A community discovery method based on nonlinear non-negative matrix factorization

Abstract

The present invention relates to a community discovery method based on nonlinear non-negative matrix decomposition, comprising the following steps: S1: formalizing a social network to generate an undirected graph; S2: based on the undirected graph, introducing a graph convolutional network GCN and an NMF model to construct a unified loss function; S3: optimizing the unified loss function to obtain a local optimal solution; S4: extracting community discovery results based on the local optimal solution. The graph convolutional network GCN and the NMF model are introduced to construct a unified loss function, and joint training is performed to give NMF the learning ability of nonlinear features, thereby improving the performance of the community discovery method based on NMF.

Description

Technical Field

[0001] The present invention relates to the field of convolutional networks, and more specifically, to a community discovery method based on nonlinear non-negative matrix decomposition. Background Art

[0002] Social networks commonly used by Internet users (such as WeChat, QQ, and Weibo) have community structures. Users in the same community are more closely connected to each other, while users in different communities are sparsely connected to each other. Community discovery on social networks can not only understand the characteristics of network structure, but also discover similar user groups, which has important application value for accurately recommending user products and services.

[0003] At present, different types of community discovery methods have been proposed, among which community discovery methods based on non-negative matrix factorization (NMF) are becoming more and more widely used, because they have the characteristics of easy interpretation of results, strong versatility, and the ability to integrate multi-source information to discover high-quality communities. However, NMF is essentially a linear model with weak expression ability. It is difficult to learn the multi-level complex features implicit in online social networks, and has certain limitations in improving community discovery performance.

[0004] Chinese invention patent CN111400652A discloses "A non-negative matrix community discovery method and a movie community discovery method", with a publication date of July 10, 2020: including the following steps S11 data collection and calculation to form similarity matrices X and L; S12 non-negative matrix decomposition of X, X≈UV; S13 constructing an objective function O containing L; and S14 based on the objective function, obtaining the iterative formula of non-negative matrix decomposition, iterating, and completing community division. This invention applies dual-attribute non-negative matrix decomposition to community discovery, which improves the accuracy of the community decomposition module, but does not have the learning ability of linear features and cannot improve the community discovery performance. Summary of the invention

[0005] In order to solve the technical defects of the existing NMF-based community discovery method, which is a linear model with weak expression ability, the present invention provides a community discovery method based on nonlinear non-negative matrix decomposition.

[0006] In order to achieve the above invention purpose, the technical solution adopted is:

[0007] A community detection method based on nonlinear non-negative matrix decomposition includes the following steps:

[0008] S1: Formalize the social network and generate an undirected graph;

[0009] S2: Based on the undirected graph, we introduce the graph convolutional network GCN and NMF model to build a unified loss function;

[0010] S3: Optimize the unified loss function to obtain the local final solution;

[0011] S4: Extract community discovery results based on the local final solution.

[0012] In the above scheme, the social network is formalized to generate an undirected graph, the graph convolutional network GCN and the NMF model are introduced to construct a unified loss function, and joint training is performed to give NMF the ability to learn nonlinear features, thereby improving the performance of the NMF-based community discovery method.

[0013] Preferably, in step S1, the undirected graph is G = (V, E), where V = {v1, ..., v n} is a set of n network user nodes, E = {e ij |v i ∈V∧v j ∈V} is represented as the set of connecting edges between user nodes.

[0014] Preferably, an n×n symmetric adjacency matrix A=[A ij ] n×n Represents the connection edge information between user nodes. If e ij ∈E, then A ij =A ji =1, otherwise A ij =A ji =0.

[0015] Preferably, an n×n unit matrix I is used as the node feature input matrix X, and the matrix X is normalized using the ReLU nonlinear activation function and the softmax function to output the node feature matrix Z=[Z ij ] n×d , d is the feature dimension, and the output of the graph convolutional network GCN is expressed as follows:

[0016] Z = softmax(P ReLU(P XW 0 )W 1 ) (1)

[0017] in is the degree matrix of matrix A+I, W 0 and W 1 are the parameter matrices learned at each layer; after obtaining Z, the similarity matrix between node pairs is obtained through the sigmoid function

[0018]

[0019] It can be regarded as the reconstructed structural feature matrix of G. The Euclidean distance between A and GCN is defined as the loss function L GCN :

[0020]

[0021] also is a non-negative matrix and a symmetric matrix. Based on the symmetric NMF model, the community affiliation indicator matrix H = [H ij ] n×k , k is the number of pre-specified communities, and the corresponding loss function is:

[0022]

[0023] Among them, H has a non-negative value constraint. Combining the loss functions of GCN and NMF, a unified loss function L of the two can be constructed as follows:

[0024]

[0025] Where α is a trade-off parameter.

[0026] Preferably, the graph convolutional network GCN adopts a three-layer structure, including an input layer, a hidden layer and an output layer; the input layer uses an n×n unit matrix I as the node feature input matrix X, the input layer to the hidden layer uses the ReLU nonlinear activation function, and the hidden layer to the output layer uses the softmax function to normalize the output node feature matrix Z=[Z ij ] n×d , d is the feature dimension

[0027] Preferably, the purpose of the graph convolutional network GCN is to extract the spatial features of the topological graph. There are two main types, namely, graph convolutional networks based on the spatial domain or vertex domain and graph convolutional networks based on the frequency domain or spectral domain.

[0028] Preferably, in step S3, a local final solution of the community belonging relationship indication matrix H is obtained by optimizing and minimizing the unified loss function L.

[0029] Preferably, the loss function of GCN is first fixed, and under the constraint that the community affiliation relationship indicator matrix H has a non-negative value, the iterative update rule of the community affiliation relationship indicator matrix H is derived by combining the Lagrange multiplier method and the Karush-Kuhn-Tucker optimization condition:

[0030]

[0031] Then, substitute equation (6) into equation (5) to obtain the loss function for training GCN, and use the gradient descent optimization method to train GCN. Finally, by setting the number of iterations t, repeatedly execute equation (6) and train GCN to obtain the local final solution of the community affiliation indicator matrix H.

[0032] Preferably, in step S4, the k communities to be discovered are C = {c1, ..., c k}, obtain the community division result according to the community affiliation indicator matrix H.

[0033] Preferably, the specific process of step S4 is: for any user node v i , where h il represents the belonging strength of node i to community l, It means that the community number of node i with the maximum belonging strength is assigned to q, that is, c q =c q ∪{v i}; According to the above rules, the final community division result is obtained by scanning the row vectors of the community affiliation indicator matrix H in sequence.

[0034] Compared with the prior art, the present invention has the following beneficial effects:

[0035] The present invention provides a community discovery method based on nonlinear non-negative matrix decomposition, formalizes social networks, generates undirected graphs, introduces graph convolutional network GCN and NMF models to construct a unified loss function, and performs joint training to give NMF the learning ability of nonlinear features, thereby improving the performance of the NMF-based community discovery method and having strong expression ability. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] Figure 1 It is a schematic diagram of the method flow of the present invention;

[0037] Figure 2 It is a schematic diagram of the structural framework of the present invention;

[0038] Figure 3 A schematic diagram of a community network of the present invention;

[0039] Figure 4 This is a community division result diagram of the present invention. DETAILED DESCRIPTION

[0040] The drawings are for illustrative purposes only and should not be construed as limiting the present patent;

[0041] The present invention is further described below in conjunction with the accompanying drawings and embodiments.

[0042] Example 1

[0043] like Figure 1 and Figure 2 As shown, a community detection method based on nonlinear non-negative matrix decomposition includes the following steps:

[0044] S1: Formalize the social network and generate an undirected graph;

[0045] S2: Based on the undirected graph, we introduce the graph convolutional network GCN and NMF model to build a unified loss function;

[0046] S3: Optimize the unified loss function to obtain the local final solution;

[0047] S4: Extract community discovery results based on the local final solution.

[0048] In the above scheme, the graph convolutional network GCN and the NMF model are introduced to construct a unified loss function, and joint training is performed to give NMF the ability to learn nonlinear features, thereby improving the performance of the NMF-based community discovery method and strengthening its expression ability.

[0049] Preferably, in step S1, the undirected graph is G = (V, E), where V = {v1, ..., v n} is a set of n network user nodes, E = {e ij |v i ∈V∧v j ∈V} is represented as the set of connecting edges between user nodes.

[0050] Preferably, an n×n symmetric adjacency matrix A=[A ij ] n×n Represents the connection edge information between user nodes. If e ij ∈E, then A ij =A ji =1, otherwise A ij =A ji =0.

[0051] Preferably, an n×n unit matrix I is used as the node feature input matrix X, and the matrix X is normalized using the ReLU nonlinear activation function and the softmax function to output the node feature matrix Z=[Z ij ] n×d , d is the feature dimension, and the output of the graph convolutional network GCN is expressed as follows:

[0052] Z = softmax(P ReLU(P XW 0 )W 1 ) (1)

[0053] in is the degree matrix of matrix A+I, W0 and W 1 are the parameter matrices learned at each layer; after obtaining Z, the similarity matrix between node pairs is obtained through the sigmoid function

[0054]

[0055] It can be regarded as the reconstructed structural feature matrix of G. The Euclidean distance between A and GCN is defined as the loss function L GCN :

[0056]

[0057] also is a non-negative matrix and a symmetric matrix. Based on the symmetric NMF model, the community affiliation indicator matrix H = [H ij ] n×k , k is the number of pre-specified communities, and the corresponding loss function is:

[0058]

[0059] Among them, H has a non-negative value constraint. Combining the loss functions of GCN and NMF, a unified loss function L of the two can be constructed as follows:

[0060]

[0061] Where α is a trade-off parameter.

[0062] Preferably, the graph convolutional network GCN adopts a three-layer structure, including an input layer, a hidden layer and an output layer; the input layer uses an n×n unit matrix I as the node feature input matrix X, the input layer to the hidden layer uses the ReLU nonlinear activation function, and the hidden layer to the output layer uses the softmax function to normalize the output node feature matrix Z=[Z ij ] n×d , d is the feature dimension

[0063] Preferably, the purpose of the graph convolutional network GCN is to extract the spatial features of the topological graph. There are two main types, namely, graph convolutional networks based on the spatial domain or vertex domain and graph convolutional networks based on the frequency domain or spectral domain.

[0064] Preferably, in step S3, a local final solution of the community belonging relationship indication matrix H is obtained by optimizing and minimizing the unified loss function L.

[0065] Preferably, the loss function of GCN is first fixed, and under the constraint that the community affiliation relationship indicator matrix H has a non-negative value, the iterative update rule of the community affiliation relationship indicator matrix H is derived by combining the Lagrange multiplier method and the Karush-Kuhn-Tucker optimization condition:

[0066]

[0067] Then, substitute equation (6) into equation (5) to obtain the loss function for training GCN, and use the gradient descent optimization method to train GCN. Finally, by setting the number of iterations t, repeatedly execute equation (6) and train GCN to obtain the local final solution of the community affiliation indicator matrix H.

[0068] Preferably, in step S4, the k communities to be discovered are C = {c1, ..., c k}, obtain the community division result according to the community affiliation indicator matrix H.

[0069] Preferably, the specific process of step S4 is: for any user node v i , where h il represents the belonging strength of node i to community l, It means that the community number of node i with the maximum belonging strength is assigned to q, that is, c q =c q ∪{v i}; According to the above rules, the final community division result is obtained by scanning the row vectors of the community affiliation indicator matrix H in sequence.

[0070] Example 2

[0071] Figure 2 It is the overall model framework of the method, which includes an input layer, a graph convolution layer and an output layer. The input layer uses the adjacency matrix and the identity matrix of the graph as input, the graph convolution layer uses a 2-layer graph convolution network, and the activation function is ReLU. The output layer is responsible for outputting the graph node representation Z normalized by the Softmax function and used to reconstruct the adjacency matrix of the graph. The entire model framework is trained by constructing a graph reconstruction and NMF joint loss function.

[0072] Example 3

[0073] Combine the following Figure 3 A social network example is shown to illustrate a specific implementation of the method of the present invention:

[0074] Step 1: Formalize the social network. For the social network example, we formalize it as G = (V, E), where V = (v0, v1, v2, v3, v4, v5), E = {e 01,e 02 ,e 12, e 23, e 34 ,e 35 ,e 45}, n = 6. The corresponding adjacency matrix A and identity matrix I are:

[0075]

[0076]

[0077] Step 2: Construct a unified loss function for GCN and NMF: Here α is set to 1 and the number of communities k is set to 2.

[0078] Step 3: Unify the optimization solution of the loss function. Solve it through the following iterative update rule:

[0079]

[0080] Step 4: Obtain community discovery results. Set the number of iterations t = 10, and after the iterations converge, the result of H is:

[0081]

[0082] According to H, we can directly determine the members of the two communities c0 and c1 to be discovered. For example, for user node v0, its corresponding community belonging strength distribution vector in H is [0.63, 0.01], 0.63>0.01, so c0=c0∪{v0}. Similarly, the community belonging relationship of the remaining nodes can be determined. The final community division result is: c0={v0,v1,v2}, c1={v3,v4,v5}. The result is as follows: Figure 4 shown.

[0083] Obviously, the above embodiments of the present invention are merely examples for clearly illustrating the present invention, and are not intended to limit the embodiments of the present invention. For those skilled in the art, other different forms of changes or modifications can be made based on the above description. It is not necessary and impossible to list all the embodiments here. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the protection scope of the claims of the present invention.

Claims

1. A community discovery method based on nonlinear non-negative matrix factorization, characterized in that: The following steps are involved: S1: Formalize the social network and generate an undirected graph; S2: Based on the undirected graph, the graph convolutional network GCN and NMF model are introduced to construct a unified loss function; S3: Optimize the unified loss function to obtain the local optimal solution; In step S3, the local optimal solution of the community belonging relationship indication matrix H is obtained by optimizing and minimizing the unified loss function L; S4: Extract community discovery results based on local optimal solutions; In step S1, the undirected graph is G = (V, E), where V = {v1, ..., v n } is a collection of n network user nodes , E={e ij |v i ∈VΛv j ∈V} represents the set of connecting edges between user nodes; Using an n×n symmetric adjacency matrix A = [A ij ] n×n Represents the connection edge information between user nodes. If e ij ∈E, then A ij =A ji =1, otherwise A ij =A ji =0; An n×n unit matrix I is used as the node feature input matrix X, and the matrix X is normalized using the ReLU nonlinear activation function and the softmax function to output the node feature matrix Z = [Z ij ] n×d , d is the feature dimension, and the output of the graph convolutional network GCN is expressed as follows: Z=softmax(PRe LU(PXW 0 )W 1 ) (1) in is the degree matrix of matrix A+I, W 0 and W 1 These are the parameter matrices learned at each layer; After obtaining Z, the similarity matrix between node pairs is obtained through the sigmoid function It can be regarded as the reconstructed structural feature matrix of G. The Euclidean distance between A and GCN is defined as the loss function L GCN : also is a non-negative matrix and a symmetric matrix. Based on the symmetric NMF model, the community affiliation indicator matrix H = [H ij ] n×k , k is the number of pre-specified communities, and the corresponding loss function is: Among them, H has a non-negative value constraint. Combining the loss functions of GCN and NMF, a unified loss function L of the two can be constructed as follows: Where α is the trade-off parameter; The graph convolutional network GCN adopts a three-layer structure, including an input layer, a hidden layer, and an output layer. The input layer uses an n×n unit matrix I as the node feature input matrix X, the input layer to the hidden layer uses the ReLU nonlinear activation function, and the hidden layer to the output layer uses the softmax function to normalize the output node feature matrix Z = [Z ij ] n×d , d is the feature dimension; The purpose of graph convolutional network GCN is to extract the spatial features of topological graphs. There are two main types: graph convolutional network based on spatial domain or vertex domain and graph convolutional network based on frequency domain or spectral domain. First, the loss function of GCN is fixed. Under the constraint that the community affiliation indicator matrix H has a non-negative value, the iterative update rule of the community affiliation indicator matrix H is derived by combining the Lagrange multiplier method and the Karush-Kuhn-Tucker optimization condition: Then, substitute formula (6) into formula (5) to obtain the loss function for training GCN, and use the gradient descent optimization method to train GCN. Finally, by setting the number of iterations t, repeatedly execute formula (6) and train GCN to obtain the local optimal solution of the community affiliation indicator matrix H.

2. A community discovery method based on nonlinear non-negative matrix decomposition according to claim 1, characterized in that: In step S4, the k communities to be discovered are C = {c1,...,c k }, obtain the community division result according to the community affiliation indicator matrix H.

3. A community discovery method based on nonlinear non-negative matrix decomposition according to claim 2, characterized in that: The specific process of step S4 is: for any user node v i , where h il represents the belonging strength of node i to community l, It means that the community number of node i with the maximum belonging strength is assigned to q, that is, c q =c q U{v i }; According to the above rules, the final community division result is obtained by scanning the row vectors of the community affiliation indicator matrix H in sequence.

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