Introduction

Buses and subways play a vital role in urban public transportation, accommodating a significant number of daily trips by urban residents. They are essential for alleviating traffic congestion and enhancing travel efficiency. People often meet their daily travel needs by using buses, subways, or transferring between them1. However, in a composite network of buses and subways, certain nodes or segments often fail due to emergencies, such as local traffic congestion, traffic accidents, or temporary traffic control. This can negatively impact network connectivity, global efficiency, and reachability2. Understanding how network performance changes during interference provides a crucial reference for developing effective improvement and emergency strategies. Therefore, accurately evaluating the performance of bus and subway composite networks under disruptive conditions is crucial to ensure the convenience and reliability of residents’ travel.

Scholars have conducted extensive research on the robustness of transportation networks. In terms of network structure analysis and key station identification, Sienkiewicz et al.3 examined the bus topology of 22 cities in Poland and found that all exhibited small-world characteristics. Dimitrov et al.4 analyzed the evolution characteristics of Auckland’s bus network through its topology identification. Luo et al.5 examined the characteristics of bus and subway subnetworks as well as their composite network in Beijing, concluding that the collaboration between bus and subway maximizes public transportation efficiency. These studies show that network structure analysis can reveal the internal structure and function of the network. However, they fail to fully consider the impact of actual traffic conditions on the identification of critical sites. To address this, Guo et al.6 proposed a method for identifying key stations in urban rail transit networks across multiple scenarios, combining the CRITIC-TOPSIS empowerment method with station and city integration. Wang et al.7 considered the diversity of transportation modes and node connectivity, constructing an IWKS model to identify key nodes in the network. Cao et al.8 simulated five network failure scenarios and found that targeted attacks on key sites were more destructive to the network. The research on identifying key stations focuses on actual traffic conditions to enhance the identification accuracy and strengthen robustness evaluation. However, these studies overlook crucial factors such as spatial information and external site accessibility. These factors can significantly affect the identification of key sites.

In transportation systems, accessibility measures how easily an individual can reach a destination using a specific mode of transportation9. In accessibility research, Caschili et al.10 established the American commuter network and found a strong positive correlation between network centrality and accessibility. Ding et al.11 employed a complex network model to analyze the impact of rail network expansion on urban accessibility. Peng et al.12 constructed a rail transit network model of Wuhan and analyzed the relationship between residents’ travel characteristics and accessibility. These studies analyze the accessibility of public transportation systems through complex networks, but they neglect the impact of the actual network’s spatial structure and traffic congestion in their calculations. Building on this, Juan et al.13 analyzed road networks in five regions and found that trunk sections played a crucial role in network accessibility. Luo et al.14 constructed a public transport network weighted by travel impedance and evaluated its accessibility. Yu et al.15 evaluated the actual accessibility of the bus network using multi-source data, including bus card swiping and departure intervals. These studies demonstrate a close relationship between complex networks and accessibility, suggesting complex network theory can be applied to evaluate the accessibility of transportation networks. However, the feedback effect of reachability on complex networks has been overlooked.

Robustness refers to a network’s ability to maintain performance stability when attacked16. In robustness evaluation, Jin et al.17 integrated the subway and bus systems to construct a multi-mode bus network, assessing the inherent resilience of Singapore’s public transport system. Yang et al.18 developed a bus-subway coupling network and found that higher coupling strength between subnetworks leads to a larger scale of cascade failures. Tran et al.19 established a bus network and found that the network’s robustness under deliberate attacks was more vulnerable. The above research evaluates the network robustness by constructing composite network models, however, the coupled network approach is more complex, and the evaluation method remains relatively simple. Consequently, scholars began to explore methods for constructing composite networks and diversifying robustness evaluation indicators. Zhang et al.20 employed ArcGIS to couple bus and subway stations using a processing method and found that coupling stations enhance network robustness. Zheng et al.21 considered the operational and transfer characteristics of buses and subways, constructed the network and analyzed its robustness using the station importance index. Hatem et al.22 evaluated the robustness of bus networks across different periods using open-source bus network data. Pan et al.23 constructed a weighted composite network considering the actual operational characteristics of buses and subways, and proposed two new robustness evaluation indices: “trip success rate” and “detour coefficient”. Yin et al.24 established a traffic network model for 40 major cities in China and evaluated its robustness from both structural and functional perspectives.

Although relevant research has been conducted, most studies focus solely on the structural characteristics of the network, neglecting the correlation between the spatial distribution of the network and various attack modes. Furthermore, when applying complex network theory to analyze reachability, the potential influence of reachability on the existence of complex networks remains unexplored.

This study aims to address the above issues by proposing a method to efficiently construct a bus-subway composite network. It introduces three types of station importance indices-direct reachability, inter-site reachability, off-site accessibility, and off-site accessibility-to identify key stations, and a new robustness evaluation index, “network accessibility”, to assess both internal and external accessibility. The CRITIC weighting method is used to assign weight to different attack modes. The robustness of the composite network is evaluated by comparing the changes in the maximum connected subgraph rate, network efficiency, and network reachability before and after various attacks. This study provides theoretical guidance for advancing network robustness evaluation theory and holds practical value for improving bus-subway composite networks.

The structure of the paper is as follows: the second part outlines the network construction process and presents the station importance indices; the third part introduces the attack modes and the robustness evaluation indices; the fourth part evaluates the robustness of the bus-subway composite network in the core area of Dalian City; the fifth discusses the results, and the sixth part presents the conclusion.

Bus and subway composite network model

Bus and subway subnetwork

The bus and subway networks are represented as topological structures, where stations serve as nodes and the connections between them are represented as edges. The bus and subway sub-networks are constructed using the Space-L method, taking into account the spatial relationships between bus and subway lines and stations. The basic assumptions are as follows: (i) In each network, stations with the same name are treated as a single node. (ii) The actual route distance between stations, rather than the Euclidean distance, is used as the edge weight, without accounting for departure frequency and periods. (iii) If multiple lines exist between two adjacent sites, only a single edge is considered. This assumption simplifies the network topology by emphasizing spatial connectivity between sites rather than the operational characteristics of individual lines. It is particularly suitable for the static analysis of network robustness and reachability. Based on these assumptions, the initial bus and subway subnetwork model is developed.

Bus and subway composite network

In practice, residents often walk to transfer between buses and subways. As a result, there are varying degrees of coupling between the bus and subway subnetwork. However, most studies currently use Euclidean distance to assess the strength of the coupling between stations20, neglecting the impact of buildings, road conditions, and road network structure. To account for these factors, ArcGIS and network map services are used to gather land plot, building, and road network structure data. A method for efficiently and accurately coupling the bus and subway subnetworks is proposed. The steps are as follows:

(1) Preprocess the coordinate data of bus and subway stations, import it into ArcGIS, convert to WGS84 coordinates, and project the data.

(2) Create a walking area with a 500 m radius around each subway station21, identify bus stations within this area, establish spatial connections, and export connection files. A 500-meter walking radius defines the transfer area between stations in network coupling, representing transfer convenience. This distance aligns with the daily travel needs of residents.

(3) Considering the impact of buildings, road condition information, and road network structure, the Amap Open Platform is used to calculate the actual walking distance between subway stations and bus stations within the walking area, filtering pairs with distances less than 500 m.

(4) Merge the data files of the bus subnetwork, subway subnetwork, and connecting pairs, and develop the bus-subway composite network model in Python. The construction process is shown in Fig. 1.

Fig. 1
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Construction process of bus and metro composite network.

Importance of the site

To facilitate the subsequent evaluation of network robustness, it is crucial to define clear indices for site importance. This paper presents a total of six indices for site importance. Among these, three are traditional indices: \({K_i},{B_i}\), and\({R_i}\), representing the degree, betweenness (Bt)25, and coreness (Cn)26 of the i-th site, respectively. The degree represents the number of edges directly connected to a node and is one of the most commonly used metrics in complex networks. Betweenness (Bt) quantifies the frequency with which a node appears on the shortest paths between all node pairs in the network. Coreness (Cn) indicates the k-core level of a node, where the k-core is the largest subgraph consisting of nodes with a degree of at least k, obtained by iteratively removing nodes with a degree less than k. However, degree, betweenness, and coreness primarily focus on network topology, neglecting spatial distribution and external service capabilities. To address these issues, three station importance indicators—Direct Reachability (DR), Inter-site Reachability (ISR), and Off-site Accessibility (OSA)—are proposed, integrating the concepts of composite network layout and accessibility. DR, ISR, and OSA integrate spatial data (e.g., actual walking distances) and population distribution (e.g., WorldPop data), enhancing the identification of key stations in real-world applications.

Direct reachability

DR is used to characterize a site’s ability within a composite network to directly access other sites without requiring transfers. This index directly reflects the degree of connectivity between the site and other sites, as well as the number of lines serving the site. Direct Reachability, originally based on “Degree” in complex networks, is extended in this paper to account for direct connections along the same line, rather than all neighboring nodes. This modification more accurately reflects the role of line services in bus and subway networks and emphasizes the importance of direct access to stations in reducing transfer demand and improving network efficiency. This indicator better captures the actual role of line services in bus and subway networks, emphasizing the importance of direct stops in reducing transfer demand and enhancing network efficiency. It is calculated by counting the stations on the same line as the target station, as shown in Eqs. (1), (2).

$${l_{ij}}=\left\{ \begin{gathered} 1,{\text{site }}i{\text{ and site }}j{\text{ are on the same line}} \hfill \\ 0,{\text{site }}i{\text{ and site }}j{\text{ are on the different line}} \hfill \\ \end{gathered} \right.$$
(1)
$${D_i}=\sum\limits_{{j=1}}^{n} {{l_{ij}}\left( {i \ne j} \right)}$$
(2)

Where: N={1,2…n} is the set of sites, i and j represent sites indices, with i, j∈N, lij is a variable indicating whether sites i and j are on the same line, and Di is the DR of site i.

Inter-site reachability

ISR measures the ease of travel between sites. Interstation reachability is calculated by summing the inverses of the travel times between the station and all other stations, as shown in Eq. (3). This metric is based on proximity centrality in complex networks6. Generally, sites with high ISR are centrally located within the network, implying that travel times from these sites to others are relatively short. The travel time between stations can be obtained using bus route planning from the Amap Open Platform.

$${C_i}=\sum\limits_{{j=1}}^{n} {\frac{1}{{{t_{ij}}}}} \left( {i \ne j} \right)$$
(3)

Where: Ci represents the ISR of site I, and tij is the time required for travel from site i to site j.

Off-site accessibility

OSA characterizes the ease with which residents can access a site, with a typical service radius of 800 m27. Unlike the 500 m mentioned in the “Bus and Subway Composite Network” section, the 800 m represents the service radius of the station to the surrounding population. These separate settings align better with the analytical framework and practical needs of this paper. The more accessible the site is, the greater the number of people it can serve. An improved spatial barrier method is proposed to calculate the OSA, considering both the population surrounding the station and the number of lines connecting to it in the actual network28. When counting the number of lines at a station, both bus and subway connections are considered to fully account for the station’s connectivity and service capacity in the network. The result is obtained by calculating the inverse of the travel time for each sampling point within the site’s range, multiplying by the corresponding population, summing the values, and then multiplying by the number of lines at the site, as shown in Eqs. (4), (5). In Eq. (5), the number of station lines and the population of sampling points can be processed and adjusted using ArcGIS while walking times are calculated using the Amap Open Platform’s walking path planning.

$$a_{{i{\text{o}}}} = \left\{ \begin{gathered} 1,\;{\text{the distance between sampling point }}o{\text{ and site }}i{\text{ is less than or equal to 800 meters}} \hfill \\ 0,\;{\text{the distance between sampling point }}o{\text{ and site }}i{\text{ is more than 800 meters}} \hfill \\ \end{gathered} \right.$$
(4)
$${A_i}={m_i}\sum\limits_{{o=1}}^{f} {\frac{{{P_{io}}{a_{io}}}}{{{t_{io}}}}}$$
(5)

Where: \({a_{io}}\) is a variable indicating whether the walking distance between site i and sampling point o exceeds 800 m.\({A_i}\)represents the OSA of site i,\({m_i}\)is the number of lines serving site i, F={1,2…f} is the set of population sampling points, o is the index of sampling points where\(o \in F\),\({P_o}\) is the population of sampling point o, and \({t_{io}}\)is the walking time from site i to sampling point o.

Using the method described above, three reachability values for each station are computed and assigned to their respective nodes. Based on the initial composite network, an undirected weighted bus-subway composite network is further constructed.

Robustness evaluation

Network attack methods

During the operation of the bus system, unforeseen factors such as traffic congestion, accidents, or temporary traffic controls may cause certain stations or inter-station segments to malfunction. Simulating node attacks allows us to assess whether the complex network can maintain its original functionality under the influence of these factors. The simulated attack includes two methods: deliberate attack and random attack. Deliberate attack involves the targeted selection of important nodes, while random attack involves the random selection of nodes for removal. The specific attack process is shown in Fig. 2. In this study, deliberate attacks are conducted by selecting nodes based on their site importance.

Fig. 2
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Processes of deliberate and random attacks.

Robust evaluation metrics

The robustness evaluation index assesses the network’s ability to maintain its original performance under adverse conditions. The maximum connectivity subgraph rate (MCSR) and network efficiency (NE) are used to characterize the network’s connectivity and operational efficiency23, as shown in Eqs. (6),  (7). To quantify the network’s accessibility after an attack, specifically its inter-site connectivity and off-site service capability, a new robustness metric, network accessibility (Z), is introduced. The entropy weighting method is used to assign weights to the metrics, which are then multiplied by their respective accessibility values. Since the metrics have different units, they are normalized separately using the maximum-minimum normalization method. The normalized values are then multiplied by the weights and summed to calculate the reachability of a node. This normalization process converts the metrics into dimensionless values between 0 and 1, eliminating unit-related discrepancies. The total network accessibility (Z) is obtained by summing the accessibility of all nodes, as shown in Eq. (8).

$$S=\frac{{\left| H \right|}}{{\left| N \right|}}$$
(6)

Where: S represents the MCSR, |N| is the number of nodes in the largest connected subgraph of the original network; and |H| denotes the number of nodes in the largest connected subgraph of the attacked network.

$$E=\frac{1}{{n\left( {n - 1} \right)}}\sum\limits_{{i=1}}^{n} {\sum\limits_{{j=1}}^{n} {\frac{1}{{{d_{ij}}}}} }$$
(7)

Where: E represents the NE, and\({d_{ij}}\)denotes the path length from node i to j.

$$Z=\sum\limits_{{i=1}}^{n} {\left( {{w_1}{D_i}+{w_2}{C_i}+{w_3}{A_i}} \right)}$$
(8)

Where: Z represents the NR, and w1,w2, and w3 are the weights of the importance of the three aforementioned site indices under the entropy weight method.

The entropy weight method requires that data be as independent as possible to avoid high correlations, which can lead to double counting and distortion of the results. A Pearson correlation coefficient analysis was conducted on DR, ISR, and OSA, revealing a moderate positive correlation between DR and ISR (0.7991), a medium-high positive correlation between ISR and OSA (0.4548), and a low correlation between DR and OSA (0.3408). To verify if the results would be distorted when applying the entropy weighting method, a sensitivity analysis was performed. The Pearson correlation coefficients between the recalculated network accessibility and the original accessibility results at the node level remained around 95% after excluding OSA or DR, and the correlation coefficients of the combined post-attack change data reached 99%. These results indicate that the entropy weighting method can effectively assign weights and accurately reflect network performance even when there is a correlation between the indicators.This paper considers the different impacts of attacks on network robustness and employs both traditional and new deliberate attack methods. Traditional attacks are based on degree, betweenness, and coreness, targeting critical network nodes for removal based on descending metric values. The objective is to assess network connectivity, including metrics such as maximum connectivity subgraph rate (MCSR) and network efficiency (NE). The new deliberate attacks focus on node removal according to direct, inter-site, and off-site reachability, with an emphasis on both network topology and reachability, as well as service capability (e.g., network reachability, NR). After conducting these attacks, the CRITIC weighting method is applied to assign appropriate weights to each attack method, resulting in three robustness evaluation indexes: MCSR, NE, and NR, as shown in Eq. (9) to Eq. (11).

$${S_t}={\omega _1}{S_K}+{\omega _2}{S_B}+{\omega _3}{S_R}+{\omega _4}{S_D}+{\omega _5}{S_C}+{\omega _6}{S_A}$$
(9)
$${E_t}={\omega _1}{E_K}+{\omega _2}{E_B}+{\omega _3}{E_R}+{\omega _4}{E_D}+{\omega _5}{E_C}+{\omega _6}{E_A}$$
(10)
$${Z_t}={\omega _1}{Z_K}+{\omega _2}{Z_B}+{\omega _3}{Z_R}+{\omega _4}{Z_D}+{\omega _5}{Z_C}+{\omega _6}{Z_A}$$
(11)

Where:\({S_t},{E_t}\)and \({Z_t}\)represent the robustness indices of MCSR, NE, and NR under comprehensive attack mode, respectively. \({S_K},{S_B},{S_R},{S_D},{S_C}\)and\({S_A}\)denote the MCSRs based on Deg, Bt, Cn, DR, ISR, and OSA respectively. \({E_K},{E_B},{E_R},{E_D},{E_C}\)and\({E_A}\)refer to the NEs based on the importance of the aforementioned sites, while\({Z_K},{Z_B},{Z_R},{Z_D},{Z_C}\), and\({Z_A}\)correspond to the NRs based on the significance of some sites. \({\omega _1},{\omega _2},{\omega _3},{\omega _4},{\omega _5}\)and\({\omega _6}\)represent the weights of each attack mode respectively.

Result

Data acquisition

Taking the core area of Dalian city as an example, relevant data were collected to evaluate the robustness of the composite bus and subway network. The bus line and station data were retrieved from the Amap Open Platform, while the administrative boundary data were obtained via the Planning Cloud Platform. The original coordinates were converted to WGS84 using ArcGIS. Population raster data (100 m×100 m) for China in 2020 were downloaded from WorldPop, cropped to the study area in ArcGIS, and corrected using the seventh population census data. The “raster to point” function was then used to generate sampling points with geographic coordinates and population data. For each site, an 800-meter buffer zone was created, and sampling points within this zone were initially selected. The Amap Open Platform was employed to calculate the actual walking distances and times between the sites and the sampling points. Finally, the sampling points with walking distances under 800 m from each site were filtered.

The study area contains 72 subway stations and 3,534 bus stations, served by 4 subway lines and 140 bus lines, as illustrated in Fig. 3.

Fig. 3
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Core area bus and subway composite network.

Network characteristic analysis

Using the method outlined in “Bus and Subway Composite Network” and Python programming, an undirected weighted composite network for bus and subway in the study area was constructed, resulting in a network with 559 nodes and 1,050 edges. An equivalent random network model was also created, and the average clustering coefficient and average path length for both the real and random networks were computed. Given the uncertainty of the random network, a 100 times equivalent random network was constructed, and the construction process was shown in Fig. 4, and the average value was taken after calculating the relevant characteristic indexes, as shown in Fig. 5; Table 1. (Degree-degree correlation, average clustering coefficient, and average path length are abbreviated as DDC, ACC, and APL, respectively.).

Fig. 4
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Random network construction process.

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Distribution of random network characteristic data.

Table 1 Comparative analysis of the complexity of real and random networks.
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According to Table 1, the DDC of the actual network is 0.2159, suggesting a slight positive correlation. In other words, nodes with higher values are more likely to connect to other high-value nodes. The AAC of the real network is significantly larger than that of the random network. Moreover, the APL for both networks is of the same order of magnitude. This suggests that the real network exhibits small-world characteristics. Networks with these characteristics typically have key stations, more transfer paths, and direct connections, resulting in strong connectivity and high travel efficiency.

Key site identification

To perform simulated attacks, it is essential to identify critical sites in the network. Site accessibility is calculated using the three types of accessibility indicators proposed in the Section “Importance of the site”. The normalized data distribution is presented in Fig. 6, while the key site importance recognition results are displayed in Table 2. The network consists of 40 subway nodes and 519 bus nodes. The proportion of subway and bus nodes refers to their respective share in the top 50 key nodes, ranked by the three accessibility indicators in descending order.

Fig. 6
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Data distribution based on site importance.

Table 2 Site importance analysis of weighted composite network.
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Evaluating site importance using the three dimensions of DR, ISR, and OSA offers a more comprehensive assessment than relying on individual structural metrics. The results indicate that metro stations represent a significant proportion (30%) in ISR, highlighting their advantage in inter-site reachability due to their efficient layout and high operational efficiency. On the other hand, bus stations dominate in DR (9.63%) because of their larger number and wide coverage, making them more accessible directly. For direct reachability, the data distribution is right-skewed with substantial variance, where a few highly accessible stations increase the average, with metro stations at 0% and bus stations at 9.63%. In terms of inter-site reachability, the distribution is more concentrated, with smaller gaps, suggesting minimal differences between stations. Here, metro stations outperform bus stations with a proportion of 30%. For off-site accessibility, the data distribution is similar to direct reachability, but metro and bus stations have comparable proportions, both playing crucial roles in serving surrounding populations.

Figure 7 illustrates the spatial distribution of the top 50 key sites, categorized by varying levels of importance. Sites with high DR are distributed broadly, while sites with high ISR are predominantly concentrated in the network center. Sites with high OSA are located in the intermediate areas. Therefore, when optimizing routes, emphasis should be placed on improving connectivity in low-DR areas, while increasing service frequency in high-ISR and high-OSA areas to enhance overall travel efficiency. Additionally, certain sites rank highly across multiple importance indicators; enhancing the resilience of these sites contributes to improving network robustness.

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Spatial distribution of key stations in the core area.

Network robustness evaluation

In the context of the network model described above, MCSR, NE, and NR are used as robustness indices, with six attack modes representing both deliberate and random attacks. During the attacks, 2% of nodes were removed in each iteration. The variation in network robustness indices, resulting from the removal of the top 40% of nodes under different attack modes, is shown in Fig. 8.

Fig. 8
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Changes in network robustness metrics. (a) MCSR ,(b) NE, (c) NR.

Regarding MCSR, the effects of various deliberate attacks differ. Deg attacks are the most effective, reducing the MCSR from 1 to 0.3703 with an attack ratio of 12%. The effect of OSA is comparable to that of the Bt attack, but its impact is weaker. The DR attack exerts a strong influence initially, but its effect diminishes slowly after the attack proportion exceeds 18%. In contrast, the effect of the Cn attack changes more rapidly as the attack progresses. ISR attacks decrease from 1 to 0.3059 at 31%, after which they decrease more rapidly than DR attacks. MCSR changes most slowly under RA attacks.

Regarding NE, deliberate attacks have a substantial impact, particularly when the attack percentage is below 20%. At a 40% attack rate, network efficiency under OSA, Bt, and Deg attacks drops by 95%, rendering the network nearly ineffective. NE under Cn attack declines by 88%. NE under DR and ISR attacks decreases by approximately 75%. RA has the least disruptive impact on NE.

Regarding NR, when the attack proportion ranges from 0 to 20%, NR under all deliberate attack modes decreases by approximately 48%, with the fastest decline occurring in this stage. OSA- and DR-based attacks exhibit the steepest decline, followed by Deg, Cn, and Bt attacks. Although the impact of ISR attacks is relatively weak, it remains significantly stronger than that of RA.

Intentional attacks have a significantly greater impact on the network than random attacks, highlighting the importance of critical sites for maintaining connectivity, efficiency, and reachability. Figure 9 presents a color-coded heat map illustrating the spatial distribution of the top 40% of sites in the core area of Dalian City, ranked in descending order of DR, ISR, and OSA. In the heat map, red indicates high reachability (strong accessibility), green represents low reachability (weak accessibility), and yellow and orange denote moderate levels.

High DR sites are dispersed across the network, particularly along major bus routes, highlighting their role in direct connectivity. They also include external transit points with limited interchanges, as shown in Fig. 9a. This distribution accounts for the varied impact of DR-based attacks, as these sites encompass both central hubs and peripheral areas, resulting in differing effects on network connectivity.

Fig. 9
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Site space distribution based on accessibility. (a) DR, (b) ISR, (c) OSA.

High ISR sites are primarily concentrated in the network center, particularly at key metro stations and interchange hubs, reflecting efficient travel times between sites, as shown in Fig. 9b. However, their impact on network performance under attack is relatively low, as these central sites can often be substituted by nearby nodes, preserving overall network functionality.

High-OSA sites are primarily located in intermediary regions, concentrated around but not exclusively within network centers, and are typically situated near densely populated areas and major transportation hubs, as shown in Fig. 9c. The spatial distribution of these sites reflects their critical role in network performance. Deliberate attacks on OSA sites can cause rapid degradation of the network’s connectivity, efficiency, and accessibility, as these sites are crucial for handling high outbound travel demands, connecting multiple lines, and serving as network hubs. Although intentional attacks based on reachability may not be as effective in disrupting the network as traditional degree-, betweenness-, and coreness-based attacks, they can identify critical sites from the perspective of service capability. This improves attack flexibility and allows for a more comprehensive evaluation of network performance, which helps inform the optimization of network defenses and enhances network robustness.

Taking into account the impact of each deliberate attack on network robustness, along with the correlation and internal variability among different attack modes, the CRITIC weighting method is applied to analyze the robustness evaluation results. The weights for each attack are presented in Table 3; Fig. 10.

Table 3 Weight of site importance index based on CRITIC weighting method.
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Fig. 10
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Node importance index weight (base Table 3).

Based on the weights listed in Table 3, the values of MCSR, NE, and NR are computed for different attack proportions, with their variations depicted in Fig. 11.

Fig. 11
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Changes of network robustness index under comprehensive attack mode.

As shown in Table 3, attacks based on Deg, Bt, and OSA have higher weights and exhibit stronger effects in robustness evaluations, indicating their effectiveness in identifying vulnerable points in the network. Figure 11 shows that when the attack proportion is between 0% and 10%, NE and NR decrease rapidly, while MCSR declines more slowly. This suggests that NE and NR rely more on key nodes for information transmission, whereas MCSR benefits from redundant nodes and paths to maintain connectivity. Between 10% and 20% attack proportion, all three robustness indicators decline rapidly, indicating a period of network vulnerability. As the attack proportion increases further, the rate of decline slows, suggesting that the remaining network nodes and structures preserve some level of connectivity and functionality. Based on the above analysis, different attack modes affect network robustness in varying ways. At different stages of attack, network robustness exhibits varying behavior depending on the robustness indicators.

Discussion

This study highlights the crucial role of accessibility in urban transit systems by assessing the robustness of composite bus-subway networks. Using the core area of Dalian city as a case study, the results demonstrate that high-OSA stations have the greatest impact on network performance, and their failure leads to rapid performance degradation. The underlying reasons for these results are further analyzed in terms of network structure, spatial function, and actual traffic conditions, with their practical implications discussed in relation to the previous data.

The impact of high OSA stations on network performance arises from their spatial and functional characteristics. First, these stations are primarily located in densely populated central intermediate areas, such as commercial or residential zones in Dalian, serving a large number of residents, as shown in Fig. 9. A failure can block numerous trips, reducing the ability of nearby residents to access the network, leading to a sharp decline in NR. Second, high OSA stations are typically multi-route hubs connecting bus and subway sub-networks. Removing these nodes disrupts multi-modal paths, causing a sharp decline in NE, as shown in Fig. 8. This aligns with the cascading effects observed in hub failures in complex networks. In operations and planning, stations with high OSA characteristics should be located in densely populated areas, with priority given to redundant facilities or contingency plans to mitigate the cascading effects of their failure.

The trends of MCSR, NE, and NR under various attack modes reveal the underlying mechanism of network robustness. In Fig. 8, MCSR initially decreases slowly due to the small-world nature of the network and the presence of redundant paths, where alternative connections help maintain connectivity even when high Deg, Bt, and OSA sites are removed. However, when the attack percentage exceeds 20%, the removal of core hubs depletes redundancy and accelerates the collapse of connectivity, highlighting the limitations of topological redundancy. NE and NR are more sensitive to critical sites, both showing a significant decline in the early stagrget station, as shown in Ees of attacks. The rapid drop in NE results from the failure of high-Deg, Bt, and OSA sites, which serve as “shortcuts,” thereby increasing path lengths and reducing efficiency. Meanwhile, the sharp initial decline in NR is caused by the simultaneous deterioration of both internal connectivity and external service capability due to OSA and DR sites. In operations and planning, contingency plans should adopt a tiered response strategy. In the early stages of an attack (0-10%), resources should focus on maintaining network efficiency and accessibility, while in the middle stages (10-20%), the emphasis should shift to ensuring connectivity. This can be achieved by increasing redundant routes, constructing ring or grid networks to enhance network resilience, optimizing signal timings to minimize critical path delays, and installing additional bike-sharing stations to compensate for the loss of pedestrian accessibility.

In the construction of the composite network, this study improves upon previous methods that used only Euclidean distance for coupling sub-network sites18,20,23 (see Fig. 12) by adopting actual walking distance (see Fig. 13). This takes into account terrain and road network constraints, providing a more accurate reflection of the real-world travel scenario. However, When modeling multiple concurrent bus and subway routes together as a single side, the capacity differences between the two (e.g., high frequency and high capacity of the subway) are masked. This can lead to two errors: the magnitude of the decline in network efficiency (NE) is magnified; the role of the metro in relieving congestion in the real network is underestimated. Second, distinguishing capacity differences across routes can also reflect the role of high-capacity routes (e.g., high departure frequency or high passenger capacity) in increasing ISR and NE. Therefore, future studies can refine the network model by incorporating factors such as departure frequency and passenger capacity to expand urban case studies and enhance the credibility of the methodology and conclusions.

Fig. 12
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Euclidean distance network coupling.

Fig. 13
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Actual walking distance network coupling.

Conclusion

Based on complex network theory and incorporating reachability, an undirected weighted composite network of bus and subway systems is constructed. Three site importance indicators—DR, ISR, and OSA—along with a network robustness index, NR, are proposed. After a preliminary evaluation of network robustness using traditional attack methods, the CRITIC weighting method is applied to assign weights to different attack modes. Subsequently, the network robustness under comprehensive attacks is assessed, leading to the following conclusions.

(1) The removal of key sites identified through site importance analysis resulted in significant network damage, demonstrating the effectiveness of this method in identifying critical nodes. The NR index, constructed by weighting various reachability factors, accurately reflects changes in network performance during robustness evaluation, validating its effectiveness. Among all attack methods, intentional attacks based on OSA have the most significant impact on network destruction. The weight of OSA in the node importance index is the highest, indicating that nodes with high OSA have the most significant impact on overall network performance.

(2) In the preliminary robustness evaluation, the Deg-based deliberate attack is the most effective for MCSR, reducing it from 1 to 0.3703 with a 12% attack ratio. Deliberate attacks based on OSA, Bt, and Deg are the most effective for NE, causing a 95% reduction with a 40% attack ratio. Deliberate attacks based on OSA and DR are most effective against NR, resulting in a 70% decline with a 40% attack ratio.

(3) In the robustness evaluation across multiple attack modes, when the attack proportion is between 0% and 20%, the MCSR decreases from slow to fast, NR decreases from fast to slow, and NE decreases rapidly and uniformly. When the attack proportion is between 20% and 40%, all three factors decrease at a slower rate. NE is most influenced by key sites.

Although this study has limitations, such as focusing on a single case and using a limited number of simulated network attack scenarios, the proposed construction method for the bus and subway composite network, station importance calculation, and robustness evaluation method can still serve as a valuable reference for similar research. Future studies can further refine and validate these methods and principles by expanding the data collection scope and increasing the number of cases.