Sparse network-oriented miRNA-mRNA interaction prediction method

Abstract

The invention discloses a sparse network-oriented miRNA-mRNA interaction prediction method, which comprises the following steps: 1) screening high-confidence genes based on multi-method cross validation to obtain miRNA and mRNA expression profile data containing a tumor sample and a normal sample; 2) constructing a background regulation and control network and graph data; 3) virtual nodes are introduced to enhance graph connectivity; (4) encoder learning node embedding based on GraphSAGE (GraphSAGE) is carried out; according to the method, the problems of poor model performance, poor expandability and input data deviation caused by the sparsity of the biological regulation and control network are solved, and the accuracy, the robustness and the efficiency of prediction are improved.

Description

Technical Field

[0001] This invention relates to the fields of bioinformatics and artificial intelligence, and in particular to a method for predicting miRNA-mRNA interactions in sparse networks. Background Technology

[0002] The mutual regulation between miRNAs and mRNAs plays a crucial role in various life processes, including tumorigenesis and development. However, verifying all potential miRNA-mRNA regulatory pairs through experimental methods is costly and time-consuming. Therefore, developing efficient and accurate computational prediction models is essential for related research. Currently, computational prediction methods mainly face the following technical challenges:

[0003] 1) Network sparsity limits model performance. In known biological regulatory networks, the number of verified miRNA-mRNA interactions is far less than the total number of potential interactions, resulting in highly sparsity in the network. This sparsity severely hinders information propagation between nodes in models such as graph neural networks, making it difficult for the model to learn comprehensive and accurate node feature representations, thus affecting the accuracy of predictions.

[0004] 2) Existing models lack scalability and adaptability. Traditional methods such as Graph Convolutional Networks (GCNs) require loading the entire adjacency matrix of the graph into memory for computation when dealing with large-scale biological networks, resulting in huge computational resource consumption and difficulty in scaling. In addition, when the network structure is updated (e.g., adding nodes or regulatory relationships), these models need to be completely retrained, lacking inductive learning ability and adaptability to dynamic networks.

[0005] 3) Input data carries the risk of bias. Existing studies typically rely on a single differential expression analysis (DEA) method when screening genes for modeling. Different DEA methods are based on different statistical assumptions, and their analytical results may be biased. Gene lists screened using only a single method may contain a high number of false positives, thus affecting the reliability and biological significance of subsequent predictive models. Summary of the Invention

[0006] To address the shortcomings of existing technologies, this invention provides a miRNA-mRNA interaction prediction method for sparse networks, which solves the problems of poor model performance, poor scalability, and biased input data caused by the sparsity of biological regulatory networks, thereby improving the accuracy, robustness, and efficiency of prediction.

[0007] The objective of this invention is achieved as follows: a method for predicting miRNA-mRNA interactions in sparse networks, comprising the following steps:

[0008] Step 1) Based on high-confidence gene screening using multi-method cross-validation, obtain miRNA and mRNA expression profile data including tumor samples and normal samples;

[0009] Step 2) Construct the background regulation network and graph data;

[0010] Step 3) Introduce virtual nodes to enhance graph connectivity;

[0011] Step 4) The encoder learns node embeddings based on GraphSAGE;

[0012] Step 5) The decoder predicts potential miRNA-mRNA interactions and inputs the miRNA and mRNA node embedding vectors generated by the encoder into the decoder;

[0013] Step 6) Model training and optimization.

[0014] Furthermore, step 1) specifically includes:

[0015] Four DEA methods based on different statistical principles were used: edgeR, DESeq2, limma-voom, and Wilcoxon rank-sum test. A screening threshold was set where the fold change in gene expression (FC) satisfying |log2FC| ≥ 1 and an adjusted p-value < 0.05. The calculation method for log2FC is as follows:

[0016]

[0017] in, This represents the average gene expression level in the tumor sample. The average gene expression level in normal samples is used as the high-confidence candidate molecule set, and differentially expressed molecules identified by all DEA methods are selected to eliminate the statistical bias of a single method.

[0018] Furthermore, step 2) specifically includes: based on the miRNA-mRNA interaction database miRWalk, screening out known regulatory relationships between high-confidence miRNAs and mRNAs obtained in step 1), and constructing a background regulatory network; the background regulatory network is a bipartite graph, where nodes represent miRNAs or mRNAs and edges represent known regulatory relationships;

[0019] Then, the background modulation network structure is transformed into model input:

[0020] Adjacency matrix A is a binary matrix constructed based on the regulatory relationships in the background network. If there is an edge between node i and node j, then the corresponding position in matrix A is... ij =1, otherwise 0;

[0021] The feature matrix X performs max-min normalization on the expression spectrum data of each node, scaling the values ​​to the [0, 1] interval:

[0022]

[0023] In the formula, x represents the original data and x′ represents the normalized data. Principal component analysis (PCA) is used to reduce the dimensionality of the normalized high-dimensional expression spectrum data to form the initial feature vector of each node.

[0024]

[0025] in, This is the original feature matrix after centering, where s is the number of samples and d is the original feature dimension. It is from the front The projection matrix is ​​composed of the eigenvectors corresponding to the principal components. It is the final feature matrix after dimensionality reduction.

[0026] Furthermore, step 3) specifically includes:

[0027] 3.1) Initialize m virtual nodes, their initial characteristic matrix The elements are from the standard normal distribution The virtual nodes are generated by random sampling, where d is the feature dimension, consistent with the real mRNA nodes and miRNA nodes. The feature vectors of the virtual nodes will be used as trainable parameters of the model.

[0028] 3.2) With a preset connection ratio of 15%-25%, each virtual node is randomly connected to some real miRNA nodes and mRNA nodes; these newly added connections are added to the adjacency matrix to form an extended connected augmented graph; the virtual nodes act as "bridges" for information transmission, connecting nodes that are originally far apart or not connected in the graph.

[0029] 3.3) By constructing an augmented adjacency matrix A′ and an augmented characteristic matrix X′, virtual nodes are integrated into the graph structure. Let the original number of nodes be n and the number of virtual nodes be m, then:

[0030]

[0031]

[0032] in, It is the original adjacency matrix. It is a connection matrix between real nodes and virtual nodes. , It is a learnable virtual node feature matrix, X v The initial feature matrix for the virtual nodes is an m×d matrix, consistent with the feature dimensions of the real miRNA and mRNA nodes. This matrix is ​​randomly initialized at the start of training and continuously updated and optimized through backpropagation during model training.

[0033] Furthermore, step 4) specifically includes:

[0034] The augmented graph defined by A′ and X′ constructed in step 3) is input into a GraphSAGE-based encoder. This encoder learns the embedding representation of each node in the network by sampling and aggregating the neighborhood information of the nodes. Each layer in the encoder contains batch normalization and ReLU activation functions, which are defined as follows:

[0035]

[0036]

[0037] Where µ is the mean of the current input data, σ is the standard deviation of the current input data, and σ 2 ε represents variance; ε is the numerical stability constant (Epsilon), a minimum value added to prevent the denominator from being zero; γ is the scale parameter, a trainable parameter of the model used to recover the expressiveness of features, allowing the model to learn the optimal scale of features; β is the shift parameter, another trainable parameter of the model, allowing the model to learn the optimal distribution center of features.

[0038] Since the features of virtual nodes are learnable, during model training, virtual nodes are optimized based on the supervision signals of the prediction task, thereby capturing the potential structural information and higher-order associations in the network.

[0039] Furthermore, step 5) specifically includes: inputting the miRNA and mRNA node embedding vectors generated by the encoder into the decoder; for any miRNA-mRNA pair to be predicted, i.e., node v i and v j The decoder performs the following operations:

[0040] Concatenate embedding vectors: Combine the embeddings z of two nodes i and z j spliced ​​together to form a union representation e ij ;

[0041]

[0042] Predicting interaction probabilities: Representing the joint representation e ij The algorithm uses a linear layer and a sigmoid activation function to output a probability value p in the range [0, 1]. ij This indicates the possibility that there is a regulatory relationship between the miRNA and the mRNA;

[0043]

[0044] Where W and b are the learnable weights and biases in the decoder.

[0045] Furthermore, step 6) specifically includes: constructing a training set using a negative sampling strategy, where positive samples are known interactions and negative samples are randomly sampled non-existent interactions; and performing end-to-end training of the model by minimizing a binary cross-entropy loss function with L2 regularization.

[0046]

[0047] in, and These are the positive and negative sample edge sets, p ij It is a prediction probability. It contains all trainable parameters of the model, including features of virtual nodes. It is the regularization coefficient.

[0048] Compared with existing technologies, the beneficial effects of this invention are as follows: 1) Effectively overcomes network sparsity and improves prediction performance: By introducing an innovative learnable virtual node mechanism, the information propagation paths in the network are increased, and the connectivity of the graph is enhanced. As a global information relay station, the virtual node enables originally separated or distant nodes to indirectly exchange information, significantly improving the model's ability to capture long-distance dependencies, thereby greatly improving the accuracy of prediction.

[0049] 2) Improved reliability of input data and robustness of prediction results: Four DEA methods based on different principles were used for joint analysis, and their intersection results were taken to form a consensus-driven screening strategy. This effectively avoids the inherent bias of a single method and ensures that the miRNAs and mRNAs used for modeling are highly correlated with specific biological conditions (such as cancer), thereby improving the biological significance of the model and the robustness of the prediction results.

[0050] 3) The model is highly scalable and supports inductive learning: Using GraphSAGE as its core framework, it learns through neighborhood sampling rather than full-graph computation, significantly reducing memory and computational overhead, making it suitable for large-scale biological networks. More importantly, GraphSAGE possesses inductive learning capabilities, enabling it to predict newly discovered RNA molecules during training without retraining the entire model. This is invaluable for rapidly validating the potential roles of newly discovered molecules in existing regulatory networks. Attached Figure Description

[0051] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the provided drawings without creative effort.

[0052] Figure 1 This is a schematic diagram illustrating the principle of virtual nodes connecting miRNA and mRNA nodes in this invention.

[0053] Figure 2 The present invention provides a flowchart of a method for predicting miRNA-mRNA interactions. Detailed Implementation

[0054] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0055] A method for predicting miRNA-mRNA interactions in sparse networks includes the following steps:

[0056] Step 1) Based on high-confidence gene screening using multi-method cross-validation, obtain miRNA and mRNA expression profile data including tumor samples and normal samples;

[0057] Four DEA methods based on different statistical principles were used: edgeR, DESeq2, limma-voom, and Wilcoxon rank-sum test. A screening threshold was set where the fold change in gene expression FC satisfied |log2FC| ≥ 1 and an adjusted p-value < 0.05. The calculation method for log2FC is as follows:

[0058]

[0059] in, This represents the average gene expression level in the tumor sample. The average gene expression level in normal samples is used as the high-confidence candidate molecule set, and differentially expressed molecules identified by all DEA methods are selected to eliminate the statistical bias of a single method.

[0060] Step 2) Construct the background regulation network and graph data;

[0061] Based on the miRNA-mRNA interaction database miRWalk, known regulatory relationships between high-confidence miRNAs and mRNAs obtained in step 1) are screened out, and a background regulatory network is constructed. The background regulatory network is a bipartite graph, where nodes represent miRNAs or mRNAs and edges represent known regulatory relationships.

[0062] Then, the background modulation network structure is transformed into model input:

[0063] Adjacency matrix A is a binary matrix constructed based on the regulatory relationships in the background network. If there is an edge between node i and node j, then the corresponding position in matrix A is... ij =1, otherwise 0;

[0064] The feature matrix X performs max-min normalization on the expression spectrum data of each node, scaling the values ​​to the [0, 1] interval:

[0065]

[0066] In the formula, x represents the original data and x′ represents the normalized data. Principal component analysis (PCA) is used to reduce the dimensionality of the normalized high-dimensional expression spectrum data to form the initial feature vector of each node.

[0067]

[0068] in, This is the original feature matrix after centering, where s is the number of samples and d is the original feature dimension. It is from the front The projection matrix is ​​composed of the eigenvectors corresponding to the principal components. It is the final feature matrix after dimensionality reduction.

[0069] Step 3) Introduce virtual nodes to enhance graph connectivity, such as Figure 1 As shown;

[0070] 3.1) Initialize m virtual nodes, their initial characteristic matrix The elements are from the standard normal distribution The virtual nodes are generated by random sampling, where d is the feature dimension, consistent with the real mRNA nodes and miRNA nodes. The feature vectors of the virtual nodes will be used as trainable parameters of the model.

[0071] 3.2) With a preset connection ratio of 15%-25%, each virtual node is randomly connected to a subset of real miRNA and mRNA nodes. These newly added connections are added to the adjacency matrix, forming an expanded and more connected augmented graph. The virtual nodes act as "bridges" for information propagation, connecting nodes that were originally far apart or unconnected in the graph.

[0072] 3.3) By constructing an augmented adjacency matrix A′ and an augmented characteristic matrix X′, virtual nodes are integrated into the graph structure. Let the original number of nodes be n and the number of virtual nodes be m, then:

[0073]

[0074]

[0075] in, It is the original adjacency matrix. It is a connection matrix between real nodes and virtual nodes. , It is a learnable virtual node feature matrix, X v The initial feature matrix for the virtual nodes is an m×d matrix, consistent with the feature dimensions of the real miRNA and mRNA nodes. This matrix is ​​randomly initialized at the start of training and continuously updated and optimized through backpropagation during model training.

[0076] Step 4) The encoder learns node embeddings based on GraphSAGE;

[0077] The augmented graph defined by A′ and X′ constructed in step 3) is input into a GraphSAGE-based encoder. This encoder learns the embedding representation of each node in the network by sampling and aggregating the neighborhood information of the nodes. Each layer in the encoder contains batch normalization and ReLU activation functions, which are defined as follows:

[0078]

[0079]

[0080] Where µ is the mean of the current input data, and σ is the standard deviation of the current input data (σ²). 2 (This represents the variance). ε is the numerical stability constant (Epsilon), a minimum value added to prevent the denominator from being zero. γ is the scale parameter, a trainable parameter of the model used to recover the expressiveness of features, allowing the model to learn the optimal scale for the features. β is the shift parameter, another trainable parameter of the model, allowing the model to learn the optimal distribution center of the features.

[0081] Since the features of virtual nodes are learnable, during model training, virtual nodes are optimized based on the supervision signals of the prediction task, thereby capturing the potential structural information and higher-order associations in the network.

[0082] Step 5) The decoder predicts potential miRNA-mRNA interactions and inputs the miRNA and mRNA node embedding vectors generated by the encoder into the decoder;

[0083] The miRNA and mRNA node embedding vectors generated by the encoder are input into the decoder. For any miRNA-mRNA pair to be predicted, i.e., node v i and v j The decoder performs the following operations:

[0084] Concatenate embedding vectors: Combine the embeddings z of two nodes i and z j spliced ​​together to form a union representation e ij ;

[0085]

[0086] Predicting interaction probabilities: Representing the joint representation e ij The algorithm uses a linear layer and a sigmoid activation function to output a probability value p in the range [0, 1]. ijThis indicates the possibility that there is a regulatory relationship between the miRNA and the mRNA;

[0087]

[0088] Where W and b are the learnable weights and biases in the decoder.

[0089] Step 6) Model training and optimization;

[0090] A negative sampling strategy is used to construct the training set, where positive samples represent known interactions and negative samples represent randomly sampled non-existent interactions. The model is trained end-to-end by minimizing a binary cross-entropy loss function with L2 regularization.

[0091]

[0092] in, and These are the positive and negative sample edge sets, p ij It is a prediction probability. It contains all trainable parameters of the model, including features of virtual nodes. It is the regularization coefficient.

[0093] like Figure 2 As shown, this embodiment uses the prediction of breast cancer-related miRNA-mRNA regulatory networks as an example to explain in detail the implementation process of the method of the present invention.

[0094] Step 1) Data Preprocessing and High-Confidence Gene Screening: miRNA and mRNA expression profile data from the BRCA project were obtained from the TCGA (The Cancer Genome Atlas) database, including tumor and normal samples. Four DEA methods—edgeR, DESeq2, limma-voom, and Wilcoxon rank-sum test—were applied to the miRNA and mRNA data respectively; the screening criteria were |log2(Fold Change)| ≥ 1 and an adjusted p-value < 0.05. Finally, a Venn diagram was used to extract differentially expressed genes identified by all four methods, resulting in a list of high-confidence miRNAs and mRNAs.

[0095] Step 2) Constructing the Background Regulatory Network and Graph Data: Access the miRWalk database and extract experimentally validated miRNA-mRNA interaction pairs. Match these pairs with the high-confidence miRNA and mRNA list selected in Step 1 to construct a BRCA-related background regulatory network. Based on this network, generate an adjacency matrix A and a feature matrix X. Each row of the feature matrix X represents an RNA node, and each column is its normalized expression value in different samples. Then, perform PCA dimensionality reduction on the feature matrix X to retain more than 95% of the principal components of variance as the final initial features of the nodes.

[0096] Step 3) Introduce virtual nodes to enhance graph connectivity. Based on the background network, 32 virtual nodes are introduced; the initial feature vector of each virtual node is randomly generated using a standard normal distribution. Then, bidirectional connections are established between 20% of the real RNA nodes (including miRNA and mRNA) randomly selected for each virtual node; these connections are updated in the adjacency matrix A, forming the augmented adjacency matrix A'. The feature vectors of the virtual nodes are set as trainable parameters.

[0097] Step 4) GraphSAGE-based encoder learns node embeddings: The augmented graph (defined by A' and the augmented feature matrix X') is input into a two-layer GraphSAGE encoder. After each GraphSAGE convolution, a batch normalization (BatchNorm) layer and a ReLU activation function are connected, and Dropout (with a ratio of 0.5) is applied to prevent overfitting. The encoder ultimately generates a 32-dimensional embedding vector for all real RNA nodes and virtual nodes in the network.

[0098] Step 5) Decoder predicts potential miRNA-mRNA interactions: Design a decoder with a fully connected layer at its core. For any miRNA node i and mRNA node j to be predicted, use their 32-dimensional embedding vector z learned in step four. i and z j The vectors are concatenated to obtain a 64-dimensional joint vector. This joint vector is then input into the decoder, where it undergoes a linear transformation and a sigmoid activation function, outputting a probability value p between [0, 1]. ij This value represents the predicted interaction probability.

[0099] Step 6) The model is trained using the AdamW optimizer and the binary cross-entropy loss function, combined with a negative sampling strategy (positive to negative samples in a 1:1 ratio) for end-to-end optimization. When the loss on the validation set no longer decreases for several consecutive epochs, an early stopping mechanism is triggered to end the training.

[0100] Experimental results show that the average AUC of this embodiment reached 97.95% in breast cancer, and the average AUC of 98.08% was achieved when the same steps were applied to lung adenocarcinoma. These results are significantly better than various baseline models, including traditional machine learning methods, demonstrating the effectiveness and superiority of the method presented in this invention. Furthermore, ablation experiments further validated that the learnable virtual node mechanism and various differential expression crossover strategies are crucial for improving model performance.

[0101] The above description of the embodiments is only for the purpose of helping to understand the method and core ideas of the present invention. It should be noted that those skilled in the art can make several improvements and modifications to the present invention without departing from the principles of the present invention, and these improvements and modifications also fall within the protection scope of the claims of the present invention.

Claims

1. A method for predicting miRNA-mRNA interactions in sparse networks, characterized in that, Includes the following steps: Step 1) Based on high-confidence gene screening using multi-method cross-validation, obtain miRNA and mRNA expression profile data including tumor samples and normal samples; Step 2) Construct the background regulation network and graph data; Step 3) Introduce virtual nodes to enhance graph connectivity; Step 4) The encoder learns node embeddings based on GraphSAGE; Step 5) The decoder predicts potential miRNA-mRNA interactions and inputs the miRNA and mRNA node embedding vectors generated by the encoder into the decoder; Step 6) Model training and optimization.

2. The method for predicting miRNA-mRNA interactions in sparse networks according to claim 1, characterized in that, Step 1) specifically includes: Four DEA methods based on different statistical principles were used: edgeR, DESeq2, limma-voom, and Wilcoxon rank-sum test. A screening threshold was set where the fold change in gene expression (FC) satisfying |log2FC| ≥ 1 and an adjusted p-value < 0.

05. The calculation method for log2FC is as follows: ； in, This represents the average gene expression level in the tumor sample. The average gene expression level in normal samples is used as the high-confidence candidate molecule set, and differentially expressed molecules identified by all DEA methods are selected to eliminate the statistical bias of a single method.

3. The method for predicting miRNA-mRNA interactions in sparse networks according to claim 1, characterized in that, Step 2) specifically includes: based on the miRNA-mRNA interaction database miRWalk, screening out the known regulatory relationships between high-confidence miRNAs and mRNAs obtained in step 1), and constructing a background regulatory network; the background regulatory network is a bipartite graph, where nodes represent miRNAs or mRNAs and edges represent known regulatory relationships. Then, the background modulation network structure is transformed into model input: Adjacency matrix A is a binary matrix constructed based on the regulatory relationships in the background network. If there is an edge between node i and node j, then the corresponding position in matrix A is... ij =1, otherwise 0; The feature matrix X performs max-min normalization on the expression spectrum data of each node, scaling the values ​​to the [0,1] interval: ； In the formula, x represents the original data and x′ represents the normalized data. Principal component analysis (PCA) is used to reduce the dimensionality of the normalized high-dimensional expression spectrum data to form the initial feature vector of each node. ； in, This is the original feature matrix after centering, where s is the number of samples and d is the original feature dimension. It is from the front The projection matrix is ​​composed of the eigenvectors corresponding to the principal components. It is the final feature matrix after dimensionality reduction.

4. The method for predicting miRNA-mRNA interactions in sparse networks according to claim 1, characterized in that, Step 3) specifically includes: 3.1) Initialize m virtual nodes, their initial characteristic matrix The elements are from the standard normal distribution The virtual nodes are generated by random sampling, where d is the feature dimension, consistent with the real mRNA nodes and miRNA nodes. The feature vectors of the virtual nodes will be used as trainable parameters of the model. 3.2) With a preset connection ratio of 15%-25%, each virtual node is randomly connected to some real miRNA nodes and mRNA nodes; these new connections are added to the adjacency matrix to form an extended connected augmented graph; the virtual nodes act as "bridges" for information transmission, connecting nodes that are originally far apart or not connected in the graph. 3.3) By constructing an augmented adjacency matrix A′ and an augmented characteristic matrix X′, virtual nodes are integrated into the graph structure. Let the original number of nodes be n and the number of virtual nodes be m, then: ； ； in, It is the original adjacency matrix. It is a connection matrix between real nodes and virtual nodes. , It is a learnable virtual node feature matrix, X v The initial feature matrix for the virtual nodes is an m×d matrix that maintains the same feature dimensions as the real miRNA and mRNA nodes.

5. The method for predicting miRNA-mRNA interactions in sparse networks according to claim 4, characterized in that, Step 4) specifically includes: The augmented graph defined by A′ and X′ constructed in step 3) is input into a GraphSAGE-based encoder. This encoder learns the embedding representation of each node in the network by sampling and aggregating the neighborhood information of the nodes. Each layer in the encoder contains batch normalization and ReLU activation functions, which are defined as follows: ； ； Where µ is the mean of the current input data, σ is the standard deviation of the current input data, and σ 2 ε represents variance; ε is the numerical stability constant; γ is the scaling factor; β is the translation factor. Since the features of virtual nodes are learnable, during model training, virtual nodes are optimized based on the supervision signals of the prediction task, thereby capturing the potential structural information and higher-order associations in the network.

6. The method for predicting miRNA-mRNA interactions in sparse networks according to claim 1, characterized in that, Step 5) specifically includes: inputting the miRNA and mRNA node embedding vectors generated by the encoder into the decoder. For any miRNA-mRNA pair to be predicted, i.e., node v i and v j The decoder performs the following operations: Concatenate embedding vectors: Combine the embeddings z of two nodes i and z j spliced ​​together to form a union representation e ij ; ； Predicting interaction probabilities: Representing the joint representation e ij The algorithm uses a linear layer and a sigmoid activation function to output a probability value p in the range [0, 1]. ij This indicates the possibility that there is a regulatory relationship between the miRNA and the mRNA; ； Where W and b are the learnable weights and biases in the decoder.

7. The method for predicting miRNA-mRNA interactions in sparse networks according to claim 1, characterized in that, Step 6) specifically includes: constructing a training set using a negative sampling strategy, where positive samples are known interactions and negative samples are randomly sampled non-existent interactions; and performing end-to-end training of the model by minimizing a binary cross-entropy loss function with L2 regularization. ； in, and These are the positive and negative sample edge sets, p ij It is a prediction probability. It contains all trainable parameters of the model, including features of virtual nodes. It is the regularization coefficient.

Images