail intermodal transport network analysis method of the present invention;

[0064] Figure 13 This refers to the silhouette coefficient score in the cluster analysis of the air-high-speed rail intermodal transport network, which is an embodiment of the air-high-speed rail intermodal transport network analysis method of the present invention. Detailed Implementation

[0065] The technical solution of the present invention will be further described below with reference to the accompanying drawings and embodiments.

[0066] Unless otherwise defined, the technical or scientific terms used in this invention shall have the ordinary meaning as understood by one of ordinary skill in the art to which this invention pertains.

[0067] This embodiment simulates the real-world intermodal transport network as much as possible through topology analysis of aviation and high-speed rail networks, focusing on connections and path directions between nodes, and considering weights such as the frequency and number of trains or flights.

[0068] like Figure 1 As shown, an air-high-speed rail intermodal transport network analysis method includes the following steps:

[0069] Step S1: Unlike previous undirected models, this embodiment constructs a directed weighted network for the first time to analyze the air-high-speed rail intermodal transport network and its sub-networks. The construction of the directed weighted network is as follows:

[0070] In the high-speed rail sub-network, the city where the station is located is taken as the node. When the same high-speed train stops at two stations, it indicates that there is a connection between the two cities.

[0071] In the aviation subnetwork, the city where the airport is located is used as a node. When there are flights between two nodes, it indicates that there is a connection between the two cities.

[0072] In a directed weighted network, the edges from city A to city B, and from city B to city A, are set as two independent directed connections.

[0073] In directed networks, each connection between stations reflects the specific direction of flight and train services, which is crucial for capturing passenger flow and network structure behavior. For high-speed rail subnetworks, this approach provides a more nuanced understanding of the role of each station; they are not merely points along the line, but key nodes with different inbound and outbound connections. This perspective reveals subnetworks originating from or converging at each station, thus more clearly demonstrating the network's hierarchical structure and the strategic importance of certain lines. Therefore, the network to be constructed is a directed weighted network.

[0074] Step S2: Select network topology indicators;

[0075] In complex networks, nodes, edges, average degree, average weighted degree, network diameter, modularity, average clustering coefficient, and average path length are selected as network topology indicators to describe the network structure and dynamic characteristics of directed weighted networks.

[0076] In a network, a node (also called a vertex) represents a single entity or point connected by edges. In this embodiment, cities are used as nodes because this embodiment primarily studies the accessibility between cities and the topological characteristics of the entire network to address how travelers choose to travel to a destination city, rather than the accessibility between high-speed rail stations or airports. Furthermore, there are no line connections between different airports within the same city. Although there are different high-speed rail stations within the same city (such as Tianjin Station and Tianjin South Station), these stations are designed to gather or transport passengers within the same city, simply to provide more options for travelers from larger cities.

[0077] Therefore, in this embodiment, the method for determining a node is as follows: if a city has both a high-speed rail station and an airport, or if a city has more than one airport or high-speed rail station, then the city itself is considered a node.

[0078] An edge (or link) is a connection between nodes. These edges represent the relationship or interaction between nodes. The number of edges is the total number of direct connections between nodes in a directed weighted network. Edges can be undirected (if the connection is bidirectional) or directed (if the connection has a specific direction).

[0079] Average degree is used to measure the average number of connections of each node in a directed weighted network. It is calculated by adding the degrees of all nodes and then dividing by the total number of nodes. The degree of a node refers to the number of edges connected to that node.

[0080] In a weighted network, each edge has a weight, representing the strength or capacity of the connection. The average weighted degree is calculated by summing the weighted degrees of all nodes and then dividing by the number of nodes. The weighted degree of a node refers to the sum of the weights of the edges connected to that node. This embodiment considers both high-speed rail services and flight numbers in the sub-networks. In the integrated network, if there are both flights and high-speed rail services from city A to city B, the total number of services is used as the directional weight for the integrated network.

[0081] The network diameter represents the longest and shortest path between any two nodes in a network. This metric measures the size of the network based on the maximum distance between nodes. It is calculated by determining the shortest path between all pairs of nodes and the maximum length of these paths.

[0082] The modularity level is measured by the degree to which a network is divided into modules or communities. A network with a high degree of modularity has dense connections between nodes within modules or communities, while connections between nodes in different modules or communities are sparse. This is used to detect the community structure in the network, and a high degree of modularity indicates a strong community structure.

[0083] The average clustering coefficient is calculated as follows: it is calculated from the average of the clustering coefficients of all nodes. The clustering coefficient of a single node is the ratio of the number of connections between its neighboring nodes to the maximum possible number of connections. The clustering coefficient of a node is used to measure the degree of connection between its neighboring nodes and to describe the overall trend of nodes clustering together in the network.

[0084] Average path length refers to the average number of steps taken by all possible node pairs along the shortest path in a network. It is calculated by adding the shortest path lengths between all node pairs and then dividing by the number of such node pairs, reflecting the traffic efficiency in the network.

[0085] Data collection:

[0086] The study area is limited to China (excluding data from Taiwan Province), encompassing 187 cities with airports and 252 cities with high-speed rail stations. This embodiment uses data from 2019, with air network data sourced from OAGAnalyser and high-speed rail network data from the 12306 China Railway Network (2019). High-speed rail lines are defined as newly built railway lines with an average speed of 250 km / h or higher and renovated railway lines with an average speed of 160 km / h or higher, operating G, D, and C type trains (referred to as high-speed rail). Based on this, high-speed rail timetables for December 12, 2019, were collected. For air transport, city pairs with an average of one or more flights per day in 2019 were selected.

[0087] The network topology indicators were calculated using Gephi software (as shown in Table 1). The air-high-speed rail intermodal network consists of 328 city nodes and 6,675 edges. Among them, the high-speed rail sub-network consists of 252 cities and 5,210 routes; and the air sub-network consists of 187 cities and 2,451 direct routes.

[0088] Table 1 Network Topology Indicators

[0089] Aviation Network High-speed rail network Comprehensive network node 187 252 328 side 2451 5210 6675 average degree 13.04 19.58 29.03 Average weighting 54.02 153.52 234.96 Network diameter 4 5 4 Modular 0.127 0.496 0.45 Average clustering coefficient 0.55 0.51 0.71 Average path length 2.38 2.26 2.19

[0090] Step S3: Analyze the network topology characteristics of the aviation sub-network, high-speed rail sub-network, and aviation-high-speed rail intermodal network based on network topology indicators, and verify whether each network has small-world properties. This includes the following steps:

[0091] S31. Draw a degree distribution map of all nodes;

[0092] S32. Draw a cumulative degree double logarithmic distribution plot based on the degree distribution;

[0093] S33. Use Python and NetworkX to generate a random network with the same number of nodes, directed edges, and average weights as the existing network, and calculate the average path length and clustering coefficient of the random network.

[0094] 1) Network topology characteristics analysis in the aviation subnetwork:

[0095] The aviation subnetwork, comprised of all cities with airports in China (excluding Taiwan), contains 187 nodes and 2451 edges. The average degree is 13.043, indicating that each city is directly connected to approximately 13 other cities on average. The average weighted degree is 54.021, demonstrating a significant improvement in overall connectivity after considering weighting (flight frequency), with an average of 4 flights per day from each city to every connected city. The network diameter of 4 indicates that the longest and shortest path between any two cities in the network is 4 steps, suggesting that all cities in the aviation network require a maximum of 3 transfers. The average path length of 2.377 indicates that reaching any other city from another city requires slightly more than 2 steps on average, i.e., one stopover, indicating high connectivity within the aviation network.

[0096] The clustering coefficient focuses on the connectivity of a network and the tendency of nodes to form tightly connected clusters. The average clustering coefficient of an aviation network is 0.551, indicating that on average, about 55.1% of the nodes are also interconnected with neighboring cities. This means that if one airport is connected to two other airports, there is a 55.1% probability that there are direct flights between those two neighboring cities.

[0097] Modularity measures the overall structure of a network and the degree to which it is divided into communities or modules. High modularity indicates dense connections within modules but sparse connections between modules. The aviation network has a modularity score of 0.127, indicating a weak modular structure. This suggests that the grouping within the aviation network is not very clear; that is, the overall network is relatively well-connected and does not have strong divisions into independent regions or city clusters.

[0098] from Figure 2 Looking at the aviation subnetwork topology diagram, each node represents a city, node size indicates degree centrality, each edge represents a direct flight route between cities, and the color intensity of the edges indicates weight (average daily flight frequency). The color of the nodes represents different communities or groups detected in the network. The blue group includes most major hub cities, such as Beijing, Shanghai, and Shenzhen, and the connection density indicates strong connectivity between cities within the blue area. The orange group below mainly includes northwestern cities such as Xi'an, Urumqi, and Yinchuan, with Xi'an being the most obvious cluster hub. This cluster represents a regional group with high internal connectivity among these northwestern cities. Figure 2 At the very top, there is a red cluster centered on Hohhot, connecting cities in Inner Mongolia such as Chifeng, Ulanhot, and Hulunbuir, showcasing Inner Mongolia's air route network connecting internal and external cities through regional hubs. Similarly, in... Figure 2In the upper right corner, there is a yellow cluster with Kunming as its hub. This cluster mainly connects other provincial capitals of Yunnan Province, such as Lijiang and Dali, as well as some eastern cities closely connected with Yunnan Province, such as Zhengzhou and Quanzhou, through Kunming, the capital of Yunnan Province.

[0099] Plot the degree distribution of all cities in the aviation subnetwork as a degree distribution map. Figure 3 ).from Figure 3 As can be seen, a small number of nodes (hubs) have very high degrees, with only 20% of the nodes having a degree above 50 and only 10% having a degree above 82. This means that these nodes are connected to many other nodes, and these hubs play a crucial role in maintaining the connectivity and robustness of the network. Meanwhile, most nodes have relatively low degrees, with 80% of the nodes having a degree below 50, indicating relatively low connectivity.

[0100] Further plotting the cumulative distribution can more intuitively illustrate the power-law relationship. Figure 4 The double logarithmic plot of the cumulative degree distribution clearly divides into two parts. The first half shows an approximately linear relationship, while the latter half exhibits a faster decay rate in the tail distribution than power-law decay. The first part fits nodes with lower degrees very well, showing power-law behavior within this range. This segment indicates that nodes with fewer connections exhibit traditional scale-free behavior. The second segment, with higher-degree nodes, is steeper, showing different scaling behavior. This can be explained by capacity constraints or saturation effects that limit degree expansion, as seen in urban airports. Scarcity of capacity and saturation effects cause the degree distribution to truncate, deviating from the scale-free network patterns often observed in unconstrained networks. In scale-free networks, connectivity continues to grow, and some nodes acquire extremely high degrees. However, under capacity constraints and saturation, the degree distribution truncates or drops sharply after a certain point. This truncation reflects the practical limitations of network connectivity expansion in the real world, where infrastructure capacity imposes an upper limit on growth.

[0101] After fitting, the degree (K) equals 56 (K). C The cutoff point is at (), and the slope of the first linear fit is -0.497, R0. 2 The value is 0.969; the second segment is a quadratic function that opens downwards, with R... 2 The value is 0.952. It can be seen that the fitting relationship is very good in both segments. Therefore, the degree of the aerospace subnetwork no longer conforms to a single power-law distribution, no longer conforms to the characteristics of a scale-free network, but conforms to the characteristics of a wide-area scale network, that is, the degree distribution exhibits a power-law state at low degree values, while it transforms into a rapidly decaying state at high altitude values.

[0102] The degree distribution function of the aeronautical subnetwork is:

[0103]

[0104] Among them, K C =56.

[0105] To further verify whether the aerospace subnetwork exhibits small-world properties, a random network with the same number of nodes, directed edges, and average weights as the existing aerospace topology network was generated using Python and NetworkX. The average path length and clustering coefficient of the random network were calculated. The average path length of the weighted random network was 3.713, indicating that the effective distance between cities increases in a random network of the same size. The clustering coefficient was 0.060, indicating a very low tendency for node clustering in the weighted random network. Generally, if a network has a shorter average path length than a random network of the same size and a significantly higher clustering coefficient, then the network exhibits small-world characteristics. By comparing these metrics with the actual aerospace subnetwork, it can be confirmed that the actual aerospace subnetwork exhibits small-world characteristics.

[0106] 2) Network topology characteristics of the high-speed rail subnetwork;

[0107] The high-speed rail subnetwork has 252 nodes and 5210 edges, both more than the aviation subnetwork, indicating a wider coverage and more connections within China (excluding Taiwan). The average degree is 19.583, meaning each city in the high-speed rail subnetwork is connected to an average of 19 cities, higher than the aviation subnetwork's average. This reflects the linear operation mode of railways, where a single line with multiple stations directly connects multiple cities, while the aviation subnetwork primarily consists of direct flights with few stops. The average weighted degree of the high-speed rail subnetwork is 153.516, meaning there are nearly 8 trains per day on average to each connected city. This service density is also higher than the average of 4 flights per day for air transport, indicating a higher degree of urban interconnection along high-speed rail lines. The network diameter is 5, slightly larger than the aviation subnetwork's diameter, indicating that the shortest path between the farthest cities is slightly longer, requiring 4 transfers to reach distant cities. The average path length is 2.262, similar to the air network, indicating that it takes slightly more than two steps, or one transfer, to travel from any city to another by high-speed rail.

[0108] The average clustering coefficient of the high-speed rail subnetwork is 0.509, slightly lower than that of the aviation subnetwork, but still showing a good level of clustering. On average, about 50% of the node cities have interconnected neighboring cities, meaning that if a high-speed rail city is connected to two other high-speed rail cities, there is a 50% probability that those two neighboring cities are also connected by high-speed rail. The modularity score of 0.496 indicates that the high-speed rail subnetwork has a stronger community structure compared to the aviation subnetwork, suggesting the existence of different communities within the high-speed rail subnetwork.

[0109] from Figure 5The high-speed rail sub-network topology map shows that China's high-speed rail network is clearly divided into six groups, mainly concentrated in cities connected by high-speed rail lines. Each group basically represents a different geographical region: the North China group, the East China group, the South China group, the Southeast group, the Northeast group, and the Southwest group. Most cities within a group are connected by the same high-speed rail line. The North China group, with Beijing and Zhengzhou as its core hubs, includes regional hub cities such as Jinan, Tianjin, and Qingdao, as well as some closely connected Northwest hub cities such as Xi'an and Lanzhou (due to the rapid development of high-speed rail in the eastern region, many high-speed rail lines in the Northwest region were not yet completed or had limited operation as of 2019, thus not forming a separate regional group). The East China group, centered on Shanghai, Hangzhou, and Nanjing, includes regional hubs such as Xuzhou, Ningbo, and Hefei. The cities in this group are all located in the Yangtze River Delta region, and the dense network surrounding them indicates a high degree of regional connectivity, promoting active economic activity and regional development. Similarly, the South China cluster, centered on Guangzhou and Shenzhen, includes major cities in the Pearl River Delta region, as well as regional hubs such as Changsha, Kunming, and Nanning connecting surrounding provinces, demonstrating the region's strong development potential and vast hinterland. The three smaller clusters are the Southwest cluster centered on Chongqing and Chengdu, the Northeast cluster centered on Shenyang, and the Southeast cluster centered on Xiamen, Nanchang, and Fuzhou. Although these cities are relatively small in area, their strong connectivity with surrounding cities and high level of regional integration indicate a developed local economy and convenient transportation, highlighting the important role of high-speed rail lines as economic corridors.

[0110] Similar to the aviation subnetwork, the degrees of all cities in the high-speed rail subnetwork were also plotted as a degree distribution map. Figure 6 As can be seen, a small number of nodes have very high degrees; only 20% of the nodes have more than 56 connections, and only 10% of the nodes have more than 100 connections. In contrast, most nodes have relatively few connections and low connectivity.

[0111] Further plotting the log-cumulative distribution as follows Figure 7 As shown, the logarithmic graph of the high-speed rail subnetwork is also divided into two parts. The first half exhibits an approximately linear relationship, indicating that nodes with fewer connections exhibit traditional scale-free behavior. The second half of the distribution decays faster than a power-law distribution, which may be because the railway lines operated by the high-speed rail limit the connections to more cities; that is, a city's position in the network determines the maximum number of cities connected to it on its railway line.

[0112] After fitting, the cutoff point is at order 24. The slope of the first linear fit is -0.2458, and R0 is... 2 The first value is 0.8682; the second value is a quadratic function that opens downwards, R. 2The value is 0.9782. This shows that both fitting relationships have a high degree of fit. Therefore, the degree distribution of the high-speed rail sub-network no longer conforms to the characteristics of a single power-law distribution, nor to the characteristics of a scale-free network, but rather to the characteristics of a wide-area scale network. Overall, the degree distribution exhibits a power-law state at low degree values, but transitions to a rapidly decaying state at high degree values.

[0113] The degree distribution function of the high-speed rail subnetwork is:

[0114]

[0115] Among them, K C =24.

[0116] 3) Network topology characteristics of the air-high-speed rail intermodal transport network;

[0117] As shown in Table 1 above, the air-high-speed rail intermodal network integrates nodes and edges from both the air and high-speed rail subnetworks, totaling 328 nodes and 6675 edges. It exhibits the highest average degree and weighted degree, indicating that the network's direct connectivity and service density are significantly higher than individual subnetworks, demonstrating its high interconnectivity. The average clustering coefficient of 0.714 is the highest among the three networks, indicating strong clustering capabilities and the formation of highly interconnected groups, with a 71.4% probability of adjacent nodes connecting to each other. The modularity score of 0.45 is slightly lower than that of the high-speed rail network, suggesting that the air-high-speed rail intermodal network contains new entity clusters, forming new groups compared to individual air and high-speed rail subnetworks, reflecting new regional geographical features or node functional groupings. The average path length of 2.191 is the shortest among the three networks, highlighting the efficiency improvements brought about by network integration, making travel in the entire transportation system faster.

[0118] The integrated network is more efficient, has the highest average connectivity, and the shortest average path length. This highlights the advantages of integrating the aviation and high-speed rail sub-networks. The integrated network also has the highest average clustering coefficient, reflecting the strong overall connectivity achieved by the combination of aviation and high-speed rail nodes. At the same time, the integration also enhances the resilience and robustness of the national network.

[0119] from Figure 8As can be seen, the intermodal transport network integrates the characteristics and connectivity of various subnetworks, regenerating four larger clusters and two smaller clusters. Many cities that occupy important positions in the aviation subnetwork and those in advantageous positions in the high-speed rail subnetwork have been integrated into the intermodal transport network. A key feature of the intermodal transport network is that the aviation subnetwork is grouped by hub cities, while the high-speed rail network is grouped by geographical region. Beijing, Tianjin, Qingdao, and Jinan jointly lead the yellow group, including cities in North China, Northeast China, and Inner Mongolia with close ties to these regions. Xi'an, Chengdu, Chongqing, and Zhengzhou together form the brown group, encompassing most cities in the central and western regions. This group also includes many western cities with fewer connections and is the most densely populated group.

[0120] Guangzhou, Shenzhen, Changsha, Kunming, and Guiyang jointly lead the light blue city cluster in Central and Southwest China. This cluster boasts the most large hub cities, primarily provincial capitals in the south. These cities have balanced development in aviation and high-speed rail, driving the development of surrounding areas. The red group, led by Shanghai, Nanjing, and Hangzhou, mainly comprises cities in the Yangtze River Delta region. Although this group has fewer cities, each city has a large node, indicating that aviation and high-speed rail are highly developed in the region, and that the cities have close external connections. The other two smaller clusters are the Hubei cluster led by Wuhan and the Fujian-Jiangxi cluster led by Nanchang, Fuzhou, and Xiamen. These two regions have formed smaller individual clusters, indicating high high-speed rail connectivity in these provinces, with highly developed high-speed rail connections between cities within the province and between neighboring provinces. Their connectivity is so dense that they would not be overlooked even within an air-high-speed rail intermodal network.

[0121] Similarly, the degree distribution map of all cities in the intermodal network is also shown in this way. Figure 9 Similar to the two subnetworks, a few nodes have very high degrees, while most nodes have relatively few connections, resulting in a limited number of directly connected cities. Further plotting the log-cumulative degree distribution is as follows... Figure 10 As shown, the logarithmic graph of the intermodal network is also divided into two parts. The first half exhibits an approximately linear relationship, indicating that nodes with lower connectivity exhibit traditional scale-free behavior, meaning that the connectivity of a node increases rapidly before the inflection point, and new nodes preferentially connect to nodes with higher connectivity. The latter half of the distribution decays faster than a power-law approach. This is partly due to the capacity limitations of airports and high-speed rail lines in cities restricting connectivity to more cities, and partly because for most small and medium-sized cities, direct connectivity with approximately 50 domestic cities is already sufficient to meet local travel and trade needs. The marginal effect of direct access to more cities on local residents' travel and economic development diminishes.

[0122] Fit analysis showed that the cutoff point occurred at 34. The slope of the first linear fit was -0.4084, and R0 was... 2The value is 0.8935, while the second fit follows a downward quadratic function, R0. 2 The value is 0.9819. These results indicate a strong correlation between the two fitting results, suggesting that the intermodal network does not conform to the characteristics of a single scale-free network. Instead, it conforms to the characteristics of a wide-area scale network, where the degree distribution follows a power-law pattern at low degrees, but the growth rate of the degree distribution decays rapidly at higher degree levels.

[0123] The degree distribution function of the air-high-speed rail intermodal transport network is:

[0124]

[0125] Among them, K C =34;

[0126] As before, a random network was used to verify whether intermodal networks exhibit small-world properties. The analysis showed that the average path length of the random network (3.37) was greater than that of the intermodal network, while its clustering coefficient (0.121) was significantly lower. These findings confirm that intermodal networks do indeed possess small-world properties.

[0127] By comparing the connectivity distributions of the three networks, different growth patterns can be observed. When connectivity is less than 20, the aviation subnetwork grows the fastest, with a cumulative connectivity distribution of 0.7. This indicates that 70% of cities have fewer than 20 direct air connections to other cities. Cities with connectivity between 20 and 56 account for 10% of the total; within this range, the growth rate of connectivity distribution slows down, indicating that only a small number of small and medium-sized cities have aviation connectivity within this range. Cities with connectivity between 56 and 102 account for 16% of the total, with a relatively even distribution, indicating that the aviation connectivity of most large cities is evenly distributed within this range. There are 8 megacities with connectivity exceeding 102, accounting for only 4% of the total. These cities—Beijing, Shanghai, Xi'an, Kunming, Guangzhou, Chengdu, Chongqing, and Shenzhen—have a stable and even distribution of connectivity, with Beijing having the highest connectivity, reaching 181 domestic cities.

[0128] On the other hand, when connectivity is below 50, the urban connectivity of the high-speed rail subnetwork shows rapid and even growth. These cities are mainly located along a single high-speed rail line, accounting for 78% of all cities. When connectivity exceeds 50, the growth rate slows down significantly, with only 22% of cities located at the intersection of two or more high-speed rail lines, benefiting from increased connectivity opportunities. Among these cities, Beijing, Shanghai, Guangzhou, Zhengzhou, Changsha, and Nanjing are the main hubs, connecting more than 160 cities with high-speed rail.

[0129] The intermodal transport network combines features of both air and high-speed rail sub-networks. Growth is rapid when city connectivity is less than 56, then gradually slows. Cities with a combined connectivity of less than 56 account for 83% of the total, indicating that most cities are directly connected to no more than 56 other cities within the intermodal network. As connectivity exceeds 56 cities, the growth rate slows further, with sporadic breakpoints appearing at the tail end. The super-connected cities in the intermodal network mirror those in the high-speed rail sub-network—Beijing, Shanghai, Guangzhou, Zhengzhou, Changsha, and Nanjing—demonstrating that the high-speed rail network plays a crucial role in enhancing regional connectivity when air connectivity to major hub cities is already high.

[0130] Step S4: Calculate the network centrality index and perform cluster analysis;

[0131] The Gephi software was used to calculate the network centrality indices of the aviation sub-network, the high-speed rail sub-network, and the air-high-speed rail intermodal network. In this embodiment, five centrality indices were selected to evaluate the importance of different cities from multiple centrality perspectives and to assess the relative importance of nodes. The network centrality indices include degree centrality, weighted degree centrality, betweenness centrality, harmonic proximity centrality, and eigenvector centrality.

[0132] 1) Degree centrality is a metric in network analysis used to determine the importance or influence of a node in a network. It is one of the simplest centrality measures, calculated based on the number of direct connections a node has. Nodes with high centrality are usually considered important due to their numerous connections. The formula for calculating the degree centrality of a node is:

[0133] C D (v) = in-deg(v) + out-deg(v);

[0134] Wherein, in-deg(v) is the number of edges entering node v, and out-deg(v) is the number of edges leaving node v.

[0135] 2) Weighted degree centrality is a metric used in network analysis that quantifies the importance or influence of a node based on the sum of the weights of the edges connected to it. Unlike traditional degree centrality, which only counts the number of edges connected to a node, weighted degree centrality considers the strength or capacity of these connections. Nodes with high weighted degree centrality have many strong connections, making them potentially influential or central nodes in the network.

[0136] For a given node, the formula for calculating the weighted degree centrality is:

[0137] C w (i)=∑ j∈N(i) w ij ;

[0138] Where N(i) is the set of nodes i, and W ij It is the weight of the edge between nodes i and j.

[0139] 3) Network centrality is a metric used in network analysis to quantify the importance of a node (or edge) in facilitating communication or interaction between other nodes. Based on the concept of shortest paths, it reflects the frequency with which a node appears on the shortest path between other nodes in the network. Nodes with high centrality are crucial for communication within the network.

[0140] For a given node, the formula for calculating betweenness centrality is:

[0141]

[0142] Where, σ st σ is the total number of shortest paths from node s to node t. st (v) is the number of all shortest paths, the sum of which is taken from the sum of all node pairs s and t, excluding node v itself.

[0143] 4) Harmonic proximity centrality is another method for measuring proximity centrality, suitable for handling disconnected or sparsely connected networks. It measures a node's influence in the network by considering the inverse of the shortest path distance from a node to all other nodes, emphasizing nearby nodes rather than distant nodes. A higher value indicates a more central node in the network and a shorter average distance from all other nodes.

[0144] For a given node, the formula for calculating harmonic proximity centrality is:

[0145]

[0146] Where d(u,v) is the shortest path distance between nodes u and v; if u and v are not connected, then d(u,v) is considered infinite. It is zero.

[0147] 5) Eigenvector centrality is a method for measuring node influence that takes into account the importance of a node's neighbors. Eigenvector centrality is based on the idea that a node's centrality is determined by the centrality of its neighboring nodes. Nodes with high scores are connected to many other well-connected nodes, which are generally considered to have high influence in the network.

[0148] For a given node, the formula for calculating the eigenvector centrality is:

[0149]

[0150] Among them, CE (v) is the eigenvector centrality of node v, λ is a constant, and A is the largest eigenvalue of the adjacency matrix. uv This represents the connection between nodes u and v.

[0151] Centrality measurement results:

[0152] The centrality metrics—degree centrality, weighted degree centrality, betweenness centrality, harmonic proximity centrality, and eigenvector centrality—were calculated using Gephi software. The top 20 cities for each centrality metric were identified and ranked accordingly, as shown in Table 2-4. These rankings provide a comparative overview of the most influential cities within the network based on different centrality perspectives.

[0153] Table 2. Centrality Measurement Ranking of the Aviation Subnetwork

[0154]

[0155]

[0156] Table 3 Ranking of Centrality Measurement of High-Speed ​​Railway Subnetwork

[0157]

[0158] Table 4 Ranking of Centrality Measurement in Air-High-Speed ​​Rail Intermodal Transport Network

[0159]

[0160]

[0161] 1) Degree Center

[0162] Degree centrality quantifies the number of direct connections each city has within the network, highlighting the prominent role of key cities in maintaining national connectivity. In the air network, Beijing, Shanghai, and Xi'an are the most connected cities, serving as major national air hubs with widespread influence. Similarly, in the high-speed rail sub-network, Beijing, Shanghai, and Guangzhou have the highest centrality, indicating their crucial role in maintaining the connectivity and robustness of the integrated high-speed rail and air-rail transport network. Regionally, cities like Kunming, Guangzhou, Chengdu, and Chongqing hold pivotal positions in their respective air networks, while Zhengzhou, Changsha, Nanjing, and Hangzhou play significant roles in the high-speed rail network, ensuring regional accessibility and enhancing the cohesion of the national network. It is noteworthy that some cities with less prominent positions in the air network, such as Jinan, Xuzhou, Fuzhou, and Wuxi, are also important high-speed rail hubs. For example, Xuzhou and Wuxi, despite their smaller airports, benefit from well-developed high-speed rail connections. The degree centrality ranking of the air-high-speed rail intermodal transport network is closer to that of the high-speed rail network, indicating that in terms of the number of cities connected, cities along the railway line contribute more significantly to the overall connectivity of hub cities.

[0163] 2) Weighted Degree Centrality

[0164] Weighted centrality assesses a city's national connectivity by considering the frequency of its operational services. Assessing by weighted degree centrality, Shanghai, Beijing, and Guangzhou rank as the cities with the highest air and high-speed rail connectivity in China, not only because of their extensive connections to numerous other cities but also because of the high frequency of services to and from their destinations. This high density of service provides passengers with greater convenience and more departure time options. Interestingly, while Urumqi and Haikou rank in the top 20 for degree centrality, they do not rank in the top 20 for weighted degree centrality. This suggests that these cities prioritize expanding their networks to connect more cities rather than increasing service frequency, highlighting their strategic focus on connectivity rather than convenience. The comparison between these two centrality indicators reflects the different strategic priorities of various cities in expanding their networks. The weighted centrality ranking of intermodal networks is consistent with the characteristics of the two sub-networks, where top-ranked cities, such as Xi'an and Kunming, show higher flight frequencies, while other cities, such as Nanjing and Changsha, show higher high-speed rail frequencies.

[0165] 3) Harmonic proximity centrality

[0166] Harmonic proximity centrality assesses the independent transport capacity of a city node, indicating its ability to facilitate transport without relying on other cities. A higher harmonic proximity centrality value indicates a shorter average path length between the city and other nodes, signifying stronger inter-city connectivity. In the air transport network, besides Beijing and Shanghai, cities like Xi'an, Kunming, and Guangzhou also rank among the top five in proximity centrality. Similarly, in the high-speed rail network, Beijing and Shanghai, along with Nanjing, Changsha, and Suzhou, are among the top five. After merging these two sub-networks into an intermodal network, Beijing, Shanghai, and Guangzhou occupy the top three positions, while Zhengzhou and Xi'an rank among the top five. This indicates that these cities have the shortest average travel distances to other cities in China, possessing strong independent connectivity. Furthermore, cities like Chongqing, Chengdu, Changsha, Nanjing, and Shenzhen also demonstrate high national interconnectivity efficiency, ensuring effective integration of regional and national transportation.

[0167] 4) Betweenness centrality

[0168] Intercity centrality assesses a city's ability to act as a transit hub, facilitating transportation between other city nodes. A higher intercity centrality value indicates a stronger ability for the city to serve as an intermediate point in transport flows. In the national air and intermodal networks, Beijing, Xi'an, and Kunming are the cities with the strongest transit capabilities. In the high-speed rail network, Beijing, Shanghai, and Guangzhou occupy the top three positions, highlighting their crucial role in connecting other cities. Notably, cities like Urumqi, Hohhot, and Lanzhou have relatively low proximity and degree centrality, but high betweenness centrality. This suggests that while these cities may not have the highest direct connectivity or proximity, they are important transit hubs, connecting major cities with smaller regional centers, thereby enhancing the overall connectivity of the network. Their strategic position as intermediaries is crucial for maintaining the cohesion and efficiency of the national transport network.

[0169] 5) Eigenvector centrality

[0170] Eigenvector centrality assesses a city's influence within a network by considering the quality of its connections to other influential nodes. Cities like Beijing, Shanghai, Guangzhou, Zhengzhou, Changsha, and Xi'an not only possess strong individual connectivity but also extend their overall importance within the network through connections to other highly influential cities. These central cities exert considerable influence and reinforce their roles through connections with other key nodes. While cities like Haikou, Xiamen, Shenyang, and Xuzhou may not have high direct connectivity centrality in sub-network rankings, they enhance their influence through strategic connections to important hubs. This strategic positioning enables them to play an influential role within the broader network.

[0171] Overall, Beijing ranks highly across all centrality indicators, highlighting its role as the most important node in China's integrated air-high-speed rail network. Beijing's strengths lie in its strong connectivity (degree centrality), significant influence (eigenvector centrality), and crucial role in transit transportation (between-number centrality). Similarly, Shanghai and Guangzhou perform exceptionally well across all centrality indicators, demonstrating their status as network hubs and their ability to manage massive traffic volumes. Xi'an, Kunming, and Zhengzhou exhibit particularly high between-number centrality, underscoring their strategic importance as transit points connecting the developed eastern regions with the rest of the country, playing a vital role in integrating the less developed western regions into the national economy. Meanwhile, Chengdu and Chongqing show balanced performance across various centrality indicators, reflecting their growing importance as key nodes in China's western transportation network.

[0172] Building upon centrality analysis, cluster analysis was used to analyze the structural characteristics of the network. To further understand the roles and interactions of different cities within China's aviation and high-speed rail sub-networks and the integrated network, cluster analysis was performed for each network type. This method uses the centrality measures calculated in the previous section—degree centrality, weighted degree, harmonic proximity centrality, betweenness centrality, and eigenvector centrality—to identify patterns and groups within the network. The analysis was conducted using the XLSTAT statistical software tool, which can detect different groups based on these centrality measures.

[0173] K-means clustering in XLSTAT is a robust method for dividing a dataset into K distinct clusters based on feature similarity, maximizing inter-cluster variance while optimizing intra-cluster variance. The method involves initializing K centroids, iteratively assigning each data point to the nearest centroid (Euclidean distance), and recalculating centroids based on group membership until convergence.

[0174] Step S5: Cluster quality evaluation;

[0175] 1) Cluster analysis of aerospace subnetworks

[0176] Cluster analysis was performed on 187 cities in the Chinese aviation network using all the centrality indices calculated above. Each centrality index had the same weight. The silhouette coefficient score was used to evaluate the quality of clustering results in data analysis. The evaluation includes the similarity (cohesion) of an object to other clusters (classifications) and to its own cluster. The higher the score, the clearer the clustering.

[0177] Figure 11The results show that the two-cluster solution achieved the highest silhouette coefficient score, indicating that this configuration provides the most unique and highly separated clusters in the network. As the number of clusters increases beyond two, the silhouette coefficient decreases, suggesting that increasing the number of clusters may lead to over-segmentation, thereby reducing the clarity and cohesion of the cluster structure.

[0178] Cluster 1 comprises 11 cities: Beijing, Shanghai, Guangzhou, Chengdu, Xi'an, Kunming, Chongqing, Shenzhen, Harbin, Hohhot, and Urumqi. These cities are major hubs in the aviation network, exhibiting high centrality values ​​across multiple metrics, highlighting their role as key nodes in maintaining air traffic flow and connectivity. The cities within this cluster demonstrate significant intra-cluster variability, reflecting their diverse and influential position within the aviation network. Notably, this cluster aligns with previous topology and centrality analyses and the ten international aviation hub cities designated by the Civil Aviation Administration of China; the inclusion of Hohhot underscores its strategic importance in connecting Inner Mongolian cities to the national network.

[0179] In contrast, Group 2, comprising 176 cities, exhibits less variation within the group and a more unified network function, representing other cities with weaker connectivity and higher homogeneity. This difference highlights the hierarchical structure of China's aviation network, where a few hub cities act as key nodes, facilitating air traffic flow and connecting various regions across the country.

[0180] 2) Cluster analysis of high-speed rail subnetwork

[0181] Similarly, in the high-speed rail network analysis, the clustering results showed two distinct urban clusters, such as Figure 12 As shown, Cluster 1 includes 24 cities, including major metropolitan hubs such as Beijing, Shanghai, Guangzhou, and Xi'an, primarily located in the eastern region. These cities exhibit significantly higher intra-cluster variance and average distance to the central point, indicating substantial differences in their core roles within the network. As key nodes in the high-speed rail network, these cities play a major hub role, possessing extensive connectivity and influence. The significant differences within this cluster suggest that while all these cities are important, some have a significantly higher influence than others, and even among top-tier cities, different levels of influence exist.

[0182] In contrast, Group 2 comprises 227 cities, characterized by significantly lower intra-group variance and shorter average distances from the central point. This reflects a more homogeneous urban cluster, with cities generally occupying less significant positions within the network. These Group 2 cities primarily function as local hubs, acting as branches or termini of the broader high-speed rail network. Compared to Group 1 cities, their overall connectivity and influence are more limited, indicating a clear distinction between the central and peripheral roles played by different cities within the high-speed rail network.

[0183] 3) Cluster analysis of air-high-speed rail intermodal transport network

[0184] For the integrated air-high-speed rail network, cluster analysis also revealed a clear distinction between a small group of highly centralized cities (cluster 1) and a small group of more peripheral cities (cluster 2). For example... Figure 13 The contour coefficient scores validate that the two cluster partitions are optimal, effectively distinguishing the major central cities from the others. This analysis further emphasizes the hierarchical structure of the integrated network, characterized by a few key cities dominating in terms of centrality and connectivity, while the majority of cities play a more localized role.

[0185] Specifically, Cluster 1 in the integrated network comprises 13 cities, drawn from cities identified as Cluster 1 in two sub-networks: Shenzhen, Shanghai, Beijing, Guangzhou, Chengdu, Chongqing, Kunming, Xi'an, Nanjing, Harbin, Urumqi, Changsha, and Zhengzhou. This grouping adds three important high-speed rail cities—Nanjing, Changsha, and Zhengzhou—to the major aviation hub cities identified in the aviation network cluster analysis, while Hohhot is excluded due to its limited high-speed rail connections within Inner Mongolia. The cities in this cluster represent the core hub cities of the air-high-speed rail intermodal transport network. The variations in these city centrality indicators suggest their distinct roles within the network, demonstrating their high connectivity and influence. These cities are crucial for maintaining the network's efficiency and effectiveness, and for promoting connectivity across all regions of China.

[0186] In contrast, Group 2, comprising 315 cities, represents the more peripheral cities in the network. These cities exhibit greater consistency in centrality indicators, suggesting they primarily play a supporting role in regional connectivity rather than acting as major hubs. While they form the basic backbone of the network by providing extensive geographic coverage, individual cities in this group do not exhibit the same level of centrality as those in Group 1. This distinction reinforces the network's hierarchical structure, where a few central cities play a dominant role in national connectivity, while the majority of cities ensure regional and local connectivity.

[0187] Analysis of China's aviation, high-speed rail sub-networks, and integrated aviation-high-speed rail network reveals several patterns in their structural characteristics. All three networks exhibit small-world characteristics, characterized by high clustering coefficients and short average path lengths, which are conducive to efficient cross-regional connections. However, these networks do not conform to the traditional characteristics of scale-free networks; instead, after exceeding a certain threshold, they exhibit a faster-than-expected decline in connectivity growth. This rapid decline, exceeding the typical power-law distribution, can be attributed to several factors. First, the capacity and operational constraints of airports and high-speed rail lines in cities may limit further expansion of connections. Second, for many small and medium-sized cities, establishing direct connections with approximately 50 domestic cities may already be sufficient to meet local travel needs and economic activities, and the benefits of additional connections will gradually diminish.

[0188] One of the most interesting features of the integrated air-high-speed rail network is its ability to combine the strengths of both systems. The integrated network benefits from the highly clustered nature of the air network and direct connections between major cities, ensuring rapid and widespread nationwide coverage. Simultaneously, it combines the high-speed rail network's ability to create dense regional communities, thereby enhancing local connectivity and promoting economic integration within specific corridors. This dual functionality makes the integrated network efficient and robust, particularly in balancing long-distance high-speed connections with more localized and convenient routes.

[0189] Centrality analysis further reveals the interaction between the two sub-networks in the integrated structure. Cities such as Beijing, Shanghai, and Guangzhou continue to dominate in centrality measurements, reflecting their important role as national hubs. However, intermodal transport also enhances the importance of secondary hubs such as Xi'an, Kunming, and Zhengzhou, whose strategic position as transit points connecting major urban centers and surrounding areas. The combination of highly concentrated and strategically located secondary hubs enhances the overall resilience of the network by reducing reliance on a few key nodes and distributing traffic flow more evenly across the network.

[0190] The clustering results provide further patterns in the hierarchical and regional structures of the integrated network. The air network primarily revolves around clusters of major hub cities, while the high-speed rail network is more regionally oriented, forming clusters along economic corridors corresponding to the geographical distribution of the rail lines. The integrated network reflects a synthesis of these two clustering patterns: major air hubs serve as anchors for broader national connectivity, while the high-speed rail network promotes regional integration along specific corridors. This dual clustering structure suggests that the integrated network is well-suited to supporting national and regional economic development, providing a robust and resilient framework for future growth.

[0191] Furthermore, the integrated network exhibits a unique modular model, combining the centralized structure of the air network with the more decentralized, region-focused structure of the high-speed rail network. This creates a hybrid modular structure where some regions form close-knit communities around key nodes (as shown in the high-speed rail network), while others are loosely connected through major hub cities (as shown in the air network). This hybrid modular structure enhances the network's ability to adapt to different levels of demand and regional development needs, providing a flexible approach to managing interconnectivity across different regions of the country.

[0192] Therefore, this invention employs the aforementioned air-high-speed rail intermodal transport network analysis method. By constructing a directed weighted network and comparing the complexity characteristics of the integrated network and its sub-networks, a comprehensive complex network analysis of China's air-high-speed rail intermodal transport network is conducted. Unlike previous studies that primarily focused on single networks, this invention innovatively integrates the air and high-speed rail networks to form an intermodal transport network. Through this method, the unique and shared characteristics of the integrated network and its sub-networks are revealed, providing a new perspective on structural composition.

[0193] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the technical solutions of the present invention, and these modifications or equivalent substitutions cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A method for analyzing air-high-speed rail intermodal transport networks, characterized in that, Includes the following steps: Step S1: Construct a directed weighted network; Step S2: Select network topology indicators; Step S3: Analyze the network topology characteristics and verify whether the network has small-world properties; Step S4: Calculate the network centrality index and perform various centrality analyses; Step S5: Cluster analysis; In step S1, the construction of the directed weighted network specifically involves: In the high-speed rail sub-network, the city where the station is located is taken as the node. When the same high-speed train stops at two stations, it indicates that there is a connection between the two cities. In the aviation subnetwork, the city where the airport is located is used as a node. When there are flights between two nodes, it indicates that there is a connection between the two cities. In a directed weighted network, the edges from city A to city B and from city B to city A are set as two independent directional connections, and each connection between stations reflects the specific direction of flight and train services. In step S2, nodes, edges, average degree, average weighted degree, network diameter, modularity, average clustering coefficient, and average path length are selected as network topology indicators to describe the network structure and topological characteristics of the directed weighted network. The network topology metrics were calculated using Gephi software, specifically: A node represents a single entity or point connected by an edge. A node is determined as follows: if a city has both a high-speed rail station and an airport, or if a city has more than one airport or high-speed rail station, then the city itself is considered a node. An edge represents the relationship or interaction between nodes, and the number of edges is the total number of direct connections between nodes in a directed weighted network. Average degree is used to measure the average number of connections of each node in a directed weighted network. It is calculated by adding the degrees of all nodes and then dividing by the total number of nodes. The degree of a node refers to the number of edges connected to that node. The average weighted degree is calculated by adding the weighted degrees of all nodes and then dividing by the number of nodes. The weighted degree of a node refers to the sum of the weights of the edges connected to that node. The network diameter represents the longest and shortest path between any two nodes in the network. It is calculated by determining the shortest path between all pairs of nodes and the maximum length of these paths. The modularity level is measured by the degree to which the network is divided into modules or communities. A highly modular network has dense connections between nodes within modules or communities, while connections between nodes in different modules or communities are sparse. The average clustering coefficient is calculated as follows: it is calculated from the average of the clustering coefficients of all nodes. The clustering coefficient of a single node is the ratio of the number of connections between its neighboring nodes to the maximum possible number of connections. The clustering coefficient of a node is used to measure the degree of connection between its neighboring nodes and to describe the overall trend of nodes clustering together in the network. The average path length refers to the average number of steps taken by all possible node pairs along the shortest path in the network. It is calculated by adding the shortest path lengths between all node pairs and then dividing by the number of such node pairs. In step S4, Gephi software is used to calculate the network centrality indices of the aviation subnetwork, high-speed rail subnetwork, and aviation-high-speed rail intermodal network to evaluate the relative importance of nodes. The network centrality indices include degree centrality, weighted degree centrality, betweenness centrality, harmonic proximity centrality, and eigenvector centrality. In step S4, degree centrality is calculated based on the number of direct connections of a node, and the calculation formula is as follows: ； in, It is a node The number of edges in the direction of entry, It is a node Count the number of sides in the outward direction; The formula for calculating weighted degree centrality is: ； in, It is the set of nodes i. It is the weight of the edge between nodes i and j; The formula for calculating betweenness centrality is: ； in, From node s To the node t The total number of shortest paths, It is the number of all shortest paths, the sum of which is taken from all node pairs. and The sum, excluding nodes itself; The formula for calculating harmonic proximity centrality is: ； in, It is a node and The shortest path distance between; if and If not connected, then It is considered infinite at this time. It is zero; The formula for calculating the eigenvector centrality is: ； in, It is the eigenvector centrality of node v. A is a constant, and A is the largest eigenvalue of the adjacency matrix. Represents a node u and v The connection between them.

2. The air-high-speed rail intermodal transport network analysis method according to claim 1, characterized in that, In step S3, the network topology characteristics of the aviation sub-network, high-speed rail sub-network, and aviation-high-speed rail intermodal network are analyzed based on network topology indicators to verify whether each network has small-world properties. This specifically includes the following steps: S31. Draw a degree distribution map of all nodes; S32. Draw a cumulative degree double logarithmic distribution plot based on the node degree distribution; S33. Use Python and NetworkX to generate a random network with the same number of nodes, directed edges, and average weights as the existing network, and calculate the average path length and clustering coefficient of the random network.

3. The air-high-speed rail intermodal transport network analysis method according to claim 2, characterized in that, In step S4, the K-means clustering method in the XLSTAT statistical software is used, and cluster analysis is performed on the nodes in the network based on the calculated centrality index.

4. The air-high-speed rail intermodal transport network analysis method according to claim 3, characterized in that, In step S5, the silhouette coefficient is used to evaluate the clustering quality. The silhouette coefficient score is used to evaluate the quality of the clustering results in data analysis. The evaluation includes the similarity of an object to other clusters and to its own cluster. The higher the score, the clearer the clustering.

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