To comprehensively characterize the structural properties of nodes in urban public transportation networks, multi-dimensional node features are constructed from three perspectives: network topology, operational characteristics, and spatial distribution.

1)Networktopologyfeatures

K-core number, eccentricity, structural hole, and h-index are selected for capturing the network topology features. The k-core number [20] indicates a node’s depth within the network backbone; eccentricity [21] measures its maximum distance to other nodes; structural hole [22] quantifies the potential to occupy an intermediary position; and the h-index [23], derived from degree distribution, supplements node influence assessment within its neighborhood.

2)Operationalfeatures

The operational features include the number of serving routes, route diversity, average operating duration, whether the station is a subway station, and whether the station is a transfer station. Route diversity is defined as the number of distinct routes serving a stop, counted by grouping the route_cn field in the CPTOND stops table by stop_id and removing duplicates, measured in counts. Average operating duration is the mean operating duration across all routes serving a stop, computed from the start_time and end_time fields in the routes table (in HHMM format), with cross-day handling applied when end_time is earlier than start_time, measured in hours. Transfer station status is a binary indicator. Metro stations are marked as 1, and bus stops within 300 m of a metro station (i.e., having transfer edges) are marked as 1; otherwise 0. Transfer edges are identified by the 300 m Euclidean distance threshold in the UTM plane. The number of serving routes and route diversity jointly reflect a station’s line connectivity; average operating duration indicates its service time span; and the binary indicators of subway or transfer station status provide modal attributes, facilitating the analysis of passenger capacity variations across modes and the structural significance of coupling nodes in the two-layer network.

3)Spatialdistributionfeatures

Spatial location, station coverage, nearest neighbor distance, average neighbor distance, and spatial dispersion of neighbors are considered for capturing the spatial distribution features. These features are derived from station coordinates using UTM projection, yielding units in meters. Specifically, station coverage is computed as a weighted sum of the number of stops within 500 m, 1000 m, and 2000 m radii (weights: 0.5, 0.3, 0.2), normalized by the total number of nodes, yielding a dimensionless value between 0 and 1. Spatial dispersion of neighbors is the mean pairwise distance among all neighboring stops, measured in kilometers; if the number of neighbors is less than 2, this value is set to 0.

The above features originate from three different data domains, with significant disparities in numerical scales and distribution characteristics. Therefore, in the preprocessing stage, Z-score standardization is performed separately for each city to eliminate the influence of dimensional differences on model training. For a node v i its multi-dimensional feature vector is denoted as:

where N is the number of multi-dimensional features. The multi-dimensional feature matrix of all nodes in the transportation network is expressed as:

where n is the total number of nodes in the network.

A learnable linear projection layer is applied to map the feature matrix X m u l t i to the target embedding dimension space. Layer normalization is employed to improve training efficiency, reduce sensitivity to initial weights, enhance generalization, and mitigate overfitting. The specific computation is formulated as:

where LN ( · ) denotes the layer normalization function, W p r o j ∈ R N × d is the learnable weight matrix, b p r o j ∈ R d is the bias term, and d is the embedding dimension. The projected feature X p r o j ∈ R n × d serves as the input feature for subsequent modules.

Graph Sample and Aggregate (GraphSAGE) [24] is an inductive graph representation learning framework that updates node representations by aggregating features from their neighbors, offering high computational efficiency and strong scalability. At layer l , for a node v i , a fixed-size neighborhood set N ( i ) of size S is randomly sampled from its neighbor set N S ( i ) , and the neighborhood features are aggregated as,

Then, the node’s own features are concatenated,

where σ denotes the ReLU activation function, and W ( l ) ∈ R d × 2 d is the weight matrix of the l -th layer. The aggregator function AGGREGATE ( · ) is implemented as the mean aggregator in this paper,

A three-layer GraphSAGE network is adopted to extract structural features of nodes, followed by layer normalization to improve training stability. The overall computation can be formulated as:

where σ is the ReLU activation function, and H S A G E ∈ R n × d is the output of the GraphSAGE branch.

Through multi-layer neighbor aggregation, this pathway progressively disseminates node features to farther neighborhoods, thereby effectively capturing the global structural information of nodes.

To capture the global structural information of the entire transportation network and compensate for the information limitations that may arise from local aggregation in GNNs, a global context extraction module is designed based on learnable scoring and weighted pooling. This module takes the output features of the GraphSAGE network H S A G E ∈ R n × d as input, computes a contextual importance score for each node through a two-layer MLP with Tanh activation, and then obtains weighting coefficients via Softmax normalization. Finally, a global context vector is derived through weighted pooling and expanded to each node, enabling all nodes to share the same global perspective.

First, the fused node features are fed into a two-layer MLP with Tanh activation to compute an importance score e ∈ R n × 1 for each node,

where σ 1 denotes the Tanh activation function W 1 ∈ R d × d / 2 and W 2 ∈ R d / 2 × 1 are weight matrices, and b 1 ∈ R d / 2 and b 2 ∈ R are bias terms.

Then, the score vector e is normalized by Softmax to obtain the normalized weights w ∈ R n × 1 ,

Based on these weights, weighted pooling is applied to the node features to obtain the global context vector c ∈ R d ,

where h S A G E ( i ) ∈ R d denotes the fused feature vector of the i -th node.

Finally, the context vector is broadcast to all nodes by repeating it n times, yielding the context matrix C ∈ R n × d as the output of this module.

This module allows each node’s final representation to capture both local neighborhood information and global graph context, thereby providing a more comprehensive characterization of node roles within the network and richer features for node importance prediction.

The features H S A G E ∈ R n × d output by the GraphSAGE network module and the features C ∈ R n × d output by the global context extraction module are concatenated along the feature dimension to obtain the final feature matrix,

Subsequently, H f i n a l is fed into a two-layer MLP to obtain the prediction score y ^ ∈ R n ,

where σ 2 denotes the Sigmoid activation function, MLP ( · ) denotes a fully connected network. A higher score y ^ indicates a more important node within the network.

To generate ranking labels for node importance, a two‑stage label generation method is adopted that integrates centrality metrics and the gravity model. This method first computes node importance by fusing multiple centrality metrics, and then characterizes the influence of nodes on their surrounding nodes via the gravity model. The final result serves as the importance label for nodes. The specific steps are as follows.

1)Multi-dimensionalCentralityFusion

To characterize node importance, five classic centrality metrics are selected, Degree Centrality (DC), Eigenvector Centrality (EC), PageRank (PR), Closeness Centrality (CC) [6], and NK-shell [10]. Among them, NK-shell extends the traditional k-shell by incorporating first-order and second-order neighbor information, more comprehensively reflecting a node’s core hierarchy and local influence. To eliminate scale differences among these metrics, Min-Max normalization is applied to the above five metrics, and they are fused with equal weights to obtain the node mass for each node.

2)Gravity-basedImportanceCalculation

Node mass reflects the importance of individual nodes; however, to account for the mutual influence between nodes, the gravity model is further introduced [25], [26]. By treating nodes as mass points, the model evaluates the comprehensive influence of nodes within the network through simulating gravitational interactions between them.

The shortest path distance is adopted as the distance metric between nodes. The shortest path distances between all node pairs are computed to obtain the distance matrix D = [ d i j ] n × n , where d i j denotes the shortest path distance between nodes v i and v j . To reduce computational complexity and focus on local influence, the gravity radius is set to R = 3 , and only node pairs with d i j ≤ R are retained. The gravity importance S i of node v i is defined as,

where M i and M j denote the node masses of v i and v j , respectively. The gravity importance vector of all nodes is given by,

3)LabelGeneration

Min-Max normalization is applied to the gravity importance scores S i to obtain the final labels,

where ε is a tiny constant to prevent division by zero. The i -th component y i ∈ [ 0 , 1 ] represents the importance label of node v i .

The loss function is denoted as L o s s . In this paper, a pairwise ranking loss is adopted to enable the model to learn the relative importance ranking among nodes. This loss is consistent with the hinge loss formulation in Ranking SVM [27]. As node pairs are randomly sampled, the loss function is defined as,

where P is the set of randomly sampled node pairs, y i and y j denote the ground-truth importance scores of nodes i and j , y ^ i and y ^ j denote the predicted scores of nodes i and j , δ > 0 is a margin hyperparameter, and sign ( ⋅ ) denotes the sign function. This loss function, by minimizing L o s s , enables the model to accurately fit node importance while preserving the correctness of relative order between nodes, thereby improving the model’s performance in ranking tasks.

The proposed model for identifying key nodes in urban public transportation networks is shown in Figure 2. The model comprises six core modules: transportation network construction, node multi-feature construction, GraphSAGE-based feature extraction, global context extraction, key node identification, and node label generation. First, features from multi-source data are constructed based on three dimensions: topological structure, operational status, and spatial distribution. Next, a GraphSAGE network is developed to capture local neighborhood features of the network. Furthermore, a global context extraction module is designed to extract global contextual information of nodes through weighted pooling and repeated expansion, and a multi-layer neural network is employed to predict node importance. Meanwhile, a gravity model is used to fuse multiple centrality metrics to generate reference labels, and a ranking loss function is adopted to train the network model. Finally, the trained network is applied to identify key nodes in the urban public transportation network.

Figure 2. Framework of the proposed model.

The public transportation network data used in this paper is sourced from the China Public Transport Operation Network Dataset (CPTOND‑2025) [19]. This dataset covers the subway networks of 351 cities in China and includes extensive information such as routes, stations, and operational attributes. To date, it is the first comprehensive nationwide public transportation dataset in China, covering 351 cities, some of which have both bus and subway networks, while others have only bus networks.

A total of 9 cities were selected to construct the experimental dataset. The training set comprised 7 cities (Xining, Yinchuan, Urumqi, Hohhot, Taiyuan, Beijing, and Chengdu). Wuhan was the validation set, and Lanzhou was the test set. This ensures that the training and test data are geographically independent, thereby effectively evaluating the model’s generalization ability. In terms of network scale, taking Lanzhou as an example, the final network contains 1143 nodes (1116 bus stops and 27 metro stations) and 1884 edges (1794 bus edges, 27 metro edges, and 63 transfer edges), with an exclusion ratio of 2.47%.

1)Kendall’sτ CorrelationCoefficient

Kendall’s τ correlation coefficient [28] is a statistical measure of ordinal association between two ranked lists, defined as,

where n denotes the number of nodes, c represents the number of concordant pairs, and d represents the number of discordant pairs.

In this study, the reported Kendall’s τ measures the ordinal agreement between the predicted node rankings and the gravity-centrality surrogate labels, rather than an independent observation of real-world node importance. In the absence of ground-truth importance labels for urban public transportation networks, these surrogate labels—generated by fusing multiple centrality measures with a gravity model—serve as a reasonable proxy for node importance, providing a quantitative basis for comparing different ranking methods.

2)LargestConnectedComponentRatio

The Largest Connected Component (LCC) [29] ratio is a metric for evaluating network vulnerability. This metric assesses the effectiveness of node importance ranking methods by measuring the change in the size of the largest connected component after the network undergoes attacks. The LCC ratio is defined as,

where | C i n i t i a l | is the number of nodes in the largest connected component of the original network, and | C a f t e r | is the number of nodes in the largest connected component after removing a fraction ρ of nodes. LCC ( ρ ) ∈ [ 0 , 1 ] . This paper adopts the LCC metric to validate the Vulnerability of the key node rankings identified by the model. A faster decline in the LCC value indicates that the nodes identified by the model are more destructive to the network structure.

To evaluate the effectiveness of the proposed model, Betweenness Centrality, Clustering Coefficient, TOPSIS, GCN, FGM, GCNGM, and SOLG are selected as baseline models.

Betweenness Centrality [6]. Betweenness centrality is a global centrality metric based on shortest paths, which quantifies the frequency with which a node appears on the shortest paths between other nodes, thereby measuring the node’s bridging role in the network.

Clustering Coefficient [6]. The clustering coefficient measures the degree to which the neighbors of a node are also neighbors of each other, reflecting the local clustering and information transfer efficiency of the network.

TOPSIS [6]. TOPSIS is a multi-attribute decision-making method that ranks alternatives by calculating their relative closeness to the positive and negative ideal solutions. In node importance evaluation, nodes are treated as alternatives and centrality metrics as attributes, with a higher closeness indicating greater node importance.

GCN [30]. Graph Convolutional Network (GCN) performs feature extraction and node representation learning on graph-structured data through spectral graph convolution, enabling the simultaneous aggregation of node features and topological information.

FGM [31]. FGM is a node importance evaluation method based on the gravity model. It defines node mass by integrating multiple attributes such as degree, k-shell, and eigenvector centrality, and evaluates node importance in conjunction with shortest path distances.

GCNGM [32]. GCNGM is a gravity model-based node importance evaluation method that leverages node embedding representations. This method generates node topological embeddings via a fixed-weight graph convolutional network, uses the L2 norm of the embedding vectors as node mass, and constructs a gravity model combined with shortest path distances to evaluate node importance.

SOLG [33]. SOLG is a node importance evaluation method that integrates multi-stage topological features. It quantifies the connection strength between nodes by extending the Jaccard coefficient, and fuses first-order, higher-order, and global influences to assess node importance.

The number of GraphSAGE layers is set to 3, and all subsequent experiments are conducted under this configuration. The hidden layer dimension in GraphSAGE is set to 128. The margin hyperparameter δ is set to 0.1. The AdamW optimizer is employed with an initial learning rate of 5 × 10⁻⁴, and a weight decay of 1 × 10⁻⁴, The total number of training epochs is set to 500. For model selection, we adopted early stopping on the Wuhan validation set, and the best model parameters were saved for final testing. Lanzhou data were used exclusively as the test set and were never involved in model training, validation, or hyperparameter tuning.

For all baseline models, the hyperparameters suggested by the original authors are adopted, further adjusted based on the dataset, and all methods are ensured to be compared under identical training conditions. All experiments are conducted on a rented cloud server equipped with an NVIDIA RTX 3090 (24 GB) GPU, 14 vCPUs Intel Xeon Gold 6330 CPU, 90 GB RAM, and 80 GB SSD storage, running Ubuntu 20.04. The experiments are implemented using PyTorch 1.11.0, Python 3.8, and CUDA 11.3.

Figure 3 shows a comparison of the node importance ranking performance between the model proposed in this paper and the baseline models on the Lanzhou public transportation network. Overall, there are significant differences in ranking accuracy among the various types of methods. First, traditional centrality metrics perform poorly in the ranking task, reflecting that relying solely on the static topological structure or local features of the network makes it difficult to accurately identify key nodes in the public transportation network. Among the multi-feature fusion methods and supervised learning baseline models, the Kendall’s τ results for TOPSIS, FGM, and GCN fall in the middle range. Among these, the GCN model achieved a Kendall’s τ of 0.6703, indicating the basic suitability of graph neural networks for extracting spatial association features of nodes in urban public transportation networks. Among the unsupervised learning baseline methods, the SOLG algorithm and the GCNGM algorithm, which is based on graph convolutional and gravity models, performed relatively well, with Kendall’s τ of 0.8495 and 0.8213, respectively, significantly outperforming the previous methods. In the Lanzhou public transportation network test, the proposed model achieved the highest Kendall’s τ of 0.8568. Compared with the optimal baseline model, SOLG, represents a relative improvement of approximately 0.86%. These results indicate that the proposed model demonstrates better ranking accuracy in the task of identifying the importance of transportation nodes and can effectively identify key nodes within the network.

Figure 3. Comparison of Kendall’s τ among different methods.

To analyze the consistency between the predicted scores of each method and the ground-truth labels, Figure 4 presents the scatter plots of prediction scores versus label values for all methods on the Lanzhou dataset. In these plots, the horizontal axis denotes the node importance prediction scores produced by each method, and the vertical axis denotes the computed label values.

An examination of the scatter distributions reveals that the points of the proposed model are the most tightly clustered around the diagonal, indicating a strong linear correlation between the predicted scores and the labels. For the baseline methods SOLG and GCNGM, although the overall distribution generally aligns with the diagonal, underestimation is observed in certain regions, causing the corresponding points to fall slightly below the diagonal. The scatters of GCN and FGM also exhibit a discernible diagonal trend, but notable deviations can be observed in some areas. In stark contrast, the results of BC, CC, and TOPSIS display a scattered and stacked pattern, failing to effectively discriminate node importance. In summary, the scatter pattern of the proposed model on the Lanzhou transportation network demonstrates superior consistency compared with the baseline methods, thus validating its effectiveness in the node importance assessment task.

Figure 4. Scatter plots of predicted scores versus ground-truth labels for all methods.

To determine node importance levels, the K-means clustering algorithm (K = 12) is applied to the importance scores produced by the proposed model. Based on the average score of each cluster, the 12 clusters are further grouped into three importance levels: the cluster with the highest scores (30 nodes) is labeled as key nodes, the cluster with the second-highest scores (34 nodes) as important nodes, and the remaining 3rd to 12th clusters collectively as ordinary nodes (1079 nodes), totaling 1143 nodes. Figure 5 illustrates the spatial distribution of node importance levels in the Lanzhou public transportation network. Spatially, key nodes exhibit an east-west belt-like distribution and are highly concentrated in the core commercial areas of the main urban district, whereas the peripheral route endpoints are predominantly ordinary nodes. This distribution pattern closely matches the constrained elongated urban form of Lanzhou, shaped by the Yellow River valley, thus verifying the practical effectiveness of the model.

Table 1 presents the specific ranking of the 30 key nodes identified by the model. Among them, “Public Transport Group,” “Xizhan Shizi,” and “Wenhua Palace” are located at intersections of core commercial areas and main arterial roads; “Lanzhou West Railway Station” and “Lanzhou Railway Station” serve as key transportation transfer hubs; and “Provincial Government” and “Lanzhou University” function as essential public service and activity centers. The accurate identification of these nodes as key further validates the applicability and reliability of the proposed method in real-world public transportation networks.

Figure 5. Spatial distribution of identified key nodes.

Table 1. Top 30 key nodes identified in Lanzhou.

To verify the practical effectiveness of the proposed model in identifying key nodes, a Vulnerability analysis is conducted using a sequential node removal strategy and observing the decay of the relative size of the largest connected component (LCC). Figure 6 illustrates the curves of the LCC ratio under different node ranking methods on the Lanzhou public transportation network, where the horizontal axis represents the proportion of nodes removed in descending order of importance, and the vertical axis represents the relative size of the largest connected component.

As shown in Figure 6, in the initial node removal interval from 0% to 30%, the LCC ratio of the proposed model declines most steeply. This indicates that removing nodes sequentially according to the importance ranking identified by the proposed model can dismantle the network connectivity at the fastest rate. The unsupervised methods SOLG and GCNGM also exhibit rapid decay in the early stage, though their overall performance is slightly inferior to that of the proposed model. Notably, the decline of GCN is slightly slower than the above three methods, with a stagnation in the LCC curve observed between the removal ratios of 40% and 50%. This reflects the over-reliance of the GCN architecture on local information during feature aggregation, which hinders its ability to accurately identify key nodes connecting different regions. In contrast, the traditional single centrality metrics (BC, CC) and the multi-attribute aggregation framework TOPSIS present relatively gentle decline curves, suggesting that such methods struggle to comprehensively capture the nodes that play a critical role in the overall connectivity of transportation networks.

Overall, the Vulnerability performance of the proposed model is consistent with the conclusions drawn from the Kendall’s τ analysis, further validating its effectiveness and reliability in the task of key node identification.

Figure 6. LCC ratio versus node removal proportion for different methods.