Complex network node role hierarchy detection method based on Legendre wavelet diffusion model

Abstract

A complex network node role hierarchy detection method based on a Legendre wavelet diffusion model comprises the steps that a complex network to be analyzed is acquired, and global and local structure information of a spectrogram is captured through spectrogram transformation; constructing a Laplacian matrix of the graph, obtaining eigenvalues and eigenvectors of the Laplacian matrix through eigenvalue decomposition, and converting a graph signal from a vertex domain to a frequency spectrum domain; a multi-scale wavelet basis function is generated by combining a thermal diffusion mechanism with a Legendre wavelet basis function; generating a multi-scale wavelet coefficient matrix of the nodes based on a multi-scale wavelet basis function, extracting embedded features of each node, calculating envelope information of a real part, an imaginary part and Hilbert transformation, and generating a feature matrix of each node; obtaining the network feature embedding representation of the whole graph through the feature matrix of each node; and the network feature embedded representation of the whole graph is introduced into a classifier for node role hierarchy identification. The method can provide support for various application scenes such as network optimization, security monitoring, function division and the like.

Description

Technical Field

[0001] The present invention belongs to the field of complex network analysis, and in particular relates to a complex network node role hierarchy detection method based on a Legendre wavelet diffusion model. Background Art

[0002] With the development of information technology and network technology, complex networks have been widely used in many fields such as society, economy, communication, and biology. There are many complex relationships between nodes and edges in complex networks, and the role of nodes in the network often determines the topological structure and function of the network. How to accurately detect and identify the roles of nodes in complex networks and their hierarchical relationships has always been a difficult problem in network science and data analysis.

[0003] In the existing technology, traditional complex network node role identification methods mostly rely on graph theory and neighborhood analysis, but these methods often have problems such as poor accuracy, slow processing speed, and inability to cope with large-scale networks when facing networks with complex structures and dynamic changes. Therefore, proposing a new method that can perform multi-level node role detection based on network structure and node behavior characteristics combined with advanced wavelet diffusion models has become a hot topic in current research. Summary of the invention

[0004] In order to overcome the deficiencies in the prior art, the present invention provides a complex network node role hierarchical detection method based on the Legendre wavelet diffusion model. Through this method, the node roles in the complex network can be effectively identified and their hierarchical structure can be revealed, thereby providing more accurate information for applications such as network optimization, attack detection, and function division.

[0005] The technical solution of the present invention is as follows:

[0006] A complex network node role hierarchy detection method based on Legendre wavelet diffusion model, the method comprising the following steps:

[0007] Obtain the complex network to be analyzed and capture the global and local structural information of the graph through spectrum graph transformation;

[0008] Construct the Laplacian matrix of the graph, and obtain the eigenvalues ​​and eigenvectors of the Laplacian matrix through eigenvalue decomposition, and convert the graph signal from the vertex domain to the spectrum domain;

[0009] The heat diffusion mechanism is combined with Legendre wavelet basis function to generate multi-scale wavelet basis function;

[0010] Generate a multi-scale wavelet coefficient matrix of nodes based on the multi-scale wavelet basis function, extract the embedded features of each node, and calculate the real part, imaginary part and envelope information after Hilbert transformation to generate the feature matrix of each node;

[0011] The network feature embedding representation of the entire graph is obtained through the feature matrix of each node;

[0012] The network features of the entire graph are embedded into a representation and passed into a classifier to identify the node role level.

[0013] Furthermore, the Laplacian matrix of the graph is constructed, and the eigenvalues ​​and eigenvectors of the Laplacian matrix are obtained by eigenvalue decomposition, so as to convert the graph signal from the vertex domain to the spectrum domain. Specifically, the following steps are performed:

[0014] First, construct the Laplacian matrix L of the graph, defined as:

[0015] L=DA

[0016] Where D is the degree matrix and A is the adjacency matrix, which is obtained by eigenvalue decomposition of the Laplace matrix L:

[0017] L=UΛU T

[0018] where Λ is a diagonal matrix containing the eigenvalues ​​of the Laplace matrix L, and U is a matrix containing the corresponding eigenvectors. The columns of U are orthogonal vectors, and U T is the transposed matrix of matrix U. Through the eigenvector matrix U, the graph signal is transformed from the vertex domain to the spectrum domain.

[0019] Furthermore, the multi-scale wavelet basis function is generated by calculating the heat kernel feature and combining the Legendre polynomial, specifically:

[0020]

[0021] Among them, φ k (x) represents the value of the wavelet basis function with scale k at node x, λ j is the eigenvalue of the Laplace matrix L, V j is the eigenvector, L k (λ j ) is the Legendre polynomial at eigenvalue λ j The value at is the thermal nuclear coefficient, λ j is the jth eigenvalue of the Laplace matrix L, t is the thermal diffusion time parameter, is the normalization factor of the wavelet basis function used for orthogonalization, V j (x) is the value of the j-th column of the eigenvector matrix U at node x.

[0022] Furthermore, the multiscale wavelet coefficient matrix of the node generated based on the multiscale wavelet basis function is specifically:

[0023] For each node, its position in the graph is defined by the impulse vector, and the multi-scale wavelet coefficient matrix of the node is generated by using the intermediate matrix and combining the multi-scale wavelet basis function:

[0024]

[0025] Among them, L k (Λ) represents the Legendre polynomial at eigenvalue λ j The value at , Λ is a diagonal matrix, is the normalization factor of the orthogonalized wavelet basis function, e i is the pulse vector, U D is the intermediate matrix, U T is the transposed matrix of matrix U.

[0026] Furthermore, the real part calculation is specifically as follows:

[0027] real i =mean(cos(ψ i,k ·step))

[0028] Among them, real i is the imaginary part of node i, step is the step size, ψ i,k is the relationship between node i and the k-th multi-scale wavelet coefficient matrix.

[0029] Furthermore, the imaginary part calculation is specifically as follows:

[0030] imag i =mean(sin(ψ i,k ·step))

[0031] Among them, imag i is the imaginary part of node i, step is the step size, ψ i,k is the relationship between node i and the k-th multi-scale wavelet coefficient matrix.

[0032] Furthermore, the Hilbert transform is calculated as:

[0033]

[0034] Among them, PV represents the principal value integral, f(τ) is the value of the signal at time τ, and the Hilbert transform captures the instantaneous frequency and envelope of the signal by supplementing the imaginary part. After the Hilbert transform, the envelope information of each node signal is calculated.

[0035]

[0036] Among them, envelope i is the signal envelope of node i, is the Hilbert transform result of the real part of node i.

[0037] Further, the network feature embedding of the entire graph is expressed as;

[0038] G F =[real|imag|envelope]

[0039] G F It is a feature matrix that combines the real part, imaginary part and envelope information. real, imag and envelope are the sets of real part, imaginary part and envelope features of all nodes respectively. [] represents the matrix concatenation operation.

[0040] Furthermore, the classifier is an XGB classifier.

[0041] Compared with the prior art, the present invention has the following advantages:

[0042] High efficiency:

[0043] The node role hierarchical detection method based on the Legendre wavelet diffusion model proposed in the present invention can effectively process large-scale complex networks and avoid the performance bottleneck problem in traditional methods.

[0044] accuracy:

[0045] The present invention can better capture the diffusion characteristics of nodes in the network through a multi-level wavelet diffusion model, thereby improving the accuracy of node role identification and hierarchical structure detection.

[0046] robustness:

[0047] The present invention can adapt to complex networks of different scales and topological structures, has a high tolerance for noise or dynamic changes in the network, and has strong robustness.

[0048] Wide range of applications:

[0049] This method is not only suitable for the analysis of static networks such as social networks, communication networks, and transportation networks, but can also be used for node role identification and hierarchical structure detection in dynamic networks and time-varying networks, and has broad application prospects. BRIEF DESCRIPTION OF THE DRAWINGS

[0050] The accompanying drawings generally illustrate various embodiments by way of example and not limitation, and together with the description and claims, serve to illustrate the embodiments of the invention. Where appropriate, the same reference numerals are used throughout the drawings to refer to the same or similar parts. Such embodiments are illustrative and are not intended to be exhaustive or exclusive embodiments of the present apparatus or method.

[0051] Figure 1 A schematic diagram of the method flow of the present invention is shown;

[0052] Figure 2 The performance of the LWDiffusion of the present invention and the baseline method on different data sets is shown;

[0053] Figure 3 A schematic diagram of two-dimensional visualization results of different methods of the present invention on the USA-Airport dataset is shown. DETAILED DESCRIPTION

[0054] It should be noted that, in the absence of conflict, the embodiments and features in the embodiments of the present application can be combined with each other. The present application will be described in detail below with reference to the accompanying drawings and in combination with the embodiments.

[0055] The method for detecting the role level of a node in a complex network based on the Legendre wavelet diffusion model provided by the embodiment of the present invention comprises the following steps:

[0056] Spectral Transformation of Graphs

[0057] Step 1: In the analysis of complex networks, spectral graph transformation is an effective tool to capture the global and local structural information of the network.

[0058] For a given graph G = (V, E), its vertex set is V and edge set is E. The adjacency matrix A is used to represent the connectivity of the graph. The adjacency matrix A is an n×n matrix, where n is the number of vertices in the graph G. The element a of the matrix A is ij Indicates whether there is an edge between vertex i and vertex j:

[0059]

[0060] Step 2: In order to analyze the graph structure from a spectral perspective, first construct the Laplacian matrix L of the graph, which is defined as:

[0061] L=DA (1)

[0062] Where D is the degree matrix, which is a diagonal matrix, and D ii is the degree of vertex i. By performing eigenvalue decomposition on the Laplace matrix L, we can obtain:

[0063] L=UΛU T (2)

[0064] Among them, Λ is a diagonal matrix containing the eigenvalues ​​of the Laplace matrix L, and U is an orthogonal vector whose columns contain the corresponding eigenvectors of the matrix U. Through the eigenvector matrix U, the graph signal can be transformed from the vertex domain to the spectrum domain, thus providing a basis for the subsequent generation of multi-scale wavelet basis functions and heat diffusion mechanism.

[0065] Step 3: Heat Diffusion Mechanism

[0066] In order to capture the detailed characteristics of the network structure in a complex network, the present invention adopts the method of combining the heat diffusion mechanism with the Legendre multi-wavelet basis based on the idea of ​​capturing the network structure information from the wavelet coefficients. Figure 1 As shown in Figure 1, the heat diffusion mechanism captures the propagation mode from node to edge node by simulating the diffusion process of node signals in the network, and uses Legendre multi-wavelet basis to capture the structural details of nodes at different scales. Specifically, the heat kernel feature is calculated by eigenvalue decomposition, and its heat kernel diagonal matrix D is defined as:

[0067]

[0068] Among them, λ i is the eigenvalue of the Laplace matrix, and t is the thermal diffusion time coefficient. Multiply the eigenvector matrix U with the thermal kernel diagonal matrix D to obtain the intermediate matrix U D =UD. By combining Legendre polynomials, wavelet basis functions at different scales are generated to capture the multi-scale characteristics of the network.

[0069] Step 4: Multiscale wavelet basis

[0070] The k-degree Legendre polynomial is represented by L k (x) is described as:

[0071]

[0072] Here k = 0, 1, ..., p-1, p is the number of Legendre polynomials used, and the Legendre scaling basis function is represented by φ k (x)

[0073]

[0074] Step 5: Generation of multi-scale wavelet basis functions

[0075] After being converted into a multi-scale wavelet basis on the graph and associated with the heat diffusion mechanism, the basis function is constructed as

[0076]

[0077] φ k (x) represents the value of the wavelet basis function with scale k at node x, λ j is the eigenvalue of the Laplace matrix L. V j is the eigenvector, L k (λ j ) is the Legendre polynomial at eigenvalue λ j The value at . is the thermal nuclear coefficient, λ j is the j-th eigenvalue of the Laplace matrix L, and t is the thermal diffusion time parameter. is the normalization factor of the wavelet basis function used for orthogonalization. V j (x) is the value of the j-th column of the eigenvector matrix U at node x.

[0078] Step 6: For each node i, pass the pulse vector e i Define its position in the graph. Using the intermediate matrix U D , combined with the Legendre polynomial L k (x) Generate the multi-scale wavelet coefficient matrix ψ of node i through the wavelet basis function i :

[0079]

[0080] Among them, L k (Λ) represents the Legendre polynomial at eigenvalue λ j The value at , Λ is a diagonal matrix, is the normalization factor of the orthogonalized wavelet basis function. This method ensures that the embedding representation of each node can simultaneously reflect the global and local structural information of the network through the combination of heat diffusion and multi-scale wavelets.

[0081] Step 7: Determine the node embedding feature composition

[0082] The coefficient matrices of heat diffusion and multi-scale wavelet are obtained through the above calculations. Based on the principle that wavelet coefficients represent the characteristics of network structure, when generating the embedded representation of nodes, the multi-scale wavelet coefficient matrix ψ is firstly i,k Processing is performed to extract the real part, imaginary part and envelope information after Hilbert transformation, and a step size step is set for each part of the signal extraction. These three parts of information capture the amplitude, phase and overall energy distribution of the signal respectively, ensuring the comprehensiveness of node embedding.

[0083] The feature matrix of each node is generated by the following calculation steps:

[0084] Step 8: Real part calculation:

[0085] real i =mean(cos(ψ i,k ·step)) (8)

[0086] Among them, real captures the main amplitude component of the signal in node propagation and reflects the basic form of the network structure in subspace or time propagation.

[0087] Step 9: Imaginary part calculation:

[0088] imag i =mean(sin(ψ i,k ·step)) (9)

[0089] Among them, imag is related to the phase information of the signal, which helps to capture the changing pattern of the signal in space or time during node propagation.

[0090] Step 10: Envelope calculation: Hilbert transform is a mathematical transformation used for signal analysis, which can convert a real-valued signal into its analytical signal. i , whose Hilbert transform Defined as:

[0091]

[0092] Among them, PV represents the principal value integral. Hilbert transform makes the representation of the signal more complete by supplementing the imaginary part and can capture the instantaneous frequency and envelope of the signal. After performing Hilbert transform, we can calculate the envelope information of each node signal.

[0093]

[0094] Among them, envelope is the signal envelope calculated after Hilbert transform, which reflects the energy distribution of the signal changing with time.

[0095] Step 11: Network feature embedding representation

[0096] Finally, the embedding feature of the entire graph is represented as:

[0097] G F =[real|imag|envelope] (12)

[0098] G F It is a feature matrix that combines the real part, imaginary part and envelope information. real, imag and envelope are the collections of the real part, imaginary part and envelope features of all nodes respectively. [] represents the matrix concatenation operation. The embedded features can not only capture the global structure and local details of the network, but also provide strong support for subsequent node classification or community detection tasks. F The data is passed into the XGB classifier to achieve accurate identification of node role levels in complex networks, effectively improving the accuracy and practicality of network analysis.

[0099] Step 12: Use the above heat diffusion mechanism and Legendre polynomials to construct a multi-scale wavelet basis function to embed G to obtain G F The method of identifying node roles based on features is called LWDiffusion.

[0100] Dataset

[0101] This paper selects three main data sets: publicly available aviation network benchmark data sets (3), project cooperation networks (3) and researcher title network data sets (3) collected by open source data. These data sets are used to experimentally verify and evaluate the performance of algorithms in different types of networks. Table 1 summarizes the basic network information of these data sets. All data sets are derived from the real world, and in the experiment, the division of training sets and test sets is performed randomly.

[0102] Table 1. Unified decimal places for data sets

[0103]

[0104]

[0105] Baseline Method

[0106] In order to evaluate the performance of the role classification of the present invention in complex networks, common network embedding baseline methods were selected for comparative experiments. DeepWalk is a network embedding method based on random walks. By generating a series of random walk sequences on the graph and using the Skip-gram model for learning, a low-dimensional vector representation of nodes is achieved. LINE is an embedding method for large-scale graphs that supports first-order and second-order neighbor relationship modeling and can effectively capture local and global information in the network. Node2Vec introduces adjustable parameters p and q on the basis of DeepWalk, so that random walks can flexibly switch between breadth-first and depth-first, so as to better learn the structural information of nodes. SDNE realizes the embedded representation of the network through deep autoencoders, which can simultaneously retain the local and global structural features of the graph, especially in highly nonlinear graph structures. Struc2Vec is a model for role embedding based on structural similarity. Through a multi-level structured representation method, the structural information of the node is converted into an embedded representation, which is suitable for analyzing the structural characteristics of nodes at different scales. GraphWave is a role embedding method based on spectral wavelets, proposed by Claire D. et al. This method uses the Laplace spectrum of the graph to encode the structural information of the nodes into feature distributions through wavelet transform to capture role similarity. Role2Vec was proposed by NKAhmed et al. and focuses on role similarity modeling rather than traditional structural or neighbor features. It generates role feature vectors by extracting specific statistical features of nodes (such as degree, clustering coefficient, specific subgraph patterns, etc.), and uses the Skip-gram model to learn node embedding representations. This method can capture the similarity between nodes at the role level, such as identifying the different functions of community center nodes or edge nodes in social networks. The N2N method uses self-supervised learning to learn high-quality embeddings by aligning nodes with their neighborhoods. Although its original design focuses more on the general modeling of node representation, its neighborhood alignment mechanism can capture role information, such as distinguishing nodes with large differences in connection patterns in social networks, providing an effective embedding method for role classification tasks. NAGphormer was proposed by Chen et al. by discretizing the nodes and edges of the graph into tokens and combining the graph Transformer architecture for node classification. Although its main goal is to improve the global modeling ability in node classification tasks, its ability to capture structural characteristics and global relationships can also indirectly serve the role classification task. For example, in large-scale graph data, NAGphormer can capture the structural patterns of nodes at different scales, which is particularly important for distinguishing nodes with different role characteristics. The Polynormer proposed by Deng et al. enhances the graph structure modeling ability of Transformer by introducing polynomial kernel expression while maintaining linear time complexity.Its innovation lies in its ability to effectively model complex relationships between nodes, which is particularly suitable for role classification tasks in large-scale networks.

[0107] Experimental setup

[0108] In the present invention, the training of all algorithm models is strictly based on the methods in the original literature to ensure the comparability of the results and the fairness of the experiment. Specifically, for the DeepWalk method, the step size of the random walk is set to 10, and 80 random walks are performed for each node. In the LINE model, the second-order nearest neighbor relationship is selected to capture the deep structural features between nodes, and the dimension of the embedding vector is set to 128. For the Node2Vec method, the step size of the random walk is set to 10, and each node also performs 80 random walks. In addition, the path parameter (p) is set to 0.25, and the jump parameter (q) is set to 4. In the SDNE[] model, the hidden layer consists of two layers of neurons, which are set to 256 and 128 units respectively. For Struc2Vec, its settings are similar to DeepWalk and Node2Vec, and the step size is still selected to be 10, and each node performs 80 random walks to extract the structural feature information of the node. The experiments of GraphWave, Role2Vec, N2N and NAGphormer were configured according to the default parameters of their original codes to ensure the fairness of the experiments and the reproducibility of the results. For Polynormer, due to the consideration of training time, the values ​​of Local Epochs and Global Epochs were adjusted to 150, and the other parameters remained the same as the original code. The training set and the validation set were randomly divided in a ratio of 8:2, and the random seed was fixed to 42.

[0109] By analyzing the experimental results in Table 2, it can be concluded that the LWDiffusion algorithm has demonstrated excellent role recognition capabilities on different types of network datasets, such as Figure 2As shown, whether it is the title recognition task of aviation network, fund project leader and member network, or fund project member cooperation network. In particular, LWDiffusion performs well in networks with high density and high clustering coefficient, and can also achieve the best precision, recall and f1-score in sparse networks without triangle structure. This clearly shows that LWDiffusion can not only effectively handle complex network structures, but also maintain efficient node representation learning and classification performance in sparse networks. In addition, LWDiffusion also shows significant robustness and stability when dealing with large-scale complex networks. This advantage can be attributed to its ability to capture diffusion information between nodes and multi-scale structural features. By combining multi-scale analysis with the heat kernel diffusion mechanism, LWDiffusion can capture global and local network features at the same time, which enables it to adapt to networks with different structural characteristics and ensure efficient and accurate node role recognition in various complex situations.

[0110] The present invention (LWDiffusion) has achieved remarkable results and advantages in the field of node role identification in complex networks. The experimental results are mainly reflected in the following aspects:

[0111] ①Significant improvement in accuracy and stability:

[0112] On three different types of real data sets, namely, aviation network, fund project leader and member network, and title recognition network of fund project members, LWDiffusion's precision, recall, and F1-score indicators are significantly better than existing typical baseline methods (such as DeepWalk, Node2Vec, Struc2Vec, GraphWave, Polynormer, and NAGphormer, etc.). In particular, on some data sets (such as Data-AI-B1), the F1-score of the present invention can reach the ideal level of 1.00, showing excellent classification performance.

[0113] ② Strong robustness to adapt to different network structures:

[0114] The present invention not only performs well in complex networks with high density and high clustering coefficient, but also maintains excellent recognition performance in sparse networks without triangle structures. This proves that LWDiffusion has good adaptability and robustness to various types of network structures. As shown in Table 2, whether it is the hierarchical role identification from hub airports to transfer airports in the airport flight network, or the hierarchical division between project leaders and members or different titles in the fund cooperation network, the present invention can accurately identify the node role hierarchy in a variety of scenarios.

[0115] ③Effective integration of global and local features:

[0116] As shown in Table 2, LWDiffusion still shows extremely high classification accuracy in the leaders and members' networks of the NSFC project, even in extremely sparse networks without complex triangular structures, thanks to its ability to capture global information of the network. With the support of multi-scale feature extraction and hot nuclear diffusion mechanism, LWDiffusion can simultaneously capture the global structural information and local topological features of the network. In actual networks, this multi-scale embedding representation ensures that the algorithm can not only distinguish the differences in local neighborhood structures when identifying node roles, but also accurately locate the role hierarchy of nodes from the overall network macro-pattern, which is significantly better than traditional methods that only rely on local statistics or simple random walks.

[0117] Table 2. Experimental results of LWDiffusion and baseline methods on different datasets

[0118]

[0119]

[0120]

[0121] There are three parts here. The first part is the experimental results of three aviation networks under different methods. The second part is the network role recognition of fund project leaders and members in the fields of artificial intelligence and computer science. The third part is the member title recognition of the fund project member cooperation network in the fields of artificial intelligence and computer science. The results are given in precision, recall, and f1-score. For each data set, the best recognition situation under each data set is highlighted.

[0122] ④ Evidence from visualization and parameter sensitivity analysis:

[0123] The visualization results of the USA-Airport dataset using t-SNE show that Figure 3 As shown in the figure, the present invention can form clear role clustering boundaries in the embedding space and effectively distinguish adjacent role nodes. The parameter discussion also shows that there is a reasonable range for the heat kernel coefficient and the scale parameter. Only a small adjustment is needed to achieve robust performance in networks with different structures. This provides convenience for parameter tuning and model deployment in practical application scenarios.

[0124] In order to more clearly demonstrate the ability of LWDiffusion in adaptively mining the roles of network nodes and the advantages of multi-scale feature learning, this paper uses the t-SNE algorithm to visualize the features mined by LWDiffusion and its comparative methods in the USA-Airport dataset in two-dimensional space. As a common nonlinear data dimensionality reduction method, the t-SNE algorithm maps data to probability distribution in the form of affine transformation, which can effectively reduce the dimension of high-level feature data. Figure 3 As shown in the figure, DeepWalk, LINE, Node2Vec, etc. are mainly based on representation learning of random walks and network adjacency structures. Their clustering results are relatively loose and it is difficult to effectively distinguish different node roles. Struc2Vec and GraphWave capture the structural characteristics of node roles to a certain extent, but in multi-role and dense graph recognition scenarios, their role boundaries are not clear enough, and the distribution has a large dispersion and overlap. Recently, the clustering effect of NAGphormer and Polynormer based on autoencoders has been improved, and some roles have formed a clear separation, but some areas still overlap. In contrast, from the t-SNE visualization Figure 3 As can be seen, LWDiffusion shows obvious clustering characteristics in the node distribution of different roles. Each role node is closely distributed, which can effectively distinguish the adjacent different role nodes. This good separation ability shows that LWDiffusion can capture multi-scale features and hidden node attributes in the network, showing significant potential and robustness in identifying node role hierarchies in complex networks.

[0125] Controllability of time and space complexity:

[0126] Although the present invention theoretically requires Laplace matrix eigendecomposition and wavelet coefficient calculation, the overall calculation cost is within an acceptable range through reasonable parameter selection and parallel acceleration strategy, which makes the present invention still have practical application potential in large-scale network processing.

[0127] The above description is only a preferred specific implementation manner of the present invention, but the protection scope of the present invention is not limited thereto. Any technician familiar with the technical field can make equivalent replacements or changes according to the technical scheme and inventive concept of the present invention within the technical scope disclosed by the present invention, which should be covered by the protection scope of the present invention.

Claims

1. A complex network node role hierarchy detection method based on Legendre wavelet diffusion model, characterized in that: The method comprises the following steps: Obtain the complex network to be analyzed and capture the global and local structural information of the graph through spectrum graph transformation; Construct the Laplacian matrix of the graph, and obtain the eigenvalues ​​and eigenvectors of the Laplacian matrix through eigenvalue decomposition, and convert the graph signal from the vertex domain to the spectrum domain; The heat diffusion mechanism is combined with Legendre wavelet basis function to generate multi-scale wavelet basis function; Generate a multi-scale wavelet coefficient matrix of nodes based on the multi-scale wavelet basis function, extract the embedded features of each node, and calculate the real part, imaginary part and envelope information after Hilbert transformation to generate the feature matrix of each node; The network feature embedding representation of the entire graph is obtained through the feature matrix of each node; The network features of the entire graph are embedded into a representation and passed into a classifier to identify the node role level.

2. The complex network node role hierarchy detection method based on Legendre wavelet diffusion model according to claim 1 is characterized in that: The Laplacian matrix of the graph is constructed, and the eigenvalues ​​and eigenvectors of the Laplacian matrix are obtained by eigenvalue decomposition to convert the graph signal from the vertex domain to the spectrum domain as follows: First, construct the Laplacian matrix L of the graph, defined as: L=DA Where D is the degree matrix and A is the adjacency matrix, which is obtained by eigenvalue decomposition of the Laplace matrix L: L=UΛU T where Λ is a diagonal matrix containing the eigenvalues ​​of the Laplace matrix L, and U is a matrix containing the corresponding eigenvectors. The columns of U are orthogonal vectors, and U T is the transposed matrix of matrix U. Through the eigenvector matrix U, the graph signal is transformed from the vertex domain to the spectrum domain.

3. The complex network node role hierarchy detection method based on Legendre wavelet diffusion model according to claim 1 is characterized in that: The multi-scale wavelet basis function is generated by calculating the heat kernel feature and combining the Legendre polynomial, specifically: Among them, φ k (x) represents the value of the wavelet basis function with scale k at node x, λ j is the eigenvalue of the Laplace matrix L, V j is the eigenvector, L k (λ j ) is the Legendre polynomial at eigenvalue λ j The value at is the thermal nuclear coefficient, λ j is the jth eigenvalue of the Laplace matrix L, t is the thermal diffusion time parameter, is the normalization factor of the wavelet basis function used for orthogonalization, V j (x) is the value of the k-th column of the eigenvector matrix U at node x.

4. The complex network node role hierarchy detection method based on Legendre wavelet diffusion model according to claim 1 is characterized in that: The multiscale wavelet coefficient matrix based on the multiscale wavelet basis function generation node is specifically: For each node, its position in the graph is defined by the impulse vector, and the multi-scale wavelet coefficient matrix of the node is generated by using the intermediate matrix and combining the multi-scale wavelet basis function: Among them, L k (Λ) represents the Legendre polynomial at eigenvalue λ j The value at , Λ is a diagonal matrix, is the normalization factor of the orthogonalized wavelet basis function, e i is the pulse vector, U D is the intermediate matrix, U T is the transposed matrix of matrix U.

5. The complex network node role hierarchy detection method based on Legendre wavelet diffusion model according to claim 1 is characterized in that: The real part calculation is specifically: real i =mean(cos(ψ i,k ·step)) Among them, real i is the imaginary part of node i, step is the step size, ψ i,k is the relationship between node i and the k-th multi-scale wavelet coefficient matrix.

6. The complex network node role hierarchy detection method based on Legendre wavelet diffusion model according to claim 1 is characterized in that: The imaginary part calculation is specifically: image i =mean(that(ψ i,k ·step)) Among them, imag i is the imaginary part of node i, step is the step size, ψ i,k is the relationship between node i and the k-th multi-scale wavelet coefficient matrix.

7. The complex network node role hierarchy detection method based on Legendre wavelet diffusion model according to claim 1 is characterized in that: The Hilbert transform is calculated as: Among them, PV represents the principal value integral, f(τ) is the value of the signal at time τ, and the Hilbert transform captures the instantaneous frequency and envelope of the signal by supplementing the imaginary part. After the Hilbert transform, the envelope information of each node signal is calculated. Among them, envelope i is the signal envelope of node i, is the Hilbert transform result of the real part of node i.

8. The complex network node role hierarchy detection method based on Legendre wavelet diffusion model according to claim 1 is characterized in that: The network feature embedding of the entire graph is expressed as; G F =[real∣imag∣envelope] G F It is a feature matrix that combines the real part, imaginary part and envelope information. real, imag and envelope are the sets of real part, imaginary part and envelope features of all nodes respectively. [] represents the matrix concatenation operation.

9. The complex network node role hierarchy detection method based on Legendre wavelet diffusion model according to claim 1 is characterized in that: The classifier is an XGB classifier.

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