Complex network key node identification method based on spectral clustering

Abstract

The invention discloses a complex network key node identification method in the field of network science and computer science, which comprises the following steps: firstly, adding hyperedges for a network model through a spectral clustering algorithm to construct a hyperedge model, and then extracting local and global features for nodes based on the hyperedge model to construct feature vectors; and correcting a regression mean square error loss function by using a sorting loss function to obtain a more accurate sorting sequence, inputting an adjacent matrix of the hypernetwork model and a feature matrix of nodes into a graph convolutional neural network for training, and obtaining a calculation result of a key score as a sorting standard to obtain a key node. According to the method, deeper-level representation of mutual relation and effects among real system nodes is realized, the focus of attention is transferred from node scores to sorting, and effective identification of key nodes is realized.

Description

Technical Field

[0001] The invention relates to a key node identification method driven by feature extraction, and belongs to the technical fields of complex network science and computer science. Background Art

[0002] Complex network science reveals the organizational structure of nodes and edges in complex systems through graph theory and network analysis methods, helping us understand the topological characteristics of the system. By studying the network structure, we can reveal the inherent mechanisms of system functions, such as information dissemination, resource allocation, and system stability.

[0003] At present, the commonly used key node identification methods mainly include identification methods based on traditional structural decomposition and criticality definition and identification methods based on deep learning. Identification methods based on traditional structural decomposition and criticality definition usually use the position of the node in the network to calculate the criticality score of the node. For example, nodes that are located at the center of the network, connect two communities, and pass through the shortest path are usually considered to have a more critical role in network function and stability. However, this type of method usually has a rich club effect, and the identified key nodes have extremely high similarity, which makes the identification results fall into a situation similar to the local optimum. The identification method based on deep learning mainly uses convolutional neural network models to extract and mine the hidden structural features in the network. Most of them adopt a fully supervised learning method. This type of method requires model training, which means that a large number of sample node features and corresponding labels are required.

[0004] At present, the primary feature extraction of the network is mainly based on the traditional network model. With the development of science and technology and the deepening of research, the traditional complex network model cannot represent the broader, deeper, and higher-dimensional structural information in the real system, and realize the continuity and integrity of the connection between the system nodes, and gradually cannot meet the needs of real-world system modeling. Compared with the traditional graph data edge, which can only represent the connection and interaction relationship between pairs of nodes, the concept of hyperedge in the hypergraph allows any number of nodes to be connected on one edge, which means that the hypergraph can represent the broader, deeper, and higher-dimensional structural information in the real system, and realize the continuity and integrity modeling of the connection between the system nodes. Summary of the invention

[0005] The purpose of the present invention is to overcome the shortcomings of existing primary feature representations and provide a key node identification method driven by feature extraction, which can form richer structural features and interaction information between nodes for analysis and judgment of key nodes, effectively extract higher-order features of network nodes, and improve the identification effect of key nodes.

[0006] To achieve the above object, the present invention is implemented by adopting the following technical solutions:

[0007] In a first aspect, the present invention provides a feature extraction driven key node identification method, comprising:

[0008] (1) Establishing a traditional network model;

[0009] (2) Using spectral clustering algorithm to add hyperedges to the network model to build a hypernetwork model;

[0010] (3) Extract local and global features for nodes and construct feature vectors based on the constructed hypernetwork model;

[0011] (4) By setting each node as the initial infected node in the SIR model, the infection scale of the entire network is calculated when the model tends to the equilibrium state and sorted as the real sequence;

[0012] (5) Use the ranking loss function to correct the regression mean square error loss function to obtain a more accurate ranking sequence;

[0013] (6) The adjacency matrix of the hypernetwork model and the feature matrix of the nodes are input into the graph convolutional neural network for training to obtain the calculation results of the criticality scores. The criticality is sorted according to the calculation results to obtain the final key node identification results.

[0014] Furthermore, the spectral clustering algorithm includes:

[0015] The original network model containing N nodes is represented as the adjacency matrix A, and the degree matrix D of the network nodes is calculated. On this basis, the connection density of each node is expressed as w ij Construct the similarity matrix W. Among them, w ij It describes the probability that two nodes can reach each other through only one intermediate node k and the sum of their adjacency:

[0016]

[0017] Calculate the normalized similarity Laplace matrix L according to the formula sym , solve the eigenvalue problem of the Laplace matrix L sym v = λv, select the smallest l non-zero eigenvalues ​​and their corresponding eigenvectors to form a new matrix. l is the final number of clusters determined by the eigenvalue interval method. All eigenvalues ​​of the Laplace matrix are ordered from large to small, and the number before the largest interval eigenvalue is assigned to l.

[0018]

[0019] The matrix U is standardized, and each row is divided by its norm to obtain the standardized matrix U′. Each row of U′ represents the mapping of each data point in the new low-dimensional space. l column vectors are randomly selected as the initial cluster centers, and the distances from other nodes to all cluster centers are calculated and assigned to the cluster where the nearest cluster center is located. For each cluster, the mean of all node vectors in the cluster is recalculated, and the mean is used as the new cluster center. The above steps are repeated until the maximum number of iterations is reached or the distance between the new cluster center and the result of the previous iteration is less than a specific value.

[0020] After spectral clustering, nodes belonging to the same cluster are on the same hyperedge, and the new hypernetwork model adjacency matrix is ​​expressed as H = (h ij ) N×N .

[0021] Furthermore, based on the constructed hypernetwork model, local and global features are extracted for nodes to construct feature vectors, specifically including:

[0022] The degree centrality, closeness centrality, eigenvector centrality, invulnerability network efficiency and subgraph centrality of the traditional network model are transferred to the hypernetwork model, and the features of each node are calculated to form its eigenvector x v ={k v , b v , η v , e v , S v}.

[0023] The degree centrality of a node in the traditional network model is expressed as the number of neighboring nodes, and the hyperdegree of a node in the hypernetwork model is expressed as the number of hyperedges to which the node belongs, that is:

[0024]

[0025] In order to measure the role of nodes in information flow in the network, the number of nodes through which the shortest path between each pair of nodes passes is calculated, that is, the node's betweenness centrality b. v .

[0026]

[0027] Among them, g s，t and They represent the shortest path between nodes s and t and the number of paths passing through node v in the shortest path.

[0028] In order to measure the importance of each node's neighboring nodes and their contribution to it, the adjacency matrix H is eigendecomposed, and the v-th dimension component of the eigenvector corresponding to its maximum eigenvalue is the eigenvector centrality of the node being sought.

[0029] In order to measure the role of nodes in network invulnerability, the influence of nodes on global closeness centrality, i.e. global efficiency, is calculated. The ratio of the global efficiency of the network after each node is removed from the network to the initial global efficiency of the network is calculated to obtain the node's invulnerability network efficiency η. v .

[0030]

[0031] Among them, d st It represents the shortest path length between nodes s and t, that is, the number of hops after which one node can be reached from another node.

[0032] Subgraph centrality is used to reveal the hierarchical structure in the network and the dependency information between nodes. Based on the degree of participation of the node in the local subgraph of the network, the number of closed paths of the node is calculated and different weights are assigned to closed paths of different lengths. The shorter the closed path, the greater the weight. The super network subgraph centrality S of the node is calculated. v As shown in formula (6):

[0033]

[0034] Based on the fact that most complex systems in the real world exhibit scale-free characteristics, the cosine similarity α between the five features is calculated for BA networks with different structural characteristics. It is proved that all the features are positively correlated, and at the same time, there are some distinctions and emphasis on the focus, for example, betweenness centrality and invulnerability network efficiency show a weak correlation, and superdegree and subgraph centrality show a strong correlation.

[0035]

[0036] Among them, P and Q represent vectors composed of two different features of all nodes respectively.

[0037] Furthermore, each node is set as the initial infected node in the SIR model, and the infection scale of the entire network is calculated when the model tends to the equilibrium state and sorted as the real sequence, which specifically includes:

[0038] 0, 1, and 2 are used to represent susceptible (S), infected (I), and recovered (R), respectively. Except for the node to be calculated, all other nodes are set to state 0, and the current node is set to state 1. In each iteration, the node in state 1 will infect the node on the same hyperedge with a probability of β, and become a recovered node with a recovery probability of γ, and lose the ability to infect its neighbor nodes in the next iteration.

[0039] The iterative process of the SIR model is expressed by the following differential equation:

[0040]

[0041] The number of iterations is set to 1000, and the infection rate β of infected nodes in each iteration is 0.3. For the convenience of simulation, the recovery probability γ is set to 1, that is, the nodes in the infected state will become recovered at the end of the current iteration and lose the ability to infect their neighboring nodes in the next iteration. The final infection scale of the entire network at the end of the iteration is expressed as the proportion of nodes in the infected state and the recovered state in the network.

[0042]

[0043] The simulation was repeated 100 times and the average value was taken to obtain the infection capacity of each node, which represents the role it plays in the process of network information flow. The results were sorted as the real key sequence.

[0044] Further, the foreground saliency map and the background prior map are fused to obtain a coarse saliency map by minimizing the data and smoothness loss of the cost function, specifically including:

[0045] The training loss function is defined as the weighted sum of the pairwise ranking loss function multiplied by the correction coefficient and the mean square error loss function, in order to avoid the situation where the calculated results of the criticality of two nodes have a small error with the actual but the ranking is opposite. The ranking loss function is expressed in the form of the cross entropy of the model predicted order probability and the true label order of a pair of nodes. The sum of the loss functions of each pair of nodes is calculated and normalized according to the logarithm. The ranking loss function L rank It is expressed as:

[0046]

[0047] Among them, P i,j =σ(Y(i)-Y(j)) represents the true sorting probability of node pair (i, j), Q i，j =σ(T(i)-T(j)) represents the predicted ranking probability, σ represents the function Y(i) and T(i) represent the criticality scores of node i calculated using the SIR model and the neural network, respectively.

[0048] Mean square error loss function L MSE It is expressed as:

[0049]

[0050] Therefore, the training loss function L is expressed as:

[0051] L=ωL rank +L MSE

[0052] Where ω is the correction coefficient, which is set to 0.02.

[0053] Furthermore, the adjacency matrix and the feature matrix of the node are input into the graph convolutional neural network for training to obtain the calculation results of the criticality scores, and the criticality is sorted according to the calculation results to obtain the final key node recognition results, which specifically include:

[0054] The characteristic spectrum of the dataset network is generated. The characteristic spectrum of the scale-free network has a very high similarity. The artificially generated hybrid BA network is used as the training set sample, and its super-adjacency matrix and two-dimensional eigenvector matrix are used as input to train the graph convolutional neural network model. After the training, the critical scores of the dataset nodes are regressed and predicted, and the key nodes are identified by sorting according to the regression results.

[0055] Compared with the prior art, the present invention has the following beneficial effects:

[0056] (1) The present invention uses a spectral clustering algorithm to construct a super network model for the traditional network, which more deeply characterizes the interconnection and interaction relationship between nodes. The extracted features are analyzed by cosine similarity, which can more deeply represent the position and role of nodes in the network structure, and distinguish them from each other, avoiding the rich club phenomenon.

[0057] (2) The present invention uses the weighted pairwise loss function and the mean square error loss function to correct the training process and results, so that the results pay more attention to the sequence of node importance rather than specific values, and can more accurately identify key nodes that have greater contributions to network information dissemination and network connectivity. BRIEF DESCRIPTION OF THE DRAWINGS

[0058] Figure 1 This is a flow chart of a key node identification method driven by feature extraction provided in the first embodiment of the present invention.

[0059] Figure 2 This is a cosine similarity heat map between the features of two BA networks with different structures. Through the heat map, it is found that the five features are positively correlated and differentiated. And the similarity between features also has similar performance on BA networks with different structures. For example, closeness centrality and invulnerability network efficiency show a weak correlation, and superdegree and subgraph centrality show a strong correlation.

[0060] Figure 3 The influence of different correction coefficients on the results is demonstrated. On different data sets, when ω is 0, it is equivalent to training the neural network with the mean square error loss function. When ω is 0.02, the key node recognition accuracy of the method performs well on both data sets.

[0061] Figure 4 and Figure 5The comparison results of the Kendall coefficient of the proposed method and the real sequence with other algorithms under different infection probability β are shown. Different data sets achieve the best results when β=0.3. When β exceeds 0.5, the recognition accuracy of the algorithm is greatly reduced. This is because when the infection rate of the infected node is too high, its neighboring nodes in the susceptible state are more likely to be infected. The initial infected node can easily cause a large-scale spread, making it difficult to distinguish the criticality of the node.

Claims

1. A key node identification method based on feature extraction, characterized in that: include: (1) Establishing a traditional network model; (2) Using spectral clustering algorithm to add hyperedges to the network model to build a hypernetwork model; (3) Extract local and global features for nodes and construct feature vectors based on the constructed hypernetwork model; (4) By setting each node as the initial infected node in the SIR model, the infection scale of the entire network is calculated when the model tends to the equilibrium state and sorted as the real sequence; (5) Use the ranking loss function to correct the regression mean square error loss function to obtain a more accurate ranking sequence; (6) The adjacency matrix of the hypernetwork model and the feature matrix of the nodes are input into the graph convolutional neural network for training to obtain the calculation results of the criticality scores. The criticality is sorted according to the calculation results to obtain the final key node identification results.

2. According to the feature extraction driven key node identification method of claim 1, the spectral clustering algorithm adding hyperedges comprises: The original network model containing N nodes is represented as the adjacency matrix A, and the degree matrix D of the network nodes is calculated. On this basis, the connection density of each node is expressed as w ij Construct the similarity matrix W. Among them, w ij It describes the probability that two nodes can reach each other through only one intermediate node k and the sum of their adjacency: According to formula (2), the normalized similarity Laplace matrix L is calculated sym , solve the eigenvalue problem of the Laplace matrix L sym v = λv, select the smallest l non-zero eigenvalues ​​and their corresponding eigenvectors to form a new matrix. l is the final number of clusters determined by the eigenvalue interval method. All eigenvalues ​​of the Laplace matrix are ordered from large to small, and the number before the largest interval eigenvalue is assigned to l. The matrix U is standardized, and each row is divided by its norm to obtain the standardized matrix U′. Each row of U′ represents the mapping of each data point in the new low-dimensional space. l column vectors are randomly selected as the initial cluster centers, and the distances from other nodes to all cluster centers are calculated and assigned to the cluster where the nearest cluster center is located. For each cluster, the mean of all node vectors in the cluster is recalculated, and the mean is used as the new cluster center. The above steps are repeated until the maximum number of iterations is reached or the distance between the new cluster center and the result of the previous iteration is less than a specific value. After spectral clustering, nodes belonging to the same cluster are on the same hyperedge, and the new hypernetwork model adjacency matrix is ​​expressed as H = (h ij ) N×N .

3. According to the feature extraction driven key node identification method of claim 1, based on the constructed hypernetwork model, local and global features of the node are extracted to form the feature vector of the node, specifically comprising: The degree centrality, closeness centrality, eigenvector centrality, invulnerability network efficiency and subgraph centrality of the traditional network model are transferred to the hypernetwork model, and the features of each node are calculated to form its eigenvector x v ={k v , b v , η v , e v , S v }. The degree centrality of a node in the traditional network model is expressed as the number of neighboring nodes, and the hyperdegree of a node in the hypernetwork model is expressed as the number of hyperedges to which the node belongs, that is: In order to measure the role of nodes in information flow in the network, the number of nodes through which the shortest path between each pair of nodes passes is calculated, that is, the node's betweenness centrality b. v . Among them, g s,t and They represent the shortest path between nodes s and t and the number of paths passing through node v in the shortest path. In order to measure the importance of each node's neighboring nodes and their contribution to it, the adjacency matrix H is eigendecomposed, and the v-th dimension component of the eigenvector corresponding to its maximum eigenvalue is the eigenvector centrality of the node being sought. In order to measure the role of nodes in network invulnerability, the influence of nodes on global closeness centrality, i.e. global efficiency, is calculated. The ratio of the global efficiency of the network after each node is removed from the network to the initial global efficiency of the network is calculated to obtain the node's invulnerability network efficiency η. v . Among them, d st It represents the shortest path length between nodes s and t, that is, the number of hops after which one node can be reached from another node. Subgraph centrality is used to reveal the hierarchical structure in the network and the dependency information between nodes. Based on the degree of participation of the node in the local subgraph of the network, the number of closed paths of the node is calculated and different weights are assigned to closed paths of different lengths. The shorter the closed path, the greater the weight. The super network subgraph centrality S of the node is calculated. v As shown in formula (6): Based on the fact that most complex systems in the real world exhibit scale-free characteristics, the cosine similarity α between the five features is calculated for BA networks with different structural characteristics. It is proved that all the features are positively correlated, and at the same time, there are some distinctions and emphasis on the focus, for example, betweenness centrality and invulnerability network efficiency show a weak correlation, and superdegree and subgraph centrality show a strong correlation. Among them, P and Q represent vectors composed of two different features of all nodes respectively.

4. The feature extraction driven key node identification method according to claim 1 is characterized in that: The final infection scale is calculated based on setting each node as the initial infection state node in the SIR model, including: 0, 1, and 2 are used to represent susceptible (S), infected (I), and recovered (R), respectively. Except for the node to be calculated, all other nodes are set to state 0, and the current node is set to state 1. In each iteration, the node in state 1 will infect the node on the same hyperedge with a probability of β, and become a recovered node with a recovery probability of γ, and lose the ability to infect its neighbor nodes in the next iteration. The iterative process of the SIR model is expressed by the following differential equation: The number of iterations is set to 1000, and the infection rate β of infected nodes in each iteration is 0.

3. For the convenience of simulation, the recovery probability γ is set to 1, that is, the nodes in the infected state will become recovered at the end of the current iteration and lose the ability to infect their neighboring nodes in the next iteration. The final infection scale of the entire network at the end of the iteration is expressed as the proportion of nodes in the infected state and the recovered state in the network. The simulation was repeated 100 times and the average value was taken to obtain the infection capacity of each node, which represents the role it plays in the process of network information flow. The results were sorted as the real key sequence.

5. The feature extraction driven key node identification method according to claim 1, characterized in that: The training process is modified through the sorting loss function, including: The training loss function is defined as the weighted sum of the pairwise ranking loss function multiplied by the correction coefficient and the mean square error loss function, in order to avoid the situation where the calculated results of the criticality of two nodes have a small error with the actual but the ranking is opposite. The ranking loss function is expressed in the form of the cross entropy of the model predicted order probability and the true label order of a pair of nodes. The sum of the loss functions of each pair of nodes is calculated and normalized according to the logarithm. The ranking loss function L rank It is expressed as: Among them, P i，j =σ(Y(i)-Y(j)) represents the true sorting probability of node pair (i, j), Q i，j =σ(T(i)-T(j)) represents the predicted ranking probability, σ represents the function Y(i) and T(i) represent the criticality scores of node i calculated using the SIR model and the neural network, respectively. Mean square error loss function L MSE It is expressed as: Therefore, the training loss function L is expressed as: L=ωL rank +L MSE 6. The feature extraction driven key node identification method according to claim 1 is characterized in that: The adjacency matrix of the hypernetwork model and the feature matrix of the nodes are input into the graph convolutional neural network for training, and the calculation results of the criticality scores are obtained. The criticality is sorted according to the calculation results to obtain the final key node identification results, which include: The characteristic spectrum of the dataset network is generated. The characteristic spectrum of the scale-free network has a very high similarity. The artificially generated hybrid BA network is used as the training set sample, and its super-adjacency matrix and two-dimensional eigenvector matrix are used as input to train the graph convolutional neural network model. After the training, the critical scores of the dataset nodes are regressed and predicted, and the key nodes are identified by sorting according to the regression results.

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