A Membership Ranking Overlapping Group Identification Method Integrating Size Constraints and Structure Regularization

Abstract

This invention relates to a membership degree ranking method for overlapping group identification that integrates size constraints and structural regularization, belonging to the field of complex network analysis technology. It solves the technical problems of group size imbalance, loss of topological information, and insufficient accuracy in overlapping node identification in existing technologies. The technical solution is as follows: First, a size constraint mechanism is introduced to force a balanced group distribution; second, a graph structure regularization term is combined to maintain the original network connection characteristics, and a three-nonnegative matrix factorization model is constructed to simultaneously solve for node-group membership degrees; third, dynamic cross-community ranking is implemented based on node-group membership degree scores, and boundary nodes are adaptively identified and overlap strength is quantified based on the ranking position differences to identify the overlapping group structure. This method has beneficial effects, performing excellently in networks with different group sizes and different degrees of overlap, significantly improving the robustness and accuracy of overlapping group detection.

Description

Technical Field

[0001] This invention belongs to the field of complex network analysis technology, specifically relating to a membership degree ranking method for overlapping group identification that integrates scale constraints and structural regularization, which is particularly suitable for the discovery of overlapping groups in complex networks such as social networks and bioinformatics networks. Background Technology

[0002] Mining group structures in complex networks is a crucial task for revealing the inherent organizational forms of networks. Current research has attempted to address overlapping clustering problems using nonnegative matrix factorization, generative models, or label propagation methods. While these methods improve the expressive power of clustering to some extent, they still have the following limitations: traditional methods neglect group size balance, leading to the difficulty in identifying or even annihilating small groups; matrix factorization loses network topology; and static thresholding methods cannot accurately identify overlapping nodes. Generative nonnegative matrix factorization has excellent data characterization capabilities and can easily introduce group size constraints and structural similarity regularization, which greatly mitigates the problems of difficulty in identifying small groups and distortion of inherent structure identification in real-world scenarios. Furthermore, dynamically sorting node-group membership to identify overlapping structures can refine the detection granularity of overlapping groups, further improving the accuracy of overlapping group identification.

[0003] Based on a review of existing technologies, Wang et al. published an article titled "Overlapping Community Detection via Bayesian Nonnegative Matrix Factorization" in the international journal *IEEE Transactions on Knowledge and Data Engineering* in 2021. This proposed a nonnegative matrix factorization framework based on Bayesian inference, automatically determining the number of clusters through probabilistic modeling. While this method avoids the limitation of manually pre-setting the number of clusters, its Markov chain Monte Carlo (MCMC) sampling process significantly increases computational complexity, leading to a severe efficiency bottleneck when dealing with networks exceeding 104 nodes. Furthermore, the suppressive effect of probabilistic prior distribution on small groups (<5 nodes) limits the practicality of this method in fine-grained group detection scenarios. Chen et al. published an article titled "Deep Autoencoder-based Overlapping Community Detection" at the international conference *The Web Conference* in 2022. This method uses a graph autoencoder to learn low-dimensional node embeddings and detects overlapping groups through a clustering algorithm. Although deep learning models automatically capture high-order structural features of networks, their performance is highly dependent on the completeness of training data. The black-box nature of neural networks also makes it impossible to trace the decision-making basis for node group membership, weakening the credibility of the results. Compared to existing technologies, this patented method for identifying overlapping groups by merging size constraints and structural regularization terms not only fully considers the suppression limitations of small-scale groups but also accurately traces the decision-making basis for membership. It has broad application prospects and technological competitiveness in terms of accuracy and structural adaptability in overlapping group identification. Summary of the Invention

[0004] This invention provides a membership-based overlapping group identification method that integrates size constraints and structural regularization, belonging to the field of complex network analysis. Addressing the three major shortcomings of existing technologies—group size imbalance, loss of topological information, and insufficient accuracy in identifying overlapping nodes—this invention proposes an innovative solution: a size constraint mechanism that forces a balanced group distribution through row and sum constraints to prevent small group annihilation; a structure preservation technique that introduces a graph Laplacian regularization term to maintain the continuity of local network topology; and a dynamic ranking engine that performs cross-group ranking based on the node-group membership matrix, adaptively identifying boundary nodes and quantifying overlap strength according to ranking position differences. This method overcomes the limitations of traditional static thresholding methods and significantly improves the detection accuracy of complex overlapping regions in networks.

[0005] The inventive concept of this invention is as follows: First, a scale-constrained three-nonnegative matrix factorization model is constructed, and the group size distribution is controlled by forcing the equalization of the sum of node group membership degrees. Second, a graph structure regularization term is introduced to inject local topological features of the network to maintain the continuity of adjacent nodes in the potential space. Finally, based on the node-group membership degree matrix generated by the decomposition, cross-group dynamic sorting is implemented, and boundary regions are adaptively identified according to the differences in the sorting positions of nodes in each group, replacing the traditional static threshold segmentation method.

[0006] To achieve the above objectives, the present invention is implemented through the following technical solution: a membership degree ranking method for identifying overlapping groups that integrates scale constraints and structural regularization, comprising the following steps:

[0007] Step S1: Model the complex network as a graph structure , where the set of nodes Link set represents the set of links in a network. The graph structure is formalized into an adjacency matrix. ;

[0008] Step S2: Design the node-group membership matrix V, community size matrix Γ, and inter-group relationship matrix Ω to characterize group structure, group size, and inter-group correlation, respectively, to fit the adjacency matrix M and construct the size constraint model O. size (V, Γ, Ω);

[0009] Step S3: Introduce graph regularization terms Characterizing the similarity of neighbor structures in a network is used to adjust node-group membership to maintain the local structure of the topology, thereby constructing a fusion model O that combines size constraints and structural regularization. size (V, Γ, Ω);

[0010] Step S4: Introduce a sorting strategy to obtain edge nodes, find nodes with multiple affiliations in the sorting results and re-divide them, and merge overlapping affiliations with non-overlapping affiliations to form the final overlapping clustering structure.

[0011] Further, in step S1, the graph structure is formalized as follows: the graph structure M is represented by an adjacency matrix. ,in Represents node c i and c j There are links between them. Represents node c i and c j There are no links between them.

[0012] Further, step S2 specifically includes the following steps:

[0013] Step S21: Under the generative framework, the scale constraint model is formed by a matrix. sum matrix Construct, where v iφ Describes the selection of node c in the φ-th group i The probability of generating an edge, ω φψ The correlation between the φ-th group and the ψ-th group is described, then ω φψ v iφ v jψ Describes the selection of node c in the φ-th group respectively i With node c in the ψth group j The probability of generating an edge simultaneously, for the entire graph M, node c i With node c j The probability of generating an edge is:

[0014] (1)

[0015] Given equation (1), using a matrix Fit the adjacency matrix M of the observed data graph M, i.e. minimize With m ij The Euclidean distance between them is achieved using three nonnegative matrix decompositions, and the objective function of the basic model is constructed as follows:

[0016] (2)

[0017] Step S22: Describe the size of the community and introduce a diagonal matrix. Describe, where γ ii Representing group size weights, correcting the group association matrix Ω→ΓΩΓ T After correction, the matrix Ω only represents the group associations of irrelevant group sizes, so the objective function (2) of the basic model can be transformed into the objective function of the size-constrained model:

[0018] (3)

[0019] Where matrix V represents node-group membership, matrix Γ represents group size, and matrix Ω represents pure group association.

[0020] Furthermore, step S3 specifically includes the following steps:

[0021] Step S31: Introduce graph regularization terms By calculating the similarity between nodes, the local structure of the network topology is established, and the design is performed. Capture local topological features of the network, where s ij Represents node c i With node c j The topological similarity between them is calculated using the following formula:

[0022] (4)

[0023] in, , They are nodes c respectively i With c j The graphs have the same number of neighboring nodes and different number of neighboring nodes; based on equation (4), a similarity matrix S is designed to construct the Laplace matrix L of the graph. g It is used to measure the connectivity between nodes in graph M, and the formula is L. g =FS, diagonal matrix A matrix representing the degree of all nodes in graph M, with its diagonal elements... Represents node c i The sum of structural similarities with all nodes in graph M is used to constrain node c. i In the latent space, v is represented as... i The geometric continuity, and the smoothness of the low-dimensional representation, can be expressed as:

[0024] (5)

[0025] Where Tr(.) represents the rank operation of the matrix;

[0026] Step S32: The graph regularization term constructed by equation (5) By incorporating a scale constraint model to form a model that integrates scale constraints and structure regularization, the final objective function is as follows:

[0027] (6)

[0028] Where μ represents the proportion of graph regularization term fusion, and matrix V represents the node-group membership degree.

[0029] Furthermore, step S4 specifically includes the following:

[0030] Step S41: Obtain the node-group membership matrix V according to the final objective function (6), and sort v in descending order. iφ Get node c i For each node c i Its tendency toward group φ and its tendency toward group ψ are respectively v iφ v iψ For every two different groups, the nodes and their corresponding group preference values ​​are calculated by maximizing the preference of each node for the two groups, max{v iφ ,v iψ The sorting process is performed in descending order to obtain the sorted result, and the formula is as follows:

[0031] (7)

[0032] The sorting function sort(.) described by equation (7) is used to sort the tendency values ​​of all nodes belonging to groups φ and ψ. An adaptive threshold τ is set. If the difference in tendency of node τ to different groups φ and ψ is within the threshold τ range, then it is an overlapping node; otherwise, it is a non-overlapping node. The formula for determining overlapping nodes is as follows:

[0033] (8)

[0034] Step S42: Merge nodes that belong to only a single group with nodes that have multiple affiliations to identify the global overlapping group structure.

[0035] Compared with the prior art, the present invention has the following beneficial effects:

[0036] This invention effectively addresses the core issues of group size imbalance and the annihilation of small groups in existing technologies by introducing a size constraint mechanism. This mechanism uses a diagonal matrix Γ to characterize group size weights, corrects the group association matrix through feature augmentation, and combines a three-nonnegative matrix factorization model to force a balanced group distribution, avoiding the shortcomings of traditional methods that neglect size constraints and thus fail to identify small groups. Simultaneously, this constraint mechanism is deeply integrated with a generative framework, balancing the reasonableness of group size with the accuracy of link generation probability during the fitting of the network adjacency matrix. This ensures stable group detection performance across networks of different sizes, improving the applicability of group identification in complex networks.

[0037] This invention innovatively incorporates a graph structure regularization term, effectively compensating for the lack of topological information loss in existing matrix factorization methods. By designing a similarity matrix S to capture the topological similarity between nodes, a graph Laplacian matrix is ​​constructed to constrain the geometric continuity of the node latent space representation, alleviating the problem of model parameter distortion and preserving the original network connectivity features with high fidelity. This regularization term works synergistically with the scale constraint model to balance group distribution and maintain the local topological continuity of the network, significantly enhancing the modeling ability for sparse network regions, making the group identification results more consistent with the actual network structure, and improving the robustness and reliability of the method.

[0038] This invention employs a dynamic ranking strategy based on cross-group membership, overcoming the limitations of traditional static threshold methods in identifying overlapping nodes. Based on the node-group membership matrix, it dynamically ranks nodes according to their tendency values ​​in different groups, sets adaptive thresholds to accurately identify boundary nodes and quantify overlap intensity, effectively solving the problem of determining the affiliation of fuzzy nodes. Compared with existing methods such as Bayesian nonnegative matrix factorization and graph autoencoders, this method requires no complex sampling or large amounts of training data, resulting in higher computational efficiency. Furthermore, it allows for the traceability of node membership decision criteria and can accurately capture overlapping structures in networks with varying degrees of overlap, significantly improving the accuracy and practicality of overlapping group identification. Attached Figure Description

[0039] The accompanying drawings are provided to further illustrate the invention and form part of the specification. They are used together with the embodiments of the invention to explain the invention and do not constitute a limitation thereof.

[0040] Figure 1 This is a flowchart of a method according to an embodiment of the present invention.

[0041] Figure 2 The diagram illustrates the evaluation of the overlapping group recognition performance of the LGR-NMF embodiment of the present invention in networks with different overlapping group structures. Specifically, (a) the diagram compares the overlapping group structure recognition accuracy of the LGR-NMF embodiment of the present invention with that of the CFinder method in networks with different network sizes, different group sizes, and overlapping group structures; (b) the diagram compares the overlapping group structure recognition accuracy of the LGR-NMF embodiment of the present invention with that of the CFinder method in networks with clear group structures and different degrees of overlapping group structures; and (c) the diagram compares the overlapping group structure recognition accuracy of the LGR-NMF embodiment of the present invention with that of the CFinder method in networks with unclear group structures and different degrees of overlapping group structures. Detailed Implementation

[0042] The embodiments of the present invention are described in detail below: These embodiments are implemented based on the technical methods of the present invention, and provide detailed implementation methods and specific operating procedures. It should be noted that those skilled in the art can make many modifications and improvements without departing from the concept of the present invention, and these all fall within the protection scope of the present invention.

[0043] Example

[0044] like Figure 1 As shown, this embodiment provides a membership degree ranking method for identifying overlapping groups that integrates size constraints and structural regularization, including the following steps:

[0045] Step S1: Model the complex network as a graph structure , where the set of nodes Link set represents the set of links in a network. The graph structure is formalized into an adjacency matrix. .

[0046] The specific content of step S1 is as follows:

[0047] This embodiment uses the artificial dataset LFR network. , where the set of nodes Link set represents the set of links in a network. The graph structure M is represented by an adjacency matrix. ,in Represents node c i and c j There are links between them. This indicates that there are no links between them. The following parameters control the characteristics of the LFR network: parameter n represents the total number of vertices in the network, parameter k represents the average degree of the vertices in the network, parameter maxk represents the maximum degree of the vertices in the network, and the blend parameter mu represents the degree of vertex c. i The number of vertices c occupied by the link to the external vertices of the category j The degree ratio is defined by the parameters minc and maxc, which represent the maximum and minimum number of vertices in each class, respectively. The parameter on represents the proportion of overlapping vertices among all vertices, and the parameter om represents the average number of classes to which overlapping vertices belong. Three different LFR datasets, A, B, and C, are used:

[0048] Class A dataset: Testing the impact of network size changes. Network sizes n are set to 2000, 4000, 6000, 8000, and 10000 respectively, k is set to 15, maxk is set to 30 (twice the average degree), minc and maxc are dynamically adjusted with the size (minc=40 / maxc=200 when n=2000; minc=80 / maxc=400 when n=4000; minc=120 / maxc=600 when n=6000; minc=160 / maxc=800 when n=8000; minc=200 / maxc=1000 when n=10000), mu is fixed at 0.2, on is set to 5% of the total number of vertices, and om is fixed at 3.

[0049] Class B dataset: Testing the impact of changes in overlap under low mixing. The network size n is fixed at 5000, k is set to 15, maxk is set to 30, minc is fixed at 100, maxc is fixed at 300, mu is fixed at 0.1 (low mixing), on is set to 5% of the total number of vertices, and om is set to 2, 3, 4, 5, and 6 respectively.

[0050] Class C dataset: Testing the impact of changes in overlap under high blending. The network size n is fixed at 5000, k is set to 15, maxk is set to 30, minc is fixed at 100, maxc is fixed at 300, mu is set to 0.5 (high blending), on is set to 5% of the total number of vertices, and om is set to 2, 3, 4, 5, and 6 respectively.

[0051] Step S2: Based on the generative approach, design the node-group membership matrix V, the community size matrix Γ, and the inter-group relationship matrix Ω to characterize the group structure, group size, and their characteristics, and fit the adjacency matrix M to construct the size constraint model O. size (V, Γ, Ω).

[0052] The specific content of step S2 is as follows:

[0053] Step S21: Under the generative framework, the scale-constrained model consists of two sets of parameter matrices. sum matrix To construct, where v iφ Describes the selection of node c in the φ-th group i The probability of generating an edge, ω φψ The correlation between the φ-th group and the ψ-th group is described, then ω φψ v iφ v jψ Describes the selection of node c in the φ-th group respectively i With node c in the ψth group j The probability of generating an edge simultaneously, for the entire graph M, node c i With node c j The probability of generating an edge is

[0054] (1)

[0055] Given equation (1), using a matrix Fit the adjacency matrix M of the observed data graph M, i.e. minimize With m ij The Euclidean distance between them is achieved using three nonnegative matrix decompositions, and the objective function of the basic model is constructed as follows:

[0056] (2)

[0057] Step S22: Introduce a diagonal matrix Further describe the size of the association, where γ ii Representing group size weights, the group association matrix Ω→ΓΩΓ is modified based on the idea of ​​feature augmentation. T Therefore, since the corrected matrix Ω only represents the group associations of irrelevant group sizes, the objective function (2) of the basic model can be transformed into the objective function of the size-constrained model:

[0058] (3)

[0059] Where matrix V represents node-group membership, matrix Γ represents group size, and matrix Ω represents pure group association.

[0060] Step S3: Introduce graph regularization term Characterizing the similarity of neighbor structures in a network allows for adjusting node-group membership to maintain the local structure of the topology, thereby constructing a fusion model O that combines size constraints and structural regularization. size_gr (V, Γ, Ω).

[0061] The specific content of step S3 is as follows:

[0062] Step S31: Introduce graph regularization term By calculating the similarity between nodes, the local structure of the network topology is established, and the design is performed. Capture local topological features of the network, where s ij Represents node c i With node c j The topological similarity between them is calculated using the following formula:

[0063] (4)

[0064] in, , They are nodes c respectively i With c j Given that the graphs have the same number of neighboring nodes and different numbers of neighboring nodes, based on equation (4), a similarity matrix S is designed to construct the Laplace matrix L of the graph. g It is used to measure the connectivity between nodes in graph M, and the formula is L. g =FS, diagonal matrix A matrix representing the degree of all nodes in graph M, with its diagonal elements... Represents node c i The smoothness of the low-dimensional representation can be expressed as the sum of the structural similarities of all nodes in graph M:

[0065] (5)

[0066] Where Tr(.) represents the rank operation of the matrix;

[0067] Step S32: The graph regularization term constructed by equation (5) By incorporating a scale constraint model to form a model that integrates scale constraints and structure regularization, the final objective function is as follows:

[0068] (6)

[0069] Where μ represents the proportion of graph regularization term fusion, and matrix V represents the node-group membership degree.

[0070] Step S4: Introduce a sorting strategy to obtain edge nodes, find nodes with multiple affiliations in the sorting results and re-divide them, and merge overlapping affiliations with non-overlapping affiliations to form the final overlapping clustering structure.

[0071] The specific content of step S4 is as follows:

[0072] Step S41: Obtain the node-group membership matrix V according to the objective function (6), and sort v in descending order. iφ Get node c i For each group φ, the group preference is determined by each node c. i Its tendency toward group φ and its tendency toward group ψ are v iφ v iψ For every two different groups, the nodes and their corresponding group preference values ​​are calculated by maximizing the preference of each node for the two groups, max{v iφ v iψ The sorting process is performed in descending order to obtain the sorted result, and the formula is as follows:

[0073] (7)

[0074] The sorting function sort(.) described by equation (7) is used to sort the tendency values ​​of all nodes belonging to groups φ and ψ. An adaptive threshold τ is set (the recommended range is 0.1-0.2). If node c i If the difference in tendencies between different groups φ and ψ is within the threshold τ, then the nodes are considered overlapping nodes; otherwise, they are considered non-overlapping nodes. The formula for determining overlapping nodes is as follows:

[0075] (8)

[0076] Step S42: Merge nodes that belong to only a single group with nodes that have multiple affiliations to identify a global overlapping group structure.

[0077] The effectiveness of this example can be further illustrated by the following comparative experiment. The experimental results are as follows: Figure 2 As shown in (a), 2(b), and 2(c), the proposed new model LGR-NMF is run on three networks with different overlapping class structure features. The Overlapping Normalized Mutual Information (OverlappingNMI) metric is used to quantitatively evaluate the model's performance. In the (Class A) dataset, the proposed method demonstrates good scalability for different group sizes under different network sizes, outperforming CFinder. In scenarios with low mixing and varying overlap (Class B), the proposed method significantly outperforms the comparative algorithm CFinder, validating the ranking strategy's ability to accurately capture overlapping group structures. However, in environments with unclear group structures (Class C), the proposed method performs slightly worse than CFinder, indicating that existing methods still have shortcomings in scenarios with unclear group structures.

[0078] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A membership degree ranking method for overlapping group identification that integrates scale constraints and structural regularization, characterized in that, Includes the following steps: Step S1: Model the complex network as a graph structure , where the set of nodes Link set represents the set of links in a network. The graph structure is formalized into an adjacency matrix. ; Step S2: Design the node-group membership matrix V, community size matrix Γ, and inter-group relationship matrix Ω to characterize group structure, group size, and inter-group correlation, respectively, to fit the adjacency matrix M and construct the size constraint model O. size (V, Γ, Ω); Step S3: Introduce graph regularization terms Characterizing the similarity of neighbor structures in a network is used to adjust node-group membership to maintain the local structure of the topology, thereby constructing a fusion model O that combines size constraints and structural regularization. size (V, Γ, Ω); Step S4: Introduce a sorting strategy to obtain edge nodes, find nodes with multiple affiliations in the sorting results and re-divide them, and merge overlapping affiliations with non-overlapping affiliations to form the final overlapping clustering structure.

2. The membership degree ranking method for overlapping group identification based on fusion of scale constraints and structural regularization as described in claim 1, characterized in that, In step S1, the graph structure is formalized as follows: the graph structure M is represented by an adjacency matrix. ,in Represents node c i and c j There are links between them. Represents node c i and c j There are no links between them.

3. The membership degree ranking method for overlapping group identification based on fusion of scale constraints and structural regularization as described in claim 1, characterized in that, Step S2 specifically includes the following steps: Step S21: Under the generative framework, the scale constraint model is formed by a matrix. sum matrix Construct, where v iφ Describes the selection of node c in the φ-th group i The probability of generating an edge, ω φψ The correlation between the φ-th group and the ψ-th group is described, then ω φψ v iφ v jψ Describes the selection of node c in the φ-th group respectively i With node c in the ψth group j The probability of generating an edge simultaneously, for the entire graph M, node c i With node c j The probability of generating an edge is: (1) Given equation (1), using a matrix Fit the adjacency matrix M of the observed data graph M, i.e. minimize With m ij The Euclidean distance between them is achieved using three nonnegative matrix decompositions, and the objective function of the basic model is constructed as follows: (2) Step S22: Describe the size of the community and introduce a diagonal matrix. Describe, where γ ii Representing group size weights, correcting the group association matrix Ω→ΓΩΓ T After correction, the matrix Ω only represents the group associations of irrelevant group sizes, so the objective function (2) of the basic model can be transformed into the objective function of the size-constrained model: (3) Where matrix V represents node-group membership, matrix Γ represents group size, and matrix Ω represents pure group association.

4. The membership degree ranking method for overlapping group identification based on fusion of scale constraints and structural regularization as described in claim 1, characterized in that, Step S3 specifically includes the following steps: Step S31: Introduce graph regularization terms By calculating the similarity between nodes, the local structure of the network topology is established, and the design is performed. Capture local topological features of the network, where s ij Represents node c i With node c j The topological similarity between them is calculated using the following formula: (4) in, , They are nodes c respectively i With c j The graphs have the same number of neighboring nodes and different number of neighboring nodes; based on equation (4), a similarity matrix S is designed to construct the Laplace matrix L of the graph. g It is used to measure the connectivity between nodes in graph M, and the formula is L. g =FS, diagonal matrix A matrix representing the degree of all nodes in graph M, with its diagonal elements... Represents node c i The sum of structural similarities with all nodes in graph M is used to constrain node c. i In the latent space, v is represented as... i The geometric continuity, and the smoothness of the low-dimensional representation, can be expressed as: (5) Where Tr(.) represents the rank operation of the matrix; Step S32: The graph regularization term constructed by equation (5) By incorporating a scale constraint model to form a model that integrates scale constraints and structure regularization, the final objective function is as follows: (6) Where μ represents the proportion of graph regularization term fusion, and matrix V represents the node-group membership degree.

5. The membership degree ranking method for overlapping group identification based on fusion of scale constraints and structural regularization as described in claim 1, characterized in that, Step S4 specifically includes the following: Step S41: Obtain the node-group membership matrix V according to the final objective function (6), and sort v in descending order. iφ Get node c i For each node c i Its tendency toward group φ and its tendency toward group ψ are respectively v iφ v iψ For every two different groups, the nodes and their corresponding group preference values ​​are calculated by maximizing the preference of each node for the two groups, max{v iφ ,v iψ The sorting process is performed in descending order to obtain the sorted result, and the formula is as follows: (7) The sorting function sort(.) described by equation (7) is used to sort the tendency values ​​of all nodes belonging to groups φ and ψ. An adaptive threshold τ is set. If the difference in tendency of node τ to different groups φ and ψ is within the threshold τ range, then it is an overlapping node; otherwise, it is a non-overlapping node. The formula for determining overlapping nodes is as follows: (8) Step S42: Merge nodes that belong to only a single group with nodes that have multiple affiliations to identify the global overlapping group structure.

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