Graph neural network link prediction method based on multi-dimensional similarity

Abstract

The invention discloses a graph neural network link prediction method based on multi-dimensional similarity, and aims to improve the accuracy and generalization ability of missing link prediction in a complex network. The method is suitable for social networks, citation networks, recommendation systems and other actual scenes with isomerism, sparsity and dynamic evolution characteristics. In order to solve the problem that a traditional method only depends on local adjacency information or single topological similarity and is difficult to capture a high-order structure relation and multi-dimensional feature association, a unified measurement system fusing structure similarity, attribute similarity and path similarity is constructed, and potential association between nodes is deeply mined. By introducing a self-adaptive feature weighting mechanism, the model can dynamically adjust the multi-dimensional similarity fusion weight according to network features, and the expression and distinguishing capability of the node relationship is enhanced. On the basis, the deep representation learning advantage of the graph neural network is combined, a model structure with the selective feature fusion capability is designed, and precise modeling of a complex link generation mechanism is achieved. The method has good expandability and interpretability, and prediction deviation caused by network heterogeneity can be effectively relieved. Experimental results show that the method is obviously superior to the existing mainstream method on a plurality of real network data sets, and has high theoretical value and wide application prospect.

Description

Technical Field

[0001] This invention belongs to the field of complex networks, specifically relating to a graph neural network link prediction method based on multidimensional similarity, applicable to link prediction research in complex networks such as social networks, citation networks, and recommender systems. Background Technology

[0002] In complex network research, link prediction is a key task, aiming to predict future links in the network or identify currently unobserved but potentially existing edges. This task has broad practical value in many application scenarios, such as friend recommendation in social networks, protein-protein interaction prediction in biological networks, path optimization in transportation networks, and user-item matching in recommendation systems. As the scale and complexity of real-world networks continue to increase, the challenges of link prediction become increasingly severe.

[0003] Complex networks typically exhibit strong heterogeneity, sparse structure, and dynamic evolution, which leads to performance bottlenecks in traditional shallow link prediction methods based on adjacency matrices. These methods often rely on local features such as direct connections or common neighbors of nodes, making it difficult to effectively capture higher-order structural relationships in the network, thus limiting their prediction accuracy in sparse or dynamic networks. For example, commonly used similarity metrics such as common neighbors (CN), Jaccard coefficient, and Adamic-Adar (AA) can only uncover local connection patterns of nodes, ignoring global or implicit structural features of the network.

[0004] To address these issues, researchers have gradually introduced deep learning methods into link prediction tasks. Among them, Graph Neural Networks (GNNs), with their powerful graph structure modeling capabilities, have become the mainstream research direction in link prediction. GNNs can automatically learn the latent representations of nodes by iteratively aggregating information from their neighbors, thus enabling edge existence determination. However, most existing GNN link prediction methods are still mainly based on a single type of topological similarity or rely solely on the dot product calculation of node embeddings, lacking an effective mechanism for fusing and modeling multi-dimensional similarity features, making it difficult to comprehensively capture the complex relationships between nodes in the network.

[0005] On the other hand, nodes in complex networks may simultaneously possess multiple association features, such as structural similarity, attribute similarity, and higher-order path relationships, constituting what is known as "multidimensional similarity." In practical applications, node connections are often not based on a single similarity dimension, but rather a combination of multiple factors. For example, in a citation network, two document nodes may be connected due to similar topics (attribute dimension), shared citations of other documents (structural dimension), or the existence of multi-hop citation paths (path dimension). Therefore, constructing a link prediction model that can integrate multidimensional similarity features is of great significance for improving prediction accuracy and model generalization ability.

[0006] Some studies have attempted to use different similarity indices as input features and combine them with GNNs for feature fusion and prediction. However, these methods often have two shortcomings: first, the feature fusion methods are relatively simple and cannot characterize the weight relationships between different dimensions of similarity; second, they lack interpretability and are difficult to analyze the specific contributions of different similarity features to the prediction results. Especially in the face of dynamic changes in network structure or increased node heterogeneity, fixed fusion rules are difficult to adapt to diverse link formation mechanisms.

[0007] Therefore, how to fully utilize multidimensional similarity features to construct a graph neural network link prediction model with adaptive fusion capabilities and interpretability has become a current research hotspot and challenge. This invention is proposed against this backdrop, aiming to construct a unified multidimensional similarity measurement system by introducing multiple feature dimensions such as structural similarity, attribute similarity, and path similarity. Based on this, a graph neural network architecture with selective feature fusion capabilities is designed to improve the accuracy and generalization performance of link prediction. This method not only more accurately reflects the multi-level and multi-angle potential connections between nodes but also provides a new approach to solving the prediction bias problem caused by heterogeneity in complex networks, possessing significant theoretical research value and broad application prospects. Summary of the Invention

[0008] This invention aims to address two main problems in existing link prediction methods: first, shallow similarity metrics based on adjacency matrices struggle to capture high-order structural information in networks, neglecting multi-hop paths and potential semantic relationships; second, traditional methods lack the ability to deeply model network structures, making it difficult to effectively integrate multi-dimensional similarity features between nodes. To address these issues, this invention proposes a graph neural network-based link prediction method based on multi-dimensional similarity. This method comprehensively considers multi-dimensional features such as structural similarity, attribute similarity, and high-order path similarity, constructs an adaptive feature weighting mechanism, and deeply integrates it with the graph neural network, thereby effectively improving the accuracy and generalization ability of link prediction.

[0009] The technical solution of this invention is:

[0010] 1. Construction of multidimensional similarity features

[0011] In link prediction tasks, the existence of potential connections between nodes often depends on their structural position in the network, attribute information, and indirect path relationships. To comprehensively characterize the association strength between nodes, this invention proposes a multi-dimensional similarity feature construction method, which measures and fuses the similarity of node pairs from the following three aspects:

[0012] 1) Structural similarity:

[0013] Structural similarity reflects the degree of similarity between the positions of two nodes in a network structure. This invention uses the Common Neighbors (CN) index for measurement. This index suggests that if two nodes have more common neighbors, they are more likely to have potential connections. Specifically, the CN value is equal to the number of common neighbors between the two nodes, as shown in the following formula:

[0014] S CN (u,v)=|N u ∩N v | (1)

[0015] In the formula, N u Let N represent the set of neighbors of node u. v Let |N| represent the set of neighbors of node v. u ∩N v | represents the number of common neighbors of nodes u and v. S CN The larger the (u,v) value, the more similar the local structure of the node pair (u,v), and the higher the potential for forming a link.

[0016] 2) Attribute similarity:

[0017] Attribute similarity measures the degree of similarity between nodes in an attribute space. This invention uses a cosine similarity index, which evaluates the similarity between two node attribute vectors by calculating the cosine of the angle between them. The formula is as follows:

[0018]

[0019] In the formula, x u and x v Let x represent the feature vectors of nodes u and v respectively. u ‖ and ‖x v ‖ represents the Euclidean norm of the two vectors. The formula takes values ​​between [-1, 1]. The closer the value is to 1, the more similar the features of the nodes are, and they may belong to the same category or have the same attributes. The closer the value is to 0, the more unrelated or dissimilar the features of the nodes are.

[0020] 3) Path similarity

[0021] Path similarity reflects the degree of indirect connection between node pairs through intermediate nodes in the network. This study uses the shortest path length as a metric. The shorter the shortest path between two nodes, the higher their correlation in the network, and the greater the likelihood of forming a potential connection. The definition is as follows:

[0022]

[0023] In the formula, sp(u,v) represents the shortest path length between node u and node v, and σ is the attenuation coefficient used to control the influence of path distance on similarity weight. When sp(u,v) is small, S PATH A value close to 1 for sp(u,v) indicates strong connections between nodes, making it easier to form links; when sp(u,v) is large, S PATH A value close to 0 for (u,v) indicates a large distance between nodes, making it less likely for new links to form. This formula not only reflects the nonlinear decay of path distance but also allows for adjustment of the sensitivity to relationships between distant nodes through the parameter σ, thus enhancing the ability to model higher-order path features.

[0024] 2. Similarity fusion mechanism design

[0025] To effectively integrate multi-dimensional features such as structural similarity, attribute similarity, and higher-order path similarity, this invention designs a similarity fusion mechanism based on joint modeling of temporal attention mechanism and multilayer perceptron, realizing dynamic weighting and nonlinear combination of different types of similarity features, thereby improving the expressive power and generalization performance of link prediction.

[0026] 1) Multidimensional similarity feature input

[0027] Let the multidimensional similarity feature vector of any node pair (u,v) be:

[0028]

[0029] in, Corresponding to the aforementioned structural similarity, attribute similarity, and path similarity measures, S uv It is a 3-dimensional row vector, and the value range of each element is consistent with the value range of the corresponding similarity feature. It is used to fully characterize the association strength of node pair (u,v) in different dimensions.

[0030] 2) Time attention mechanism

[0031] To highlight the contribution of key similarity features to the prediction results, this invention introduces a temporal attention mechanism, which adaptively assigns importance coefficients to each dimension of similarity through learnable weights. Specifically, the attention weights are calculated as follows:

[0032]

[0033] Where W is a learnable linear transformation matrix that performs a linear transformation on the similarity features; a is the attention vector used to calculate the importance of each similarity; LeakyReLU(·) is the ReLU activation function with leakage; and the Softmax function ensures that all weights satisfy normalization, i.e. This attention mechanism can automatically identify the most critical feature dimension for the current prediction task based on the distribution characteristics of node pairs under different similarity indices, thereby achieving adaptive dynamic weighting between features and enhancing the robustness and generalization of the model under different network structures and time evolution conditions.

[0034] Finally, the attention-weighted multidimensional similarity features are fused into a global similarity score. The formula for its calculation is as follows:

[0035]

[0036] The fusion result comprehensively reflects the overall association strength of node pairs in the multidimensional similarity space, providing a more comprehensive and interpretable feature representation for subsequent link existence discrimination based on graph neural networks.

[0037] 3) Multilayer perceptron fusion

[0038] To further enhance the nonlinear expressive power of the model, this invention designs a Multi-Layer Perceptron (MLP) module to perform a nonlinear transformation on the weighted features mentioned above, in order to capture higher-order interaction relationships. Specifically:

[0039] The weighted fusion features of the attention mechanism output are Its nonlinear transformation process through the MLP module is expressed as follows:

[0040]

[0041] Where W1 and W2 are the learnable weight matrices of the MLP, used for linear mapping of the feature space; b1 and b2 are bias terms; σ(·) is a nonlinear activation function used to enhance the nonlinear expressive power of the features; h uv Let f be the fused embedding representation of the node pair (u,v).

[0042] 3. Graph Neural Network Modeling

[0043] After constructing and fusing multidimensional similarity features, this invention introduces a graph neural network modeling method based on Graph Convolutional Network (GCN) to further model higher-order structural relationships between nodes and improve link prediction performance. This model can effectively propagate information between nodes in a graph structure and gradually capture the local and global contextual features of nodes through multi-layer graph convolution operations.

[0044] This invention uses the classic GCN layer as the basic module. Its core idea is that in each graph convolutional layer, the node representation is aggregated and updated through information from neighboring nodes, forming a layer-by-layer evolving node embedding. Specifically, let the graph G = (V, E), where V represents the set of nodes, E represents the set of edges, and the node feature matrix is... Where N is the number of nodes and d is the original feature dimension.

[0045] The basic propagation formula for graph convolutional layers is:

[0046]

[0047] in:

[0048] · The node representation matrix of layer l (initially H) (0) =X);

[0049] · Adjacency matrix after adding self-connections;

[0050] · The degree matrix;

[0051] ·W (l) : The learnable weight matrix of the l-th layer;

[0052] ·σ(·): Activation function LeakyReLU.

[0053] The above normalization operation This effectively avoids numerical instability and improves aggregation efficiency, ensuring a balanced feature contribution of each node in neighborhoods of different sizes. Through multi-layer GCN stacking, the model can effectively capture complex high-order structural context information, generating final node embeddings h with rich semantics. uv .

[0054] In obtaining node embedding h uv Then, it is input into the link prediction module of the graph neural network to calculate the connection probability between node pairs (u,v). The prediction formula is as follows:

[0055]

[0056] Among them, w c It is a learnable weight parameter, b c The bias term is sigmoid(·), which is the activation function used to map the output value to the interval (0,1). This represents the probability that an edge exists between node pairs (u, v). A higher probability indicates a greater likelihood of a link between nodes u and v. To optimize model performance, a cross-entropy-based loss function is used for supervised learning of the prediction results. The calculation formula is as follows:

[0057]

[0058] Among them, E + and E - Let y represent the sets of positive samples (node ​​pairs with edges) and negative samples (node ​​pairs without edges), respectively. uv The values ​​represent the true labels. By minimizing this loss function, the model can effectively learn the potential structural patterns between nodes, improving the accuracy and generalization ability of link prediction.

[0059] This completes the technical solution for the proposed method.

[0060] Table 1 shows the overall process of the link prediction method MSGNN proposed in this invention.

[0061] Table 1: MSGNN Link Prediction Algorithm Flow

[0062] Attached Figure Description

[0063] To make the objectives, technical solutions, and beneficial effects of this invention clearer, the following figures are provided for illustration:

[0064] Figure 1 This is a diagram illustrating the overall framework of the MSGNN link prediction method proposed in this invention.

[0065] Figure 2 A comparison chart of AUC for link prediction using different methods on different datasets;

[0066] Figure 3 A comparison chart of AP for link prediction using different methods on different datasets;

[0067] Figure 4 Hit@20 comparison chart of link prediction for each method on different datasets. Detailed Implementation

[0068] To make the technical solution, advantages and objectives of the present invention clearer, the present invention will be described in detail below with reference to specific embodiments.

[0069] 1. Dataset

[0070] To evaluate the advantages of this invention compared to other link prediction methods, publicly available datasets were obtained from the internet and compared with representative methods. Network datasets from six different domains were used to evaluate the performance of the proposed method.

[0071] The dataset details are as follows:

[0072] 1. Cora: Citation Network (in computer science). Nodes represent papers, and edges represent citation relationships.

[0073] 2. PubMed: Citation Network (Biomedical field). Nodes represent papers, and edges represent citation relationships.

[0074] 3. Twitch: A social network (the Twitch platform). Nodes represent Twitch users, and edges represent interactions between users.

[0075] 4. Facebook: A social network (Facebook platform). Nodes represent Facebook users, and edges represent the social relationships between users.

[0076] 5. YeastPPI: Biological Network (Protein-Protein Interactions). Nodes represent proteins, and edges represent the interactions between proteins.

[0077] 6. Airports: Transportation network (USA aviation). Nodes represent airports, and edges represent flight routes between airports.

[0078] Table 2 provides a brief summary of the above dataset, where |V| represents the total number of nodes, |E| represents the total number of edges, D represents the network diameter, <k> represents the average degree of nodes, and <cc> represents the average clustering coefficient.

[0079] Table 2: Dataset Introduction

[0080]

[0081]

[0082] 2. Evaluation Indicators

[0083] This study uses three metrics to evaluate model performance: area under the ROC curve (AUC), average precision (AP), and hit rate (Hits@20).

[0084] AUC: The probability that the model ranks the score of the true missing link before the score of a random negative sample. It measures the model's ability to distinguish between positive and negative samples. The formula is defined as follows:

[0085]

[0086] Where n is the number of independent comparisons, n′ is the number of missing nodes with higher scores, and n″ is the number of comparisons with the same scores. A higher AUC indicates a stronger ability of the model to accurately distinguish between node pairs with and without links, and AUC = 1 represents perfect classification.

[0087] AP: The average prediction precision of the model at different recall rates, which measures the overall quality of positive samples in the ranking results. The formula is defined as follows:

[0088]

[0089] Where P n For accuracy, R n This is the recall rate at the nth threshold. AP focuses on the predictive ability of positive samples (true edges); a higher value indicates a more accurate model.

[0090] Hits@20: True links appear in the top 20 results with the highest model prediction scores. This measures the model's ability to hit positive samples among the top 20 candidates. The formula is defined as follows:

[0091]

[0092] Where V test This is the test set, and rank(t) is the ranking of t. A higher Hits@20 value indicates that the model is more capable of finding real links in high-confidence predictions, reflecting the model's ranking coverage ability.

[0093] 3. Comparison Model

[0094] To comprehensively evaluate the effectiveness of the proposed method, this study selected several representative link prediction methods as comparative experimental models, covering various paradigms such as graph neural networks, autoencoders, and subgraph modeling. The specific methods are described below:

[0095] 1. GCN: A classic graph neural network model that learns representations by aggregating features of neighboring nodes and has wide applications in various graph tasks.

[0096] 2. VGAE: An unsupervised graph autoencoder model that uses variational inference to learn the latent representations of nodes, suitable for modeling the latent space structure of graphs.

[0097] 3. SEAL: Based on the idea of ​​extracting locally closed subgraphs, it combines graph neural networks to model structural patterns, which effectively improves the ability to recognize complex topological structures.

[0098] 4. GraphSAGE: A scalable graph neural network model that learns representations by sampling and aggregating features from a fixed number of neighbors, making it suitable for processing large-scale graphs.

[0099] 5. NSMLLP: A link prediction algorithm that integrates node centrality, similarity metrics, and machine learning classifiers. It improves consistency across different networks by utilizing combined node features for link prediction.

[0100] 6.NCSM: A hybrid model that integrates node structure features and similarity features, combining heuristic methods with neural networks to improve prediction accuracy while ensuring efficiency.

[0101] 4. Parameter settings

[0102] To construct the model input data, this invention divides the positive edges in the network into training, validation, and test sets. During training, an equal number of negative edges are generated through random sampling to form negative samples. On six different types of network datasets, 5% of the edges are randomly removed from the original graph as positive samples for the validation set, and 10% of the edges are removed as positive samples for the test set. An equal number of non-existent edges are generated for both the validation and test sets as negative samples. The model structure consists of four layers, each containing 256 neurons, trained using the Adam optimizer with a learning rate of 0.01 and 200 training epochs. The settings of other baseline models also follow the implementation scheme of the original study. Since link prediction is essentially a binary classification problem, this invention uses binary cross-entropy as the loss function. Evaluation metrics include AUC, AP, and Hits@20, and each experiment is repeated 10 times, recording the average value. All graph neural network models and the three similarity measurement methods are implemented in the Python environment using PyTorch Geometric, Scikit-learn, and NetworkX toolkits, respectively.

[0103] 5. Results and Analysis

[0104] Tables 3, 4, and 5 present the comparison results of AUC, AP, and Hit@20 metrics for link prediction using various methods on different datasets (bold indicates the best performance of all methods). For ease of presentation and comparison, all evaluation metrics were multiplied by 100 and presented as percentages. The results in the tables show that the MSGNN method proposed in this invention outperforms the baseline methods in all three performance metrics.

[0105] Table 3: AUC of link prediction for each method on different datasets

[0106]

[0107]

[0108] Table 4: AP of link prediction for each method on different datasets

[0109]

[0110] Table 5: Hit@20 for link prediction by each method on different datasets

[0111]

[0112] In terms of AUC, MSGNN achieved the highest score across all datasets. For example, it achieved 97.89 and 98.16 on Cora and PubMed, respectively, significantly outperforming the second-best model, NCSM (95.92 and 95.53), indicating that MSGNN can more accurately capture potential link structures in academic citation networks. MSGNN also performed well on the more challenging Twitch, Facebook, and YeastPPI networks, achieving AUCs of 95.74, 99.24, and 94.34, respectively, demonstrating its good adaptability to heterogeneous structures and complex interactions.

[0113] In terms of AP (Average Performance) metrics, MSGNN continues to lead. For example, it achieved APs of 97.93 and 99.08 on PubMed and Facebook, respectively, representing improvements of 2.68% and 0.15% over NCSM. On the YeastPPI biological network dataset, MSGNN also outperformed all models with an AP of 95.23, demonstrating its advantage in capturing low-frequency edges and sparse connection patterns. In contrast, other models such as GCN, VGAE, and SEAL showed significant performance gaps across multiple datasets.

[0114] MSGNN's advantage is even more significant on the Hits@20 metric. Especially on core datasets such as Cora, PubMed, and Facebook, it far surpasses other methods. For example, on Cora, MSGNN improves upon NCSM by 5.07%, and on the Airports dataset, it achieves a hit rate of 65.34, demonstrating strong ranking capabilities in high-ranking link prediction tasks. Even on the structurally complex Twitch and YeastPPI networks, MSGNN achieves excellent scores of 55.47 and 51.23 respectively, further validating its generalization ability.

[0115] In summary, MSGNN achieves a comprehensive lead in AUC, AP, and Hits@20 metrics. Its performance advantage is mainly attributed to: (1) the fusion of similarity from multiple dimensions; and (2) the introduction of an adaptive feature weighting mechanism. This design significantly enhances the model's ability to express complex network structures and effectively pushes the performance ceiling of link prediction tasks.

[0116] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any transformations or substitutions that can be conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of the present invention.

Claims

1. A graph neural network link prediction method based on multidimensional similarity, characterized in that, Includes the following steps: (1) Input an unweighted, undirected network G = (V, E), where V is the set of nodes and E is the set of edges. Initialize the connection probabilities of all candidate node pairs (u, v). (2) Perform the following operations for each candidate node pair (u,v): a) Calculate the structural similarity S between node pairs (u,v). CN (u,v), attribute similarity S COS (u,v) and path similarity S PATH (u,v); b) Construct a multidimensional similarity feature vector for node pairs (u,v): in, Corresponding to the aforementioned structural similarity, attribute similarity, and path similarity measures, S uv It is a 3-dimensional row vector. c) Introduce a temporal attention mechanism, which adaptively assigns importance coefficients to the similarity across dimensions using learnable weights. Specifically, the attention weights are calculated as follows: Where W is a learnable linear transformation matrix that performs a linear transformation on the similarity features; a is the attention vector used to calculate the importance of each similarity; LeakyReLU(·) is a leaky ReLU activation function; and the Softmax function ensures that all weights satisfy normalization, i.e. d) Calculate attention-weighted fusion features: e) Construct a Graph Convolutional Network (GCN), using the original feature matrix X of the nodes as the initial input, and aggregate neighbor information layer by layer through GCN layers. The GCN layer propagation formula is: in: · The node representation matrix of layer l (initially H) (0) =X); · Adjacency matrix after adding self-connections; · The degree matrix; ·W (l) : The learnable weight matrix of the l-th layer; ·σ(·): Activation function LeakyReLU. f) Finally, based on the node representation h output by GCN. uv The connection probability is calculated using the Sigmoid function, as shown in the following formula: Among them, w c It is a learnable weight parameter, b c The bias term is sigmoid(·), and the activation function is sigmoid(·). Used to map output values ​​to the interval (0,1). This represents the probability that there is an edge between node pair (u,v). The higher the probability, the greater the likelihood of a link between node u and node v. (3) Output missing or potentially existing links based on the prediction results to achieve link prediction.

2. The method according to claim 1, characterized in that, The structural similarity mentioned in step (2)a) is a local topological index of the number of common neighbors (CN). The calculation formula is as follows: S CN (u,v)=|N u ∩N v | (6) In the formula, N u Let N represent the set of neighbors of node u. v Let |N| represent the set of neighbors of node v. u ∩N v | represents the number of common neighbors of nodes u and v. S CN The larger the (u,v) value, the more similar the local structure of the node pair (u,v), and the higher the potential for forming a link.

3. The method according to claim 1, characterized in that, Step (2)a) The attribute similarity mentioned above is calculated based on the node's label, category, text information, or other attribute information using the cosine similarity measurement method. The formula is as follows: In the formula, x u and x v Let x represent the feature vectors of nodes u and v respectively. u ‖ and ‖x v ‖ represents the Euclidean norm of the two vectors. The formula takes values ​​between [-1, 1]. The closer the value is to 1, the more similar the features of the nodes are, and they may belong to the same category or have the same attributes. The closer the value is to 0, the more unrelated or dissimilar the features of the nodes are.

4. The method according to claim 1, characterized in that, Step (2)a) The path similarity is obtained by calculating the number of multi-hop paths between candidate node pairs, the path length weighted value, or the similarity between path embedding vectors. The formula is as follows: In the formula, sp(u,v) represents the shortest path length between node u and node v, and σ is the attenuation coefficient used to control the influence of path distance on similarity weight. When sp(u,v) is small, S PATH A value close to 1 for sp(u,v) indicates strong connections between nodes, making it easier to form links; when sp(u,v) is large, S PATH A value close to 0 for (u,v) indicates that the nodes are far apart and the likelihood of a new link being formed is low.

5. The method according to claim 1, characterized in that, The specific parameters of the Graphical Convolutional Network (GCN) in step (2)f) are as follows: (1) GCN contains 3 layers, and the initial node representation matrix H (0) =X (X is a |V|×d dimensional matrix, where d is the original feature dimension of the node); (2) Except for the last layer, all other layers use ReLU as the activation function. The last layer does not use an activation function to preserve the original feature information. (3) Learnable weight matrix W for each layer (l) The dimensions satisfy: Let d be the output dimension of the l-th layer. l+1 (where l+1 is the input dimension of the layer) and is initialized using a random normal distribution.

6. The method according to claim 1, characterized in that... The MLP structure used to calculate the connection probability in step (2)g) is as follows: it contains two hidden layers (256 units per layer), the activation function is ReLU, and the output layer dimension is 1; the Sigmoid function maps the MLP output to the [0,1] interval to achieve binary classification probability prediction.

7. The method according to claim 1, characterized in that, It also includes the model training steps, specifically: (1) Dataset partitioning: The network edge set E is divided into a training set (85%), a validation set (5%), and a test set (10%). An equal number of negative samples are generated for the training set and the validation set (randomly selected non-existent edges). ); (2) Training parameter settings: The Adam optimizer was used, the learning rate was set to 0.01, the number of training rounds was 200, and the binary cross-entropy was used as the loss function. (3) Model optimization: Update W through backpropagation s a s b s GCN weight W (l) And MLP weights, minimize the loss function.

8. The method according to claim 1, characterized in that, The model performance was evaluated using the area under the ROC curve (AUC), average precision (AP), and hit rate (Hits@20), calculated as follows: Where n is the number of independent comparisons, n′ is the number of missing nodes with higher scores, and n″ is the number of comparisons with the same scores. A higher AUC indicates a stronger ability of the model to accurately distinguish between node pairs with and without links, and AUC = 1 represents perfect classification. Where P n For accuracy, R n This is the recall rate at the nth threshold. AP focuses on the predictive ability of positive samples (true edges); a higher value indicates a more accurate model. Where V test This is the test set, and rank(t) is the ranking of t. A higher Hits@20 value indicates that the model is more capable of finding real links in high-confidence predictions, reflecting the model's ranking coverage ability.

9. The method according to claim 1, characterized in that, The method can be applied to real-world complex network scenarios such as social networks, citation networks, e-commerce recommendation systems, biological networks, and transportation networks. The overall process of the link prediction method MSGNN proposed in this invention is given below. Table 1: MSGNN Link Prediction Algorithm Flow

Images