Complex network key node identification method based on attention mechanism and multi-scale feature fusion

Abstract

The invention provides a complex network key node identification method based on an attention mechanism and multi-scale feature fusion. The method comprises the following steps: S1, generating a feature matrix; s2, inputting the data into a training model composed of a multi-scale convolution branch and a channel attention branch; and S3, outputting an identification result by using the trained model. According to the method, sparse network representation can be solved through a dynamic mechanism, so that more robust and accurate key node identification is realized.

Description

Technical Field

[0001] This invention relates to the field of social network technology, and in particular to a method for identifying key nodes in complex networks based on attention mechanisms and multi-scale feature fusion. Background Technology

[0002] In the digital age, complex networks are ubiquitous in areas such as social interaction, infrastructure operation, and biological regulation. Identifying key nodes in these networks has significant theoretical and practical value. For example, in social networks, accurately locating key nodes can optimize marketing resource allocation or suppress the spread of rumors. In power grids, identifying critical substations allows for proactive maintenance to prevent system failures. In biological networks, mining key proteins can provide targets for disease diagnosis and drug development. Therefore, the identification of key nodes in complex networks remains a research hotspot in network science and data mining. Summary of the Invention

[0003] This invention aims to at least solve the technical problems existing in the prior art, and in particular, it innovatively proposes a method for identifying key nodes in complex networks based on attention mechanism and multi-scale feature fusion.

[0004] To achieve the above-mentioned objectives of this invention, this invention provides a method for identifying key nodes in complex networks based on attention mechanisms and multi-scale feature fusion, comprising the following steps:

[0005] S1, generate the social network feature matrix;

[0006] S2, input the data into the training model consisting of multi-scale convolutional branches and channel attention branches;

[0007] S3 uses the trained model to output the identification results of key nodes in the social network.

[0008] In a preferred embodiment of the present invention, step S1 includes:

[0009] ,

[0010] in, Represents a node The sum of the neighboring degrees;

[0011] Represents a node The set of neighbors;

[0012] Represents a node The degree;

[0013] ,

[0014] in, A matrix representing the local information of a node;

[0015] Indicates the 0th row, 1st row. Elements in the adjacency matrix of columns;

[0016] Represents a node The sum of the neighboring degrees;

[0017] Indicates the number of nodes in the local subgraph;

[0018] Indicates the first The element in the adjacency matrix at row 0 and column 0;

[0019] Represents a node The sum of the neighboring degrees;

[0020] Indicates the first Line number Elements in the adjacency matrix of columns;

[0021] ,

[0022] in, Represents a node The number of communities connected;

[0023] This indicates finding the number of nodes in a set.

[0024] Represents a node The set of neighbors;

[0025] Represents a node The community group to which it and its neighbors belong;

[0026] ,

[0027] in A matrix representing the community information of nodes;

[0028] Indicates the 0th row, 1st row. Elements in the adjacency matrix of columns;

[0029] Represents a node The number of communities connected;

[0030] Indicates the number of nodes in the local subgraph;

[0031] Indicates the first The element in the adjacency matrix at row 0 and column 0;

[0032] Represents a node The number of communities connected;

[0033] Indicates the first Line number Elements in the adjacency matrix of columns;

[0034] ,

[0035] Represents a node The eigenvector center value;

[0036] Indicates a constant factor;

[0037] This represents the total number of nodes in the network;

[0038] Indicates the first Line number Elements in the adjacency matrix of columns;

[0039] It is a node The eigenvector center value;

[0040] ,

[0041] in A matrix representing the global information of a node;

[0042] Indicates the 0th row, 1st row. Elements in the adjacency matrix of columns;

[0043] Represents a node The eigenvector center value;

[0044] Indicates the number of nodes in the local subgraph;

[0045] Indicates the first The element in the adjacency matrix at row 0 and column 0;

[0046] Represents a node The eigenvector center value;

[0047] Indicates the first Line number The elements in the adjacency matrix of the columns.

[0048] In a preferred embodiment of the present invention, step S2 includes:

[0049] The multi-scale convolution branch consists of three cascaded multi-scale convolutional layers, each deploying convolutional kernels of three different scales: 3×3, 5×5, and 7×7. The first convolutional layer has 3 input channels, 10 output channels, a stride of 1, and padding of 1. The second convolutional layer has 10 input channels, 20 output channels, a stride of 1, and padding of 2. The third convolutional layer has 20 input channels, 30 output channels, a stride of 1, and padding of 3.

[0050] The channel attention branch includes three 5×5 convolutional layers for feature extraction, followed by parallel global average pooling and global max pooling of the features. The resulting two feature vectors are then fed into a shared 1×1 convolutional layer for processing. Finally, channel weights are generated by element-wise addition and non-linear activation.

[0051] In a preferred embodiment of the present invention, step S2 further includes:

[0052] ,

[0053] express It follows a normal distribution with a mean of 0 and a variance of 1;

[0054] Let represent the learnable parameter at the position of the i-th row and j-th column, with the superscript (0) indicating the initial value of the parameter;

[0055] ,

[0056] in, This indicates that the size is increased to L feature matrix by using zero padding;

[0057] Representation matrix Element;

[0058] Represents the original characteristic matrix;

[0059] Indicates the number of nodes in the local subgraph;

[0060] ,

[0061] in, Represents the learnable parameter matrix and the original characteristic matrix Merged matrices;

[0062] Represents the mask matrix;

[0063] This represents element-wise multiplication;

[0064] This indicates that the size is increased to L feature matrix by using zero padding;

[0065] Represents a matrix of all 1s;

[0066] This represents the learnable parameter matrix.

[0067] In a preferred embodiment of the present invention, step S2 further includes:

[0068] ,

[0069] in, Indicates partial derivative;

[0070] Represents the loss function;

[0071] Represents the learnable parameter matrix;

[0072] Indicates the number of nodes in the local subgraph;

[0073] This indicates a logical condition;

[0074] Represents the learnable parameter matrix and the original characteristic matrix Merged matrices;

[0075] This indicates that the size is increased to L feature matrix by using zero padding;

[0076] ,

[0077] in, Indicates the first The learnable parameter matrix updated after each training iteration;

[0078] Indicates the first The parameter matrix for the next iteration;

[0079] Indicates the learning rate;

[0080] Indicates partial derivative;

[0081] This represents the loss function.

[0082] In a preferred embodiment of the present invention, the method further includes evaluating the model, the evaluation comprising:

[0083] ,

[0084] in, Represents a node The degree;

[0085] Represents a node The degree;

[0086] This represents the total number of nodes in the network.

[0087] In a preferred embodiment of the present invention, the evaluation further includes:

[0088] ,

[0089] in, Represents a node The betweenness centrality value;

[0090] This represents the total number of elements in each sequential list;

[0091] Represents a node and nodes Between nodes The number of shortest paths;

[0092] Represents a node and nodes The number of shortest paths between them.

[0093] In a preferred embodiment of the present invention, evaluation indicators are further included, including:

[0094] ,

[0095] in, Kendall's coefficient;

[0096] Indicates the number of coordination pairs;

[0097] Indicates the number of incoherent pairs;

[0098] This represents the total number of elements in each sequential list.

[0099] The present invention also discloses a computer system, comprising:

[0100] processor;

[0101] Memory used to store processor-executable instructions;

[0102] The processor is configured to implement the complex network key node identification method based on attention mechanism and multi-scale feature fusion when executing the executable instructions.

[0103] The present invention also discloses a computer-readable storage medium, comprising:

[0104] A memory on which computer programs are stored;

[0105] A processor is configured to execute the program in the memory to implement the method for identifying key nodes in complex networks based on attention mechanisms and multi-scale feature fusion.

[0106] In summary, by adopting the above technical solutions, this invention can solve the sparse network representation through a dynamic mechanism, thereby achieving more robust and accurate key node identification.

[0107] Additional aspects and advantages of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. Attached Figure Description

[0108] The above and / or additional aspects and advantages of the present invention will become apparent and readily understood from the description of the embodiments taken in conjunction with the following drawings, in which:

[0109] Figure 1 This is a schematic diagram of the DAMSCNN framework of the present invention.

[0110] Figure 2 This invention relates to the node ranking capability.

[0111] Figure 3 This is a schematic diagram illustrating the impact of network size on the field of this invention.

[0112] Figure 4 This is a schematic diagram of the experimental ablation analysis of the DAMSCNN of this invention.

[0113] Figure 5 This is a schematic diagram illustrating the correlation between the normalized propagation impact scores predicted by DAMSCNN and RCNN on nine real networks in this invention.

[0114] Figure 6This is a schematic diagram illustrating the correlation between the normalized propagation impact scores predicted by DAMSCNN and MRCNN on nine real networks in this invention.

[0115] Figure 7 This is a schematic diagram comparing the node features of RCNN, MRCNN, and DAMSCNN on the NS network. Detailed Implementation

[0116] Embodiments of the present invention are described in detail below. Examples of these embodiments are shown in the accompanying drawings, wherein the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout. The embodiments described below with reference to the accompanying drawings are exemplary and are only used to explain the present invention, and should not be construed as limiting the present invention.

[0117] 1 Introduction

[0118] Traditional methods for identifying critical nodes primarily rely on network topology-based centrality metrics. For example, degree centrality (DC) measures importance by the number of a node's direct neighbors, betweenness centrality (BC) focuses on a node's mediating role in the shortest path, eigenvector centrality (EC) considers the importance propagation of neighboring nodes, and k-shell decomposition identifies core nodes by removing low-degree nodes layer by layer. These methods are computationally simple but have significant limitations: DC only focuses on local neighbor information, ignoring the node's position in the global network; BC and EC have high computational complexity for large-scale networks, making them unsuitable for networks with millions of nodes; and k-shell decomposition tends to group nodes of different core degrees into the same level, resulting in insufficient resolution. Furthermore, some methods, such as semi-local centrality (SLC), attempt to integrate local and neighborhood information, but still rely on manually designed feature combination rules, resulting in limited generalization ability and a significant performance drop in networks with significant community structures or sparse topologies.

[0119] As networks grow in scale and topology complexity, data-driven methods have become a research hotspot due to their ability to automatically learn network features. These methods can be broadly divided into two branches: machine learning-based and deep learning-based, which overcome the limitations of traditional methods by focusing on feature engineering optimization and deep mining of topological information, respectively. Machine learning-based methods emphasize extracting discriminative features from the network using domain knowledge and then using classic machine learning algorithms to build key node identification models. Examples include Least Squares Support Vector Machine (LS-SVM), Machine Learning Exponential (MLI), and Ensemble Learning (EL). These methods offer strong interpretability and perform stably in small-scale structured networks, but they heavily rely on manual feature design experience. Deep learning-based methods automatically extract network topology information through neural networks, generating deep feature representations of nodes. They also combine this with the SIR model to simulate node propagation influence and generate ranking labels for model training. The algorithms include: InfGCN utilizes the neighborhood feature aggregation capability of graph convolutional networks to capture the topological relationships of nodes through multi-layer convolution; RCNN introduces convolutional neural networks to extract a fixed-size neighborhood network through breadth-first search, combines the adjacency matrix with node degree to construct a single-channel feature input, and learns the influence of nodes; MRCNN extends RCNN to a three-channel input to improve feature diversity.

[0120] While existing deep learning methods have made significant progress in key node identification tasks, they still face challenges in dealing with complex and varied network topologies. Traditional sampling subgraph representation methods with zero-padding strategies disrupt structural continuity and introduce noise, making them difficult to adapt to the topological characteristics of sparse networks. Existing algorithms often fail to fully integrate local, community, and global structural information, thus limiting their generalization ability and accuracy. To address these limitations, this invention proposes DAMSCNN, which solves the sparse network representation problem through a parameterized adaptive padding mechanism. It employs a dual-branch architecture with multi-scale convolution and attention enhancement to achieve cross-level feature extraction and innovatively introduces an attention fusion module. To verify the effectiveness of DAMSCNN, this study compares it with existing methods on nine real-world complex networks. Experimental results show that DAMSCNN achieves significant performance across different network types.

[0121] The main contributions of this invention patent are as follows:

[0122] To address the challenge of inconsistent sizes in the neighborhood subgraphs of nodes in sparse networks, a learnable padding method is designed. This method dynamically generates padding values ​​through trainable parameters, avoiding the information loss caused by traditional zero padding and effectively improving the model's generalization ability in networks of different densities.

[0123] A dual-branch network structure integrating multi-scale convolution and channel attention was constructed, which realized the collaborative learning and complementary enhancement of multi-level topological features of nodes, effectively improving the model's ability to capture complex structural patterns.

[0124] An attention fusion module was introduced, which enables the model to automatically adjust the contribution ratio of the two types of features for different network topologies, realizing the shift from "fixed allocation" to "dynamic allocation", which significantly improves the model's adaptability and expressive power.

[0125] A key node identification method based on attention mechanism and multi-scale fusion of dual-branch convolutional neural networks is proposed. The effectiveness and robustness of the proposed method are verified on multiple real network datasets, and a new solution is provided for node importance analysis in complex networks.

[0126] The remaining parts of this invention patent are structured as follows: Section 2 introduces some related work, including traditional methods and deep learning methods. Section 3 systematically describes the method of this invention patent. Section 4 introduces the experimental design and results. Finally, Section 5 summarizes the work.

[0127] 2. Related work

[0128] Key node identification is a core research problem in network science, aiming to identify nodes in complex networks that have a crucial impact on the network's structure or function. Based on the core technologies employed, existing research methods can be broadly categorized into two types: traditional methods and data-driven deep learning methods.

[0129] 2.1 Traditional Key Node Identification Methods

[0130] Traditional key node identification methods are mainly based on network topology. According to the topology range they depend on, they can be systematically divided into two categories: one takes the perspective of local structure and utilizes the neighborhood information of nodes; the other takes the perspective of global structure and considers the position and connection pattern of nodes in the entire network.

[0131] Local structure-based methods primarily utilize direct neighborhood information to assess a node's importance. Degree centrality is a typical example of this type of method, with the core assumption that a node's influence is proportional to the number of its neighbors. However, this local perspective ignores the overall structural features of the network, leading to significant biases in the assessment of node importance. To overcome this limitation, researchers have explored two main directions: "expanding the neighborhood scope" and "enriching feature dimensions." Regarding expanding the neighborhood scope, Kitsak et al.'s SLC significantly improves recognition accuracy while maintaining low computational complexity by introducing second-order neighbor information. Regarding enriching feature dimensions, Hirsch, drawing inspiration from the H-index in academic evaluation, proposed H-index centrality, innovatively incorporating the quality of neighboring nodes into the evaluation dimension. Furthermore, clustering coefficient centrality, by measuring the density of a node's neighborhood, and gravity model centrality, by fusing node degree and distance information, both improve the utilization of local information to varying degrees.

[0132] As research deepened, scholars gradually recognized the limitations of local information and began to shift towards methods based on global structure. Freeman's BC and compactness centrality established node importance evaluation systems based on shortest paths from the perspectives of information control capability and propagation efficiency, respectively. However, these methods still offered a limited characterization of network structure. Bonacich et al.'s eigenvector centrality broke through this limitation by introducing the recursive idea that "node importance depends on the importance of its neighbors," establishing a more comprehensive evaluation framework. Building upon this theory, researchers have made several extensions. Brin et al.'s PageRank algorithm effectively solved the importance propagation problem in directed graphs by introducing a random walk process. Katz's Katz centrality, by introducing a decay factor and weighting the sum of long and short paths, assigned higher weights to close connections, addressing the potential failure of eigenvector centrality in undirected networks.

[0133] While traditional centrality methods demonstrate their advantages in specific scenarios, they also have limitations: they rely on a single topological perspective and have high computational complexity. To address these limitations, researchers have proposed hybrid centrality methods. These methods improve recognition performance by fusing multi-dimensional features. The fusion of indicators is achieved primarily through two paths: First, integrating existing centrality indicators based on multi-attribute decision theory, such as Hu et al. using TOPSIS technology to comprehensively evaluate multiple centrality measures; and Chen et al. using a genetic algorithm to adaptively determine the weights of each indicator. Second, constructing novel hybrid indicators, such as the escape velocity centrality proposed by Ullah et al., inspired by physics, which organically combines local features with global topological characteristics. These methods not only overcome the limitations of a single indicator perspective but also significantly improve the accuracy of key node identification while maintaining reasonable computational complexity.

[0134] 2.2 Data-Driven Key Node Identification Method

[0135] While traditional physics-based methods offer advantages such as simplicity, speed, and no need for training, their performance is often unsatisfactory in most scenarios. Therefore, a shift towards data-driven approaches (such as machine learning and deep learning) has emerged. Although these methods require time to train the model, they typically achieve superior performance.

[0136] The potential of machine learning technology in identifying key nodes in complex networks is attracting increasing attention. Related methods aim to improve the accuracy and efficiency of node ranking through automated feature extraction and network topology analysis. Existing research generally relies on classic machine learning algorithms (such as logistic regression, support vector machines, and K-nearest neighbors) for key node identification. For example, Wen et al. proposed a node importance evaluation model trained using least squares support vector machines, while Win et al. proposed an evaluation model based on the analytic hierarchy process (AHP) and used least squares support vector machines to construct a regression model between traditional centrality indices and AHP evaluation results. To enhance the completeness of network representation, Wei et al. introduced a method based on network representation learning, integrating overlapping community structures and underlying network features. Li et al. proposed an attack algorithm targeting the robustness of complex networks. Its core idea is to construct an effective node importance evaluation model, identify and attack key nodes in the network using machine learning methods, and evaluate the network's robustness against specific attack strategies.

[0137] In recent years, deep learning technology has become an important technical approach for identifying key nodes in complex networks. The core of these methods lies in leveraging the multi-layered nonlinear transformation capabilities of deep learning to aggregate multi-order neighbor information of nodes layer by layer through message passing mechanisms, thereby automatically learning high-order representations that can simultaneously encode local and global topological features of nodes. For example, the InfGCN method proposed by Zhao et al. utilizes classic structural features such as degree centrality and compactness centrality to construct node feature vectors and uses a graph convolutional network (GCN) to learn the mapping relationship between these vectors and the real labels generated by the SIR model. In addition, research has explored unsupervised or self-supervised methods such as graph autoencoders (GAE) and depth graph Infomax (DGI) to alleviate the dependence on expensive labeled data. Meanwhile, methods based on subgraph sampling and convolution transform graph-structured data into regular grid data, leveraging the powerful spatial feature extraction capabilities of convolutional neural networks (CNNs) to transform the key node identification problem into an image style regression problem. Specifically, the RCNN method proposed by Yu et al. samples a fixed-size local subgraph for each target node using breadth-first search (BFS) and treats the adjacency matrix of the subgraph as a single-channel "image" input into the CNN for training. Zhang et al.'s CGNN enhances the node feature matrix generation technique and adds two GNN layers after the CNN network to improve experimental results. Notably, Ou et al.'s MRCNN method significantly extends RCNN by constructing a multi-channel input feature matrix, enabling the CNN to learn node features from multiple complementary dimensions. This method marks the evolution of key node recognition from a "single view" to "multi-view fusion," capturing the topological characteristics of nodes more comprehensively. Besides MRCNN, subsequent research has explored other multi-view fusion strategies, such as using adjacency matrices, node degree sequences, and K-shell sequences as different channels, or utilizing features generated by graph embedding algorithms like Node2Vec as additional channels.

[0138] Most existing deep learning methods rely on incomplete network information to generate node representations, failing to fully utilize the inherent topological information of deeply fused networks. Their feature extraction and fusion mechanisms remain relatively static and simplistic. They generally lack adaptive processing for sparse networks and struggle to adaptively capture multi-scale structural features from local to global perspectives. To address these limitations, this invention proposes the DAMSCNN method. This method utilizes learnable padding and a dual-branch structure, not only solving the problem of sparse network representation through dynamic mechanisms but also considering core challenges such as multi-level topological features and adaptive feature fusion, thereby achieving more robust and accurate key node identification.

[0139] Figure 1In the diagram: (a) extracts the nodes and edges of the real network; (b) uses the BFS algorithm to obtain the adjacency matrix of the nodes; (c) transforms the adjacency matrix into a three-channel feature matrix using the principles of local, community, and global information and a learnable filling algorithm; (d) uses the feature matrix as input to train the model, where the labels are generated by the SIR model and the loss function is mean squared error.

[0140] 3. The proposed method

[0141] DAMSCNN framework diagram as follows Figure 1 As shown, the process consists of two core steps: feature matrix generation and model training. First, breadth-first search (BFS) is used to transform network nodes into an adjacency matrix. Then, the sum of neighbor degrees (ND) and the number of node communities (Q) are used. Com The eigenvector centrality (EC) and learnable padding transform the local, community, and global information of a node into a feature matrix with three dimensions: local, community, and global. During the model training phase, the feature matrix is ​​input into a dual-branch parallel structure consisting of a multi-scale convolutional module and a channel attention module. Subsequently, an attention fusion mechanism is introduced to adaptively integrate the output features of the two branches. Training labels generated by SIR propagation simulation are used, and the loss function adopts mean squared error (MSE). Finally, the predicted influence score of the node is output.

[0142] 3.1 Basic Definitions and Assumptions

[0143] set up For a network graph, the set of nodes is represented as... The set of edges is represented as , Indicates the number of nodes. Indicates the number of edges. Network graphs can use... adjacency matrix To represent, where matrix elements A value of 1 indicates a node With nodes There are edges connecting them, with a value of 0 indicating no connection. In this invention patent, the following assumptions are used:

[0144] Assumption 1: The network is static and undirected.

[0145] To simplify model complexity, this study limits the network to static undirected networks. First, discussing the applicability of algorithms to all network types (such as temporal and directed networks) within a single framework would make the problem overly complex; second, algorithms designed based on static undirected networks often provide a foundation for extending to more complex network types.

[0146] Assumption 2: The network has a non-overlapping community structure.

[0147] The Louvain community detection algorithm used in this study is based on modularity optimization. Its core principle requires the existence of a clear community structure in the network, and assumes that there is no overlap between communities. To ensure the consistency of feature extraction, it is assumed that both the training and testing networks meet this condition, that is, they have a clear non-overlapping community structure.

[0148] 3.2 Generating the Adjacency Matrix

[0149] In constructing the input representation of the graph convolutional neural network, this study employs a breadth-first search strategy based on degree ranking to generate local adjacency matrices for network nodes. The detailed process is described in Algorithm 1. Specifically, this process systematically selects at most L-1 neighboring nodes within the neighborhood of the target node: the algorithm prioritizes first-order neighbor nodes with higher degree centrality; when the number of first-order neighbors is insufficient, it iteratively expands to second-order and higher-order neighbor spaces. To avoid infinite loops, the upper limit of the number of iterations is set to 50. If L-1 neighboring nodes are not collected after reaching the upper limit, a local subgraph is constructed using all currently available neighbors; at this point, the total number of nodes in the subgraph may be less than the preset value L. After determining the node set, a local adjacency matrix is ​​constructed based on a fixed node order relationship: the target node is always at the first position, and the remaining nodes are arranged in descending order of degree centrality. Let the final subgraph contain k( If there are ) nodes, then construct the corresponding 3D adjacency matrix .

[0150]

[0151] 3.3 Generating the Feature Matrix

[0152] After obtaining the adjacency matrix, it needs to be transformed into a feature matrix to create a three-channel input for the DAMSCNN model. Three low-complexity algorithms are used to extract node features at the local, community, and global levels: the sum of neighbor degrees (ND), the number of communities connected to the node, and the number of neighbor degrees (ND). Eigenvector centrality (EC).

[0153] In terms of local structure, Locality-Neutralization (ND) is chosen to describe the microstructural attributes of nodes. ND is a local centrality index that considers not only the degree of a node itself but also the influence of its neighbors. It can distinguish "potentially influential nodes" that have low degrees but connect to multiple high-degree nodes. These nodes may play a key role in information propagation. The equation for ND is:

[0154] (1)

[0155] in, Represents a node The sum of the neighboring degrees;

[0156] Represents a node The set of neighbors;

[0157] Represents a node The degree;

[0158] in Represents a node The degree, It is a node The set of neighbors of a node. The local information of a node is transformed into a local information representation matrix using the following formula:

[0159] (2)

[0160] in, A matrix representing the local information of a node;

[0161] Indicates the 0th row, 1st row. Elements in the adjacency matrix of columns;

[0162] Represents a node The sum of the neighboring degrees;

[0163] Indicates the number of nodes in the local subgraph;

[0164] Indicates the first The element in the adjacency matrix at row 0 and column 0;

[0165] Represents a node The sum of the neighboring degrees;

[0166] Indicates the first Line number Elements in the adjacency matrix of columns;

[0167] in The matrix represents the local information of a node. This represents the number of nodes in the local subgraph.

[0168] Regarding community structure, the Louvain algorithm is used for community partitioning to quantify the importance of nodes within the community structure. The community structure attribute of a node is defined as follows:

[0169] (3)

[0170] in, Represents a node The number of communities connected;

[0171] This indicates finding the number of nodes in a set.

[0172] Represents a node The set of neighbors;

[0173] Represents a node The community group to which it and its neighbors belong;

[0174] in Represents a node The set of communities to which each node belongs. The community information of a node is transformed into a community information representation matrix using the following formula:

[0175] (4)

[0176] in A matrix representing the community information of nodes;

[0177] Indicates the 0th row, 1st row. Elements in the adjacency matrix of columns;

[0178] Represents a node The number of communities connected;

[0179] Indicates the number of nodes in the local subgraph;

[0180] Indicates the first The element in the adjacency matrix at row 0 and column 0;

[0181] Represents a node The number of communities connected;

[0182] Indicates the first Line number Elements in the adjacency matrix of columns;

[0183] Regarding the global structure, the eigenvector centrality (EC) algorithm is used to obtain information. Eigenvector centrality quantifies the influence of nodes from a global perspective. Its basic principle is that the importance of a node depends on the importance of the other nodes it connects to. The EC calculation formula is as follows:

[0184] (5)

[0185] Represents a node The eigenvector center value; can also be expressed as ;

[0186] Indicates a constant factor;

[0187] This represents the total number of nodes in the network;

[0188] Indicates the first Line number Elements in the adjacency matrix of columns;

[0189] It is a node The eigenvector center value;

[0190] in It is a node The center value of the eigenvector, It is a constant factor. Indicates the first Line number The elements in the adjacency matrix of the columns, It is a node The eigenvector center values. The global information of a node is transformed into a global information representation matrix using the following formula:

[0191] (6)

[0192] in A matrix representing the global information of a node;

[0193] Indicates the 0th row, 1st row. Elements in the adjacency matrix of columns;

[0194] Represents a node The eigenvector center value;

[0195] Indicates the number of nodes in the local subgraph;

[0196] Indicates the first The element in the adjacency matrix at row 0 and column 0;

[0197] Represents a node The eigenvector center value;

[0198] Indicates the first Line number Elements in the adjacency matrix of columns;

[0199] Based on the above method, three feature matrices were successfully constructed. , and However, the generated feature matrix dimension does not meet the input requirements (insufficient feature matrix size). The problem, namely To address this issue, a learnable padding method is introduced. The core of this method is the use of a trainable template matrix as the padding basis. During forward propagation, the original feature matrix is ​​placed in the top left corner of this template matrix, serving as the network input. During backpropagation, the padding region receives gradients and participates in parameter updates, thus achieving end-to-end learning. The complete flow of the learnable padding step is shown below:

[0200] First, define the learnable parameter matrix: (The size of the matrix is) The matrix is ​​initialized using a standard normal distribution, a common parameter initialization strategy in deep learning. This strategy ensures that the initial values ​​of the parameters avoid symmetry problems. The formula for the standard normal distribution is:

[0201] (7)

[0202] express It follows a normal distribution with a mean of 0 and a variance of 1;

[0203] Let represent the learnable parameter at the position of the i-th row and j-th column, with the superscript (0) indicating the initial value of the parameter;

[0204] in Each element in the matrix is ​​an independent trainable parameter, which is updated during backpropagation based on the gradient of the loss function. The original feature matrix... Overlay it onto the generated learnable parameter matrix The top left corner of the original feature matrix Fill the target size L to obtain the matrix. :

[0205] (8)

[0206] in, This indicates that the size is increased to L feature matrix by using zero padding;

[0207] Representation matrix Element;

[0208] Represents the original characteristic matrix;

[0209] Indicates the number of nodes in the local subgraph;

[0210] in For matrix The elements of the matrix. As input for forward propagation, matrix operations are then performed:

[0211] (9)

[0212] in, Represents the learnable parameter matrix and the original characteristic matrix Merged matrices;

[0213] Represents the mask matrix;

[0214] This represents element-wise multiplication;

[0215] This indicates that the size is increased to L feature matrix by using zero padding;

[0216] Represents a matrix of all 1s;

[0217] Represents the learnable parameter matrix;

[0218] in This represents element-wise multiplication. It is a matrix of all ones. In a mask matrix, the positions of elements with a value of 1 correspond to the valid regions of the original feature matrix, and the positions of elements with a value of 0 correspond to the regions that need to be filled. For the matrix... Perform backpropagation, and let the loss function be... The gradient propagation rule is:

[0219] (10)

[0220] in, Indicates partial derivative;

[0221] Represents the loss function;

[0222] Represents the learnable parameter matrix;

[0223] Indicates the number of nodes in the local subgraph;

[0224] This indicates a logical condition;

[0225] Represents the learnable parameter matrix and the original characteristic matrix Merged matrices;

[0226] This indicates that the size is increased to L feature matrix by using zero padding;

[0227] Then, the learnable parameters are updated using the gradient descent algorithm:

[0228] (11)

[0229] in, Indicates the first The learnable parameter matrix updated after each training iteration;

[0230] Indicates the first The parameter matrix for the next iteration;

[0231] Indicates the learning rate;

[0232] Indicates partial derivative;

[0233] Represents the loss function;

[0234] in It's the learning rate. This represents the number of training iterations. The gradient of the loss function with respect to the learnable imputation parameters is calculated through backpropagation, and this gradient is used to iteratively update the numerical distribution of the imputation matrix; this generates a uniform-sized feature matrix that contains both the original topological structure information and the learned imputation representation. Unlike traditional zero-padding, the feature matrix generated through learnable padding contains richer information.

[0235] Using the methods in this section, the adjacency matrix In this process, three entirely new feature matrices were constructed. , and These information, which includes local, community, and global information, serve as inputs.

[0236] 3.4 Generate Tags

[0237] This study treats node propagation influence identification as a regression problem and uses a supervised CNN. This requires each node to have a corresponding propagation influence metric as a label, but the core challenge is that the precise influence in the real world is inherently unobservable. To address this fundamental difficulty, we utilize the SIR propagation model to calculate the average infection scale of nodes in a simulated scenario, generating an approximate, computable surrogate label for each node, thus providing data support for subsequent model training.

[0238] Specifically, in the SIR model, each node can be in one of three states: susceptible (S), infected (I), or recovered (R). During simulation initialization, only the target node is set to the infected state, while all other nodes are in the susceptible state. During propagation, at each time step, susceptible nodes will increase by an infection rate... It is infected by its infected neighboring nodes; at the same time, the infected node itself will be infected with a probability. The process transitions to a recovery state. When no infected nodes remain in the network, the propagation process terminates. At this point, the total number of nodes that were in both the infected and recovery states is defined as the overall influence of the target node. To reduce the randomness of a single simulation, each node is independently simulated 1000 times, and the average of its propagation influence is taken as the node's final label value. This invention's patented label is based on an infection rate... The simulation of the SIR model is generated under the following conditions, where It is the propagation threshold of the network; Indicates the degree of a node; This indicates taking the average value.

[0239] 3.5 Model Training

[0240] After the feature matrix and labels are generated, an attention mechanism and a multi-scale bi-branch convolutional network are used to train and predict the propagation effect of nodes. The network consists of a multi-scale convolutional branch, a channel attention branch, and an attention fusion module.

[0241] The multi-scale convolution branch consists of three cascaded multi-scale convolutional layers, each deploying convolutional kernels of three different scales: 3×3, 5×5, and 7×7. The first convolutional layer has 3 input channels, 10 output channels, a stride of 1, and padding of 1; the second convolutional layer has 10 input channels, 20 output channels, a stride of 1, and padding of 2; the third convolutional layer has 20 input channels, 30 output channels, a stride of 1, and padding of 3. All convolutional layers use the ReLU activation function. After each convolutional layer, 2×2 max merging is applied to reduce spatial dimensionality, ensuring effective feature downsampling while preserving key information.

[0242] In the channel attention branch, basic features are first extracted using three standard 5×5 convolutional layers. Then, a channel attention module is used, which performs global average pooling and global max pooling on the features in parallel. The resulting two feature vectors are fed into a shared 1×1 convolutional layer for processing. Finally, channel weights are generated through element-wise summation and non-linear activation, enabling the model to focus on the most relevant features for the current task while suppressing irrelevant or redundant information. To integrate the information from the two branches, an attention-based fusion strategy is introduced. This strategy first inputs the output features of each branch into a small fully connected network to learn the importance coefficients of each branch. Then, these coefficients are normalized using the Softmax function to ensure that the sum of the weights of each branch is 1. Finally, the normalized weights are used to perform a weighted summation of the features from both branches.

[0243] The model, with node labels generated by the SIR model, was trained for 500 epochs on a scale-free BA network. Mean squared error was used as the loss function during training, and the Adam optimizer was employed for optimization. After training, the model was applied to feature matrices generated by various unlabeled real networks. Node importance scores were calculated and ranked. Compared to the actual rankings obtained from the real SIR model, the proposed DAMSCNN method demonstrated superior performance on different real networks, validating its robustness and effectiveness.

[0244] 4. Experimental Results and Analysis

[0245] 4.1 Dataset

[0246] In the dataset, nine real-world networks were used to validate the performance of the DAMSCNN model. These networks cover social / information networks, power grid networks, routing networks, and protein networks. The training network was a BA network with 1000 nodes and an average degree of 4. Brief descriptions of these networks are as follows:

[0247] BA(n,k) network: A scale-free network with n nodes and an average of k nodes generated by the BA model.

[0248] NS: A collaborative network of scientists dedicated to network theory and experimentation.

[0249] USAir: Flight network between U.S. airports in 2010.

[0250] Sex: A two-sided online community of sexual activity, where nodes are men (sex buyers) and women (sex sellers), and a boundary is established when a man posts indicating that he is having sex with a woman.

[0251] PowerGrid: A network that contains information about the power grid in the western United States.

[0252] Vidal: An initial version of the network of a proteome proportion map of human binary protein-protein interactions.

[0253] GrQC: A research collaboration network of authors in the fields of general relativity and quantum cosmology, where nodes are authors and edges are established when two authors co-author a paper.

[0254] Hep: A collaborative research network for authors in the field of high-energy physics, where nodes are authors and edges are created when two authors co-author a paper.

[0255] Routers: A network consists of interconnected autonomous systems on the Internet.

[0256] Stelzl: Networks represent interacting protein pairs in humans.

[0257] The statistical properties of the network dataset used are shown in Table 1.

[0258] Table 1 Statistical properties of real networks

[0259]

[0260] Where n and m are the total number of nodes and edges in the network, respectively. <k>k is the average degree of the network. max is the maximum degree of the network, C is the average clustering coefficient of the network, and d is the density of the network.

[0261] 4.2 Baseline Method

[0262] To evaluate the effectiveness of DAMSCNN, two baseline methods—traditional methods and deep learning-based methods—were compared with the method of this invention:

[0263] DC: DC defines the importance of a node by the number of its first-order neighbors, as follows:

[0264] (12)

[0265] in, Represents a node The degree;

[0266] Represents a node The degree;

[0267] This represents the total number of nodes in the network;

[0268] in Represents a node The degree, This represents the total number of nodes in the network.

[0269] k-shell value: The k-shell value reflects the position of a node. Specifically, the k-shell value is obtained by recursively removing nodes based on their degree to divide the node into different layers.

[0270] BC: Betweenness centrality defines a node that acts as a bridge between two unconnected populations as a node that propagates influence. Its definition is as follows:

[0271] (13)

[0272] in, Represents a node The betweenness centrality value;

[0273] This represents the total number of elements in each sequential list;

[0274] Represents a node and nodes Between nodes The number of shortest paths;

[0275] Represents a node and nodes The number of shortest paths between them;

[0276] in Represents a node and nodes The number of shortest paths between them Represents a node and nodes Between nodes The number of shortest paths.

[0277] VC: The VC (V-community) index measures a node's propagation influence by calculating the number of communities each node is connected to. The more communities a node is connected to, the higher its propagation influence.

[0278] RCNN: RCNN constructs a feature matrix using the BFS algorithm as input to a convolutional neural network, and calculates the importance score of nodes through two layers of convolutional neural networks.

[0279] MRCNN: MRCNN constructs a three-channel convolutional neural network input and obtains the importance score of a node through a network of two convolutional layers, two pooling layers and one fully connected layer.

[0280] 4.3 Evaluation Indicators

[0281] Kendall's Coefficients: Kendall's coefficients are used to evaluate the performance of different ranking methods. As a ranking correlation coefficient, it is defined as follows:

[0282] (14)

[0283] in, Kendall's coefficient;

[0284] Indicates the number of coordination pairs;

[0285] Indicates the number of incoherent pairs;

[0286] This represents the total number of elements in each sequential list;

[0287] in and These represent the number of coordinated and non-coordinated pairs, respectively. It is the total number of elements in each sequential list. (Kendall's) The closer the coefficient is to 1, the more similar the two ranking lists are. This coefficient is widely used to test the ranking accuracy of node identification algorithms by the propagation effect. For example, let (x1, y1), (x2, y2), ..., (x n y n () is a joint team of X and Y, if x i > x j and y i > y j or x i < x j and y i < y j Then (x i y i ) and (x j y j ) are harmonious pairs; if x i > x j and y i < y j or x i < x j and y i > y j Then these two are considered discordant. If x i = x j or y i = y j Therefore, they are neither harmonious pairs nor unharmonious pairs.

[0288] 4.4 Node ranking capability

[0289] To verify the ranking accuracy of DAMSCNN in the key node identification task, this study uses Kendall's method. The coefficients were used as evaluation metrics to systematically compare the ranking performance of the proposed method with that of various baseline methods on nine real networks. All deep learning methods (including RCNN, MRCNN, and DAMSCNN) were trained on a uniformly configured BA scale-free network (containing 1000 nodes with an average degree of 4) to control experimental conditions. Table 2 shows the Kendall's coefficients between the rankings generated by each method and the real rankings simulated by the SIR model. coefficient, Figure 2 The correlation analysis results between the scores of each method and the actual scale of infection were further presented.

[0290] Table 2 Node sorting ability

[0291]

[0292] This invention patent uses experimental results from 7 different methods on 9 real networks, with a neighborhood size L=24 and an infection probability. The highest result is shown in bold, and the second highest result is shown in underline.

[0293] Experimental results show that DAMSCNN exhibits the best node ranking ability on all nine real-world network datasets. Compared to the suboptimal benchmark methods, DAMSCNN achieves significant improvements across various networks, specifically Kendall's... The average improvement rates of the coefficients are as follows: GrQC network 1.94%, Hep network 2.45%, NS network 10.15%, PowerGrid network 4.29%, Router network 4.2%, Sex network 3.03%, Stelzl network 5.24%, USALair network 2.75%, and Vidal network 2.88%. In summary, DAMSCNN not only significantly outperforms traditional centrality methods DC, KS, VC, and BC, but also surpasses RCNN and MRCNN, demonstrating higher node ranking accuracy and stronger cross-network robustness.

[0294] Figure 2 The ranking scores of DC, KS, BC, VC, RCNN, MRCNN, and DAMSCNN were compared with those of the SIR model simulated on nine real networks at different infection rates. By comparing the ranking scores below, Kendall's coefficients, where The extended threshold is used for training RCNN, MRCNN, and DAMSCNN with L = 24.

[0295] 4.5 Impact of Neighborhood Network Size

[0296] To systematically evaluate the impact of the neighborhood size parameter L on the performance of DAMSCNN, this study conducted multiple controlled experiments on nine real networks. By adjusting the value of L, and within an infection rate range of 1× Up to 1.9× The internal simulation uses the SIR model to calculate the average Kendall's score for each network under different L values. The coefficients are used to comprehensively evaluate the sensitivity of the model's ranking performance to changes in neighborhood scale.

[0297] Experimental results are as follows Figure 3 The results show that, in the vast majority of networks, DAMSCNN's ranking performance exhibits good robustness to variations in the domain network size L. For example... Figure 3 Kendall's diagram of the Sex network, NS network, GrQC network, and Router network shown in Figure (a) The coefficients fluctuate less with L, while PowerGrid fluctuates more significantly. This phenomenon can be attributed to the extremely low topological density of the PowerGrid network and its average degree being lower than the average degree of the training network BA. Its sparse structure results in limited structural information gain during neighborhood expansion, making it more sensitive to the value of L. Figure 3 As shown in (b), the performance of DAMSCNN is not significantly affected by changes in L in the USAL, Hep, Vidal, and Stelzl networks, further verifying the model's adaptability under different topological characteristics. In summary, DAMSCNN can adapt to complex networks of different sizes, indicating that its structural design has good parameter robustness and cross-network generalization ability, making it suitable for complex network analysis tasks with diverse structures.

[0298] Figure 3 In the middle: (a) the neighborhood size L has little effect on networks other than PowerGrid; this is because the PowerGrid network has a low topology density and the average degree of the network is lower than the average degree of the training network; (b) the neighborhood size L has little effect on the networks in USAL, Hep, Vidal and Stelzl.

[0299] 4.6 Ablation

[0300] To evaluate the performance contribution of the dual-branch structure and attention fusion mechanism in DAMSCNN, this study designed a systematic ablation experiment. The complete model was deconstructed into three key variants for comparative analysis: a simplified model containing only multi-scale convolutional branches (MS), a simplified model containing only attention mechanism branches (AT), and a model fusing dual-branch features using a common concatenation method (AT+MS). With a fixed neighborhood size parameter L=24, each model was tested on multiple real-world network datasets. The SIR model was used to generate a node influence benchmark ranking under different infection rates, and Kendall's algorithm was used as the benchmark. The coefficients were used as performance evaluation metrics to systematically compare the performance differences between the various variants and the full DAMSCNN.

[0301] Experimental results on nine real-world complex networks are as follows: Figure 4 As shown, the complete DAMSCNN model achieves state-of-the-art performance on networks such as Sex, Stelzl, and USAL, and its Kendall's... The coefficients are significantly higher than those of the various ablation variants. This result validates the design effectiveness of the dual-branch structure and attention fusion mechanism: the multi-scale convolutional branch can effectively extract local and global topological features of nodes under different receptive fields, while the channel attention branch enhances the expressive power of key features through dynamic weight allocation. The deep fusion of the two not only achieves comprehensive perception of the multi-scale structure of the network, but also improves the model's recognition accuracy of key topological patterns through an adaptive feature selection mechanism, thereby maintaining stable recognition performance in complex and ever-changing network environments.

[0302] Figure 4 In the Chinese model: MS uses only multi-scale convolution, AT uses only the attention mechanism module, and AT+MS is a model that uses a normal concatenation method to fuse features from two branches.

[0303] 4.7 Node Ranking Correlation Analysis

[0304] To systematically evaluate the differences in node influence recognition performance between DAMSCNN, MRCNN, and RCNN, this study employs a ranking score correlation analysis method for comparative research. Under a unified experimental setting (neighborhood size parameter L=24), we trained and compared the performance of each method on multiple real networks. In the figure, the horizontal and vertical axes represent the normalized propagation influence scores of the benchmark method and DAMSCNN for nodes, respectively, and the color depth of each scatter point corresponds to the node's infection rate. The actual propagation impact scale was simulated using the SIR model. The diagonal reference line marked in the figure provides an intuitive benchmark for performance comparison: the point set below the diagonal line indicates that DAMSCNN scores higher than the comparison method for the corresponding node, while the point set above the diagonal line reflects a higher score for the comparison method. By observing the distribution and color characteristics of the points on both sides of the diagonal line, the performance differences of each method in identifying the propagation impact of nodes can be effectively compared.

[0305] The correlation between the ranking scores of DAMSCNN and RCNN is as follows: Figure 5 As shown in the figure, the analysis reveals a significant positive correlation between the ranking scores of the two methods, proving that both methods can effectively distinguish the importance of nodes. Further comparison reveals that in the Sex, Router, and USALearning networks, DAMSCNN can identify key nodes that RCNN failed to capture, indicating that the proposed DAMSCNN significantly outperforms RCNN in key node identification.

[0306] Depend on Figure 6 As can be seen, the ranking scores of DAMSCNN and MRCNN also show a significant positive correlation, further demonstrating the effectiveness of DAMSCNN in distinguishing node importance. In Hep, NS, USAL, and PowerGrid networks, DAMSCNN can identify key nodes that MRCNN failed to capture. This advantage stems from DAMSCNN's stronger feature extraction capabilities. Its multi-scale convolution and channel attention mechanism in its dual-branch structure can more comprehensively extract important correlation information between nodes, thus performing better in key node identification.

[0307] Figure 5 In terms of infection rate Below, the position of each point in the figure reflects the difference in the propagation scores assigned to nodes by RCNN and DAMSCNN: points located above the diagonal indicate that RCNN scores higher than DAMSCNN, and vice versa; the color of each point is determined by the diffusion effect obtained by simulating the SIR model.

[0308] Figure 6 In terms of infection rate Below, the position of each point in the figure reflects the difference in the propagation scores assigned to nodes by MRCNN and DAMSCNN: points located above the diagonal indicate that MRCNN scores higher than DAMSCNN, and vice versa; the color of each point is determined by the diffusion effect obtained by simulating the SIR model.

[0309] 4.8 Node Feature Comparison

[0310] To verify the superiority of the DAMSCNN model in network node feature learning and key node identification, the multi-dimensional topological features of network nodes are transformed into "image-like" features. RCNN (single-channel) and MRCNN (3-channel) are used as benchmarks to examine whether DAMSCNN is more suitable for network data characteristics in the 3-channel feature fusion strategy, whether it can more accurately capture node differences and more efficiently identify key network nodes, and thus provide a better model selection for deep learning analysis of complex networks.

[0311] Comparison based on node features Figure 7 The results show that the DAMSCNN model outperforms other models in terms of stability, robustness, and feature learning quality. Specifically, under the same noise conditions, the feature distribution output by DAMSCNN is the most concentrated and compact, with significantly lower volatility than MRCNN, demonstrating excellent noise suppression capabilities. Its integrated dual-attention mechanism and multi-scale fusion architecture enable it to accurately focus on key topological features, effectively identify important nodes that are highly consistent with the network's ground truth centrality, and achieve the clearest node distinction in the feature space. Overall, DAMSCNN demonstrates a clear advantage in balancing feature richness and stability, providing a reliable solution for network representation learning and key node discovery tasks.

[0312] Figure 7 The five most important nodes are marked in red, and the rest are marked in blue.

[0313] 5. Conclusion

[0314] This invention proposes a key node identification method called DAMSCNN. This method effectively improves the accuracy and robustness of key node identification through three core innovations: First, addressing the limited representational ability of traditional feature matrix construction methods in sparse networks, a learnable parameterized imputation method is proposed. This method can dynamically generate the imputation content of the feature matrix according to the network topology, thereby enhancing the model's adaptability to sparse and irregular network structures while preserving structural continuity. Second, a dual-branch network architecture is designed to collaboratively extract multi-level topological features of nodes. This parallel dual-branch design achieves complementary learning of local details and global context. Finally, a dynamic attention fusion mechanism is innovatively introduced. This mechanism can automatically learn and adjust the fusion weights of the output features of the multi-scale convolutional branch and the channel attention branch according to the specific topological characteristics of the input network, thereby significantly improving the model's generalization ability to represent networks with different structures. To verify the effectiveness of DAMSCNN, we conducted systematic experiments on nine real-world complex network datasets and compared it with six baseline methods. Evaluation results show that DAMSCNN achieves the best performance in the key node identification task. On different networks, the ranking relevance improvement of DAMSCNN compared with the second-best model ranges from 1.94% to 10.15%, which fully demonstrates its superior performance in capturing node influence and accurately identifying key nodes.

[0315] Although embodiments of the invention have been shown and described, those skilled in the art will understand that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the claims and their equivalents.< / k>

Claims

1. A method for identifying key nodes in complex networks based on attention mechanisms and multi-scale feature fusion, characterized in that, Includes the following steps: S1, generate the feature matrix; S2, input the data into the training model consisting of multi-scale convolutional branches and channel attention branches; S3 outputs the recognition results using the trained model.

2. The method for identifying key nodes in complex networks based on attention mechanism and multi-scale feature fusion according to claim 1, characterized in that, Step S1 includes: ， in, Represents a node The sum of the neighboring degrees; Represents a node The set of neighbors; Represents a node The degree; ， in, A matrix representing the local information of a node; Indicates the 0th row, 1st row. Elements in the adjacency matrix of columns; Represents a node The sum of the neighboring degrees; Indicates the number of nodes in the local subgraph; Indicates the first The element in the adjacency matrix at row 0 and column 0; Represents a node The sum of the neighboring degrees; Indicates the first Line number Elements in the adjacency matrix of columns; ， in, Represents a node The number of communities connected; This indicates finding the number of nodes in a set. Represents a node The set of neighbors; Represents a node The community group to which it and its neighbors belong; ， in A matrix representing the community information of nodes; Indicates the 0th row, 1st row. Elements in the adjacency matrix of columns; Represents a node The number of communities connected; Indicates the number of nodes in the local subgraph; Indicates the first The element in the adjacency matrix at row 0 and column 0; Represents a node The number of communities connected; Indicates the first Line number Elements in the adjacency matrix of columns; ， Represents a node The eigenvector center value; Indicates a constant factor; This represents the total number of nodes in the network; Indicates the first Line number Elements in the adjacency matrix of columns; It is a node The eigenvector center value; ， in A matrix representing the global information of a node; Indicates the 0th row, 1st row. Elements in the adjacency matrix of columns; Represents a node The eigenvector center value; Indicates the number of nodes in the local subgraph; Indicates the first The element in the adjacency matrix at row 0 and column 0; Represents a node The eigenvector center value; Indicates the first Line number The elements in the adjacency matrix of the columns.

3. The method for identifying key nodes in complex networks based on attention mechanism and multi-scale feature fusion according to claim 1, characterized in that, Step S2 includes: The multi-scale convolution branch consists of three cascaded multi-scale convolutional layers, each deploying convolutional kernels of three different scales: 3×3, 5×5, and 7×7. The first convolutional layer has 3 input channels, 10 output channels, a stride of 1, and padding of 1. The second convolutional layer has 10 input channels, 20 output channels, a stride of 1, and padding of 2. The third convolutional layer has 20 input channels, 30 output channels, a stride of 1, and padding of 3. The channel attention branch includes three 5×5 convolutional layers for feature extraction, followed by parallel global average pooling and global max pooling of the features. The resulting two feature vectors are then fed into a shared 1×1 convolutional layer for processing. Finally, channel weights are generated by element-wise addition and non-linear activation.

4. The method for identifying key nodes in complex networks based on attention mechanism and multi-scale feature fusion according to claim 1, characterized in that, Step S2 also includes: ， express It follows a normal distribution with a mean of 0 and a variance of 1; Let represent the learnable parameter at the position of the i-th row and j-th column, with the superscript (0) indicating the initial value of the parameter; ， in, This indicates that the size is increased to L feature matrix by using zero padding; Representation matrix Element; Represents the original characteristic matrix; Indicates the number of nodes in the local subgraph; ， in, Represents the learnable parameter matrix and the original characteristic matrix Merged matrices; Represents the mask matrix; This represents element-wise multiplication; This indicates that the size is increased to L feature matrix by using zero padding; Represents a matrix of all 1s; This represents the learnable parameter matrix.

5. The method for identifying key nodes in complex networks based on attention mechanism and multi-scale feature fusion according to claim 1, characterized in that, Step S2 also includes: ， in, Indicates partial derivative; Represents the loss function; Represents the learnable parameter matrix; Indicates the number of nodes in the local subgraph; This indicates a logical condition; Represents the learnable parameter matrix and the original characteristic matrix Merged matrices; This indicates that the size is increased to L feature matrix by using zero padding; ， in, Indicates the first The learnable parameter matrix updated after each training iteration; Indicates the first The parameter matrix for the next iteration; Indicates the learning rate; Indicates partial derivative; This represents the loss function.

6. The method for identifying key nodes in complex networks based on attention mechanism and multi-scale feature fusion according to claim 1, characterized in that, It also includes evaluating the model, including: ， in, Represents a node The degree; Represents a node The degree; This represents the total number of nodes in the network.

7. The method for identifying key nodes in complex networks based on attention mechanism and multi-scale feature fusion according to claim 1, characterized in that, The assessment also included: ， in, Represents a node The betweenness centrality value; This represents the total number of elements in each sequential list; Represents a node and nodes Between nodes The number of shortest paths; Represents a node and nodes The number of shortest paths between them.

8. The method for identifying key nodes in complex networks based on attention mechanism and multi-scale feature fusion according to claim 1, characterized in that, It also includes evaluation indicators, which include: ， in, Kendall's coefficient; Indicates the number of coordination pairs; Indicates the number of incoherent pairs; This represents the total number of elements in each sequential list.

9. A computer system, characterized in that, include: processor; Memory used to store processor-executable instructions; The processor is configured to implement the method for identifying key nodes in complex networks based on attention mechanism and multi-scale feature fusion as described in any one of claims 1 to 8 when executing the executable instructions.

10. A computer-readable storage medium, characterized in that, include: A memory on which computer programs are stored; A processor is configured to execute the program in the memory to implement the method for identifying key nodes in complex networks based on attention mechanism and multi-scale feature fusion as described in any one of claims 1 to 8.

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