A Community Discovery Method Based on Triangular Structure for Fusing Node Features and Network Topology

Abstract

The present invention discloses a community discovery method based on fusing node features and network topology with a triangular structure, comprising the following steps: Step S1, a quality index is defined based on the high-order information of fusing node features and network topology with a closed triangular structure; Step S2, two-level regularization constraint terms are formulated according to the low-order information of node features and network topology; Step S3, the proposed index and constraint terms are combined as an optimization objective function, and algorithm optimization is formulated from a local perspective to determine the non-overlapping and overlapping community memberships of each node. The present invention not only improves the topological compactness of each community, but also improves the feature consistency of each community, thereby effectively alleviating the unbalanced community problem caused by simply optimizing the quality index, and the community structure found by this method is more accurate and reasonable.

Description

Technical Field

[0001] The present invention belongs to the technical field of the Internet, and particularly relates to a community discovery method based on a triangular structure that fuses node features and network topology. Background Art

[0002] Extensive research work has achieved community discovery by optimizing quality indicators and proposed many generally recognized community discovery methods, but there are still some major problems to be solved: First, the vast majority of methods only focus on the low-order structures in the network, that is, considering node features and network topology from the perspective of points and edges to achieve community discovery. A large number of studies have shown that high-order structures in the network, such as triangles, subgraphs, motifs, etc., are more helpful for revealing the internal mechanism of the network. Although existing work has considered high-order structures, it often only focuses on network topology and ignores node features. Second, the communities found by simply optimizing quality indicators are often unbalanced. For example, there are several super-large communities, a large number of small-scale communities accompanied by a small number of large-scale communities, etc. Although the optimization degree of the quality indicator is very high, it is difficult to fully reveal the true community structure in the network. In fact, large-scale communities still contain many nodes and edges, and their internal structures need to be further analyzed. Finally, as the network scale becomes larger and larger, it becomes very difficult to access the global information of the network. Community discovery algorithms based on global models are more likely to face the dilemma of low computational efficiency, and thus there will be performance bottlenecks. Summary of the Invention

[0003] The purpose of the present invention is to provide a community discovery method based on a triangular structure that fuses node features and network topology, which is used to solve the technical problem that in the prior art, due to the too large network scale, it becomes very difficult to access the global information of the network, and community discovery algorithms based on global models are more likely to face low computational efficiency.

[0004] The described community discovery method based on a triangular structure that fuses node features and network topology includes the following steps:

[0005] Step S1: Define a quality indicator based on the high-order information of fusing node features and network topology in a closed triangular structure;

[0006] Step S2: Formulate two-level regularization constraint terms according to the low-order information of node features and network topology;

[0007] Step S3: Combine the proposed indicator and constraint terms as an optimization objective function and formulate algorithm optimization from a local perspective to determine the non-overlapping and overlapping community affiliations of each node.

[0008] Preferably, in the step S1, it includes, arbitrarily given a node v i and a community C k , specifically define the quality indicator WCC* (v i ,C k ) is as follows:

[0009]

[0010] Among them, NC is a set composed of the neighbor nodes of node v i and the nodes in community C k . tf(v i , C k ) represents the number of feature and topological triangles formed by v i and the nodes in C k . tf(v i , NC) represents the number of feature and topological triangles formed by v i and the nodes in NC. vtf(v i , NC) represents the number of nodes in NC that can form at least one feature or topological triangle with v i . |C k -{v i}| represents the size of community C i after excluding node v k .

[0011] Preferably, based on the definition of the quality index WCC * (v i , C k ), the quality index WCC(C k ) value of a community C k is defined as follows:

[0012]

[0013] The quality index WCC(C) value of a single community partition is defined as follows:

[0014]

[0015] Among them, K represents the total number of communities, |v| represents the total number of nodes in the network, v is the set of nodes in the network, and |C k | represents the size of C k .

[0016] Preferably, in the two-level regularization constraint term formulated in step S2, the first regularization term improves the topological compactness of each community by considering the connection strength between nodes; for any given node v i and a community C k , the first-level constraint is defined as follows:

[0017]

[0018] Among them, d i represents the degree of v i , that is, the number of edges connected to this node, |C k | represents the size of C k , represents whether there is an edge between the node v k in the community C i and v j . If so, otherwise is 0. The more nodes v i is connected to in C k , the larger it is; by maximizing T(v i , C k ), the nodes added to C k will strengthen the topological compactness of the community.

[0019] Preferably, in the two - level regularization constraint terms formulated in step S2, the second regularization term improves the feature consistency of each community by considering the feature differences between nodes; for any given node v i and a community C k , the second - level constraint is defined as follows:

[0020]

[0021] Among them, p represents the number of features of the node, |C k | represents the size of C k , f i and f j represent the feature vectors of the nodes v i and v j respectively, |f i - f j | represents the difference between the feature vectors of the nodes v i and v j . The smaller the value, the more similar the features of the nodes; by minimizing H(v i , C k ), the nodes added to C k will strengthen the feature consistency of the community.

[0022] Preferably, in step S3, a local learning framework is established. By combining the quality index with two - level regularization, the proposed index and constraint terms are combined as the optimization objective function, and the maximized objective function is defined as follows:

[0023]

[0024] Based on this utility function, the node v iThe utility gain Δu of updating the community label from k to k′ ik→ik′ Is quantified, and based on this, a local learning framework for community discovery is formulated. The basic process of step S3 includes inputting the network, initializing, corresponding to the process of the local learning framework for community discovery, and then determining whether the number of rounds reaches the set threshold. If it reaches, the corresponding community structure is output; otherwise, it returns to the loop of stage 1 of the local learning framework for community discovery and runs until the number of rounds reaches the set threshold.

[0025] Preferably, the local learning framework includes a local learning framework for non-overlapping community discovery, and the specific stages are as follows:

[0026] Stage 1: Expected utility update. Each node updates its own expected utility by considering historical information and current information; assume Denotes the expected utility of node v i With label k at the t-th round, then given a label k′ of a neighbor node, v i Will update the expected utility of node v with label k′ at the (t + 1)-th round based on the utility gain obtained by changing the current label and the expected utility at the t-th round: i Where α ∈ [0, 1] is a weighting factor;

[0027]

[0028] Among them, α ∈ [0, 1] is a weighting factor;

[0029] Stage 2: Update of the candidate community label set. Based on the expected utility updated in stage 1, each node updates its own candidate label set:

[0030] Stage 3: Non-overlapping community label update. Given a perturbation factor η i , node v i Updates its own membership vector u i As follows: (1) If the current label k belongs to the candidate label set Then u i Remains unchanged, and η i Decays by 0.05; (2) If the current label k does not belong to Then generate a random number. When the random number is greater than the perturbation factor η i Select the largest As the k′-th element in u i , and the rest of the elements are all 0; otherwise, randomly select one As the k′-th element in u i , and the rest of the elements are all 0.

[0031] Preferably, the local learning framework includes a local learning framework for overlapping community discovery. For overlapping community discovery, since each node can belong to multiple communities, the local learning framework for overlapping community discovery introduces a label set to save the community label of node v i in the t-th round. The specific stages are as follows:

[0032] Stage 1: Compromise utility update. Given a perturbation factor η i , if the generated random number is greater than η i , then select the largest element from the membership vector u i of v i as Otherwise, randomly select an element as Suppose represents the compromise utility of node v i with label k in the i-th round. Then, given a label k' of a neighbor node, v i will update the compromise utility of node v with label k' in the (t + 1)-th round based on the utility gain obtained by changing the current label and the compromise utility in the t-th round: i where α ∈ [0, 1] is a weighting factor;

[0033]

[0034]

[0035] Stage 2: Candidate label set update. Based on the compromise utility updated in Stage 1, each node updates its own candidate label set:

[0036]

[0037] Stage 3: Overlapping community label update. Node v i updates its own membership vector u i and label set as follows: (1) If the largest element is selected in Stage 1, then η i decays by 0.05. When only contains k, u i remains unchanged, and is used as Otherwise, randomly select several k' from to form For each k″ in , select as the k″-th element in u i , and all other elements are 0; (2) If a randomly selected element is selected in Stage 1, then randomly select several k' and k from and to form For​ For each k″ in select as u i the k″-th element in, if from then select as u i the k″-th element in, and all other elements are 0.

[0038] The present invention has the following advantages:

[0039] 1. First, an internal quality index is formulated for community discovery and community evaluation based on closed features and topological triangle structures. This index can not only evaluate the quality of high-order structures but also be used as an optimization objective function to achieve community discovery.

[0040] 2. Then, when the proposed index is used as the objective function, two-level regularization constraints are developed to strengthen its function, which can improve both the topological compactness of each community and the feature consistency of each community, thus effectively alleviating the unbalanced community problem caused by simply optimizing the quality index.

[0041] 3. Finally, a local learning framework is designed to optimize the proposed internal index and secondary constraints to simultaneously achieve non-overlapping and overlapping community discovery. The local learning framework is adjusted for overlapping communities, and the results show that the community structure found by the proposed method is more accurate and reasonable. Description of the Drawings

[0042] Figure 1 is the basic flowchart of a community discovery method based on triangle structure fusion of node features and network topology according to the present invention.

[0043] Figure 2 is the schematic diagram of a community discovery method based on triangle structure fusion of node features and network topology according to the present invention.

[0044] Figure 3 is the flowchart of step S3 of the local learning framework corresponding to non-overlapping community discovery in the present invention.

[0045] Figure 4 is the flowchart of step S3 of the local learning framework corresponding to overlapping community discovery in the present invention.

[0046] Figure 5 is the process schematic diagram of a specific embodiment of the present invention taking an undirected unweighted network as an example. Detailed Embodiment

[0047] The following is a more detailed description of the specific embodiments of the present invention with reference to the drawings and through the description of the embodiments, so as to help those skilled in the art have a more complete, accurate and in-depth understanding of the inventive concept and technical solution of the present invention.

[0048] As shown Figure 1-2 in the figure, the present invention provides a community discovery method based on fusing node features and network topology with a triangular structure. First, a quality index is defined based on the high-order information of fusing node features and network topology with a closed triangular structure. Then, two-level regularization constraint terms are formulated according to the low-order information of node features and network topology. Finally, the proposed index and constraint terms are combined as an optimization objective function, and algorithm optimization is carried out from a local perspective to determine the non-overlapping and overlapping community memberships of each node. The results show that the community scale and structure found by the above method are more reasonable and accurate.

[0049] The community discovery method based on fusing node features and network topology with a triangular structure includes the following steps.

[0050] Step S1: Define a quality index based on the high-order information of fusing node features and network topology with a closed triangular structure.

[0051] In this step, a quality index based on the triangular structure is defined, and at the same time, considering the closed triangular structure composed of node features and the closed triangular structure composed of network topology, a quality index WCC * (v i , C k ) is formulated.

[0052] Given any node v i and a community C k , the quality index WCC * (v i , C k ) is specifically defined as follows:

[0053]

[0054] Among them, NC is a set composed of the neighbor nodes of node v i and the nodes in community C k . tf(v i , C k ) represents the number of characteristic and topological triangles formed by v i and the nodes in C k . tf(v i , NC) represents the number of characteristic and topological triangles formed by v i and the nodes in NC. vtf(v i , NC) represents the number of nodes in NC that can form at least one characteristic or topological triangle with v i . |C k -{v i}| represents excluding node v i from community C kThe size.

[0055] Based on this, a community C k The quality index WCC(C k ) is defined as follows:

[0056]

[0057] Based on this, the quality index WCC(C) value of a community partition is defined as follows:

[0058]

[0059] where K represents the total number of communities, |v| represents the total number of nodes in the network, v is the set of nodes in the network, and |C k | represents the size of C k The size.

[0060] Step S2: Two-level regularization constraints are formulated according to the node characteristics and the low-order information of the network topology.

[0061] This step formulates two-level regularization constraints. For the "unbalanced community" problem in community discovery that optimizes the quality index, the so-called two-level regularization constraints are formulated.

[0062] The first regularization term improves the topological compactness of each community by considering the connection strength between nodes. Generally speaking, the larger the degree of a node, the more influential it is. Once its community membership is determined, the community membership of the nodes connected to it is naturally determined; conversely, if the community membership of a highly influential node deviates, it may affect the final result. Accordingly, for any given node v i and a community C k , the first-level constraint is defined as follows:

[0063]

[0064] where represents whether there is an edge between nodes v k and v i and v j in community C. If there is, then Conversely, if not, then is 0. In other words, the more nodes v i is connected to in C k , the larger . By maximizing T(v i , C k ), the nodes added to C k will strengthen the topological compactness of the community. In the formula, d i represents the degree of v i , that is, the number of edges of this node, |Ck | represents C k 's size.

[0065] The second regularization term enhances the feature consistency of each community by considering the feature differences between nodes. Generally speaking, nodes in the same community should have more similar feature distributions. Accordingly, given any node v i and a community C k , the second-level constraint is defined as follows:

[0066]

[0067] where f i and f j represent the feature vectors of nodes v i and v j respectively, |f i - f j | represents the difference between the feature vectors of nodes v i and v j , the smaller the value, the more similar the features of the nodes. By minimizing H(v i , C k ), the nodes added to C k will strengthen the feature consistency of the community. p represents the number of features of the nodes, and |C k | represents the size of C k .

[0068] Through the two-level regularization constraints of this step, this method not only improves the topological compactness of each community but also improves the feature consistency of each community, making the community scale and structure more reasonable and accurate.

[0069] Step S3: Combine the proposed metrics and constraint terms as the optimization objective function and formulate algorithm optimization from a local perspective to determine the non-overlapping and overlapping community memberships of each node.

[0070] This step establishes a local learning framework by combining the quality metrics with two-level regularization and combines the proposed metrics and constraint terms as the optimization objective function. The objective function maximized in the present invention is defined as follows:

[0071]

[0072] where the meanings of the parameters such as node v i , community C k , the first-level constraint T(v i , C k ), the second-level constraint H(v i , C k ), and the total number K of communities are the same as those in the foregoing formulas.

[0073] Assume node v i has a community label of k (i.e., v i is in the k-th community C k ), then according to the above formula, the utility function for v i can be formulated as follows:

[0074] u ik = WCC * (v i , C k ) + T(v i , C k ) - H(v i , C k )

[0075] Based on this utility function, the utility gain Δu i for the community label of node v ik→ik′ updated from k to k′ can be quantified. The present invention accordingly formulates the following local learning framework for non-overlapping community discovery:

[0076] Phase 1: Expected utility update. Each node updates its own expected utility by considering historical information and current information. Assume represents the expected utility of node v i with label k in the t-th round. Then, given a label k′ of a neighbor node, v i will update the expected utility of node v i with label k′ in the (t + 1)-th round based on the utility gain obtained by changing the current label and the expected utility in the t-th round:

[0077]

[0078] where α ∈ [0, 1] is a weighting factor, which can be set to 0.2 in the present invention.

[0079] Phase 2: Update of the candidate community label set. Based on the expected utility updated in Phase 1, each node updates its own candidate label set:

[0080] Phase 3: Non-overlapping community label update. Given a perturbation factor η i = 0.8, node v i updates its membership vector u i as follows: (1) If the current label k belongs to the candidate label set then u i remains unchanged and η i decays by 0.05. (2) If the current label k does not belong to then a random number is generated. When the random number is greater than the perturbation factor η i , the largest As the k'-th element in u i and all other elements are 0; conversely, randomly select one as the k'-th element in u i and all other elements are 0.

[0081] For overlapping community detection, since each node can belong to multiple communities, the present invention introduces a label set to save the community label of node v i in the t-th round and adjusts the above three stages to formulate the local learning framework for overlapping community detection as follows:

[0082] Stage 1: Compromise utility update. Given a perturbation factor η i = 0.8, if the generated random number is greater than η i , then select the largest element from the membership vector u i of v i as Conversely, randomly select one element as The subsequent process is the same as that in Stage 1 of non-overlapping community detection. Assume represents the compromise utility of node v i with label k in the t-th round. Then, given a label k' of a neighbor node, v i will update the compromise utility of node v i with label k' in the (t + 1)-th round based on the utility gain obtained by changing the current label and the compromise utility in the t-th round:

[0083]

[0084] where α ∈ [0, 1] is a weighting factor, which can be set to 0.2 in the present invention.

[0085] Stage 2: Candidate label set update. The process is the same as that in Stage 2 of non-overlapping community detection. Based on the compromise utility updated in Stage 1, each node updates its own candidate label set:

[0086]

[0087] Stage 3: Overlapping community label update. Node v i updates its own membership vector u i and label set as follows: (1) If the largest element is selected in Stage 1, then η i decays by 0.05. When has only k, u i remains unchanged, and is used as Conversely, randomly select several k' from to form For each k″ in select i as the k″-th element of u and randomly select several k′ and k from For each k″ in if it comes from then select i as the k″-th element of u if it comes from then select i as the k″-th element of u, and set the remaining elements to 0.

[0088] The flowcharts of step S3 of the above-mentioned local learning framework for non-overlapping community discovery and the local learning framework for overlapping community discovery are shown in Figure 3 and Figure 4 respectively. For the local learning framework for non-overlapping community discovery, the basic process of step S3 includes inputting the network, initializing, going through the three-stage process of the local learning framework for non-overlapping community discovery, and then determining whether the number of rounds reaches the set threshold. If it reaches, output the non-overlapping community structure; otherwise, return to the first stage of the local learning framework for non-overlapping community discovery and loop until the number of rounds reaches the set threshold. Similarly, for the local learning framework for overlapping community discovery, the basic process of step S3 also includes inputting the network, initializing, going through the three-stage process of the local learning framework for overlapping community discovery, and then determining whether the number of rounds reaches the set threshold. If it reaches, output the overlapping community structure; otherwise, return to the first stage of the local learning framework for overlapping community discovery and loop until the number of rounds reaches the set threshold.

[0089] The invention is applicable to both unweighted undirected and weighted undirected networks. Figure 5 Taking an unweighted undirected network as an example, the specific implementation process of the present invention is illustrated. Given a network as input, the network contains 8 nodes, and the connections between the nodes are as shown in the figure. Initially, all nodes belong to different communities. After t rounds of learning, nodes {2, 3, 4} belong to community C1, nodes {6, 7, 8} belong to community C2, and the community affiliations of nodes {1} and {5} are not determined yet.

[0090] Taking the determination of the non-overlapping community affiliation of node {5} as an example, the implementation process of the proposed method is described below. In the expected utility update stage, first, calculate the utility gains of node {5} joining communities C1 and C2 respectively. The calculation formula is: Δu ik→ik′ = u ik′ - u ik. Among them, u ik = WCC * (v i , C k ) + T(v i , C k ) - H(v i , C k )u ik′ has the same calculation formula as u ik . Then, calculate the expected utility, and the calculation formula is: In the candidate community label update stage, compare the expected utility of node {5} joining communities C1 and C2 to generate a set of candidate community labels. In the community label update stage, if the current community label of node {5} is in the candidate label set, the community affiliation of the node remains unchanged, and the perturbation factor value decays by 0.05. If the current community label of node {5} is not in the candidate label set, first compare the perturbation factor value with a random number. When the perturbation factor value is less than or equal to the random number, select the community label with the highest expected utility from the candidate community label set as the community affiliation of the node at the t+1 stage; when the perturbation factor value is greater than the random number, randomly select a community label from the candidate community label set and assign it to the node.

[0091] As the value of the perturbation factor gradually decays, the probability that the random number is greater than the perturbation factor value becomes higher and higher. In other words, as the community label with the highest expected utility is continuously selected and assigned to the node, the node will finally obtain the community label with the highest global expected utility. After that, in each round, the candidate community label set only has the current community label (because the global expected utility is the highest), and the node will no longer update the community label. As the perturbation factor value decays to 0, the algorithm convergence stops, and the final result is output.

[0092] The present invention has been described exemplarily above in conjunction with the accompanying drawings. Obviously, the specific implementation of the present invention is not limited by the above-mentioned manner. As long as various non-substantive improvements are made by adopting the inventive concept and technical solution of the present invention, or the inventive concept and technical solution of the present invention are directly applied to other occasions without improvement, they are all within the protection scope of the present invention.

Claims

1. A community discovery method based on fusing node features and network topology with a triangular structure, characterized in that: Including the following steps: Step S1: Define a quality metric based on fusing node features and high-order information of network topology in a closed triangle structure; Step S2: Formulate two-level regularization constraint terms according to node features and low-order information of network topology; Step S3: Combine the proposed metric and constraint terms as an optimization objective function and formulate algorithm optimization from a local perspective to determine the non-overlapping and overlapping community memberships of each node; In the step S1, an arbitrary node υ is given i and a community C k , and a quality index WCC * (υ i , C k ) is defined as follows: Among them, NC is a set composed of the neighbor nodes of node υ i and the nodes within community C k . tf(υ i , C k ) represents the number of characteristic and topological triangles formed by υ i and the nodes within C k . tf(υ i , NC) represents the number of characteristic and topological triangles formed by υ i and the nodes within NC. υtf(υ i , NC) represents the number of nodes within NC that can form at least one characteristic or topological triangle with υ i . |C k - {υ i}| represents the size of community C i after excluding node υ k ; In the two-level regularization constraint terms formulated in step S2, the first regularization term enhances the topological compactness of each community by considering the connection strength between nodes; given any node υ i and a community C k , the first-level constraint is defined as follows: Among them, d i represents the degree of υ i , that is, the number of edges connected to this node, |C k | represents the size of C k . represents whether there is an edge between the node υ k in the community C i and v j . If so, otherwise is 0. The more nodes ν i is connected to within C k , the larger it is. By maximizing T(υ i , C k ), the nodes added to C k will strengthen the topological compactness of the community; In the two-level regularization constraint terms formulated in step S2, the second regularization term improves the feature consistency of each community by considering the feature differences between nodes; for any given node υ i and a community C k , the second-level constraint is defined as follows: Among them, p represents the number of features of a node, and |C k | represents the size of C k , f i and f j respectively represent the feature vectors of nodes υ i and υ j . |f i - f j | represents the difference between the feature vectors of nodes υ i and υ j . The smaller the value, the more similar the features of the nodes. By minimizing H(υ i , C k ), the nodes added to C k will strengthen the feature consistency of the community; In step S3, a local learning framework is established. By combining the quality metric with two-level regularization, the proposed metric and constraint terms are combined as an optimization objective function, and the maximized objective function is defined as follows: Based on this utility function, for node υ i the utility gain Δu when the community label of ik→ik′ it is updated from k to k′ is quantified, and a local learning framework for community discovery is formulated accordingly. The process of step S3 includes inputting the network, initializing, corresponding to the process of the local learning framework for community discovery. Then, it is judged whether the number of rounds reaches the set threshold. If it reaches, the corresponding community structure is output; otherwise, it returns to loop and run in stage 1 of the local learning framework for community discovery until the number of rounds reaches the set threshold.

2. The community discovery method based on fusing node features and network topology with a triangular structure according to claim 1, wherein: Based on the quality metric WCC * (v i , C k ) is defined, the quality metric WCC(C k ) value of a community C k ) is defined as follows: The quality metric value WCC(C) of a single community partition is defined as follows: Among them, K represents the total number of communities, |v| represents the total number of nodes in the network, υ is the set of nodes in the network, and |C k | represents the size of C k .

3. A community discovery method based on fusing node features and network topology with a triangular structure according to claim 1, characterized in that: The local learning framework includes a local learning framework for non-overlapping community discovery, and the specific stages are as follows: Phase 1: Expected utility update, where each node updates its own expected utility by considering historical information and current information; assume represents the expected utility of node υi with label k at the t-th round. Then, given a label k′ of a neighbor node, υ i will update the expected utility of node υ at the (t + 1)-th round with label k′ based on the utility gain obtained from changing the current label and the expected utility at the t-th round i of label k′: where α ∈ [0, 1] is a weighting factor; Phase 2: Update of the candidate community label set. Based on the expected utility updated in Phase 1, each node updates its own candidate label set: Stage 3: Non-overlapping community label update, given the perturbation factor η i , node υ i updates its membership vector u i as follows: (1) If the current label k belongs to the candidate label set then u i remains unchanged, and η i decays by 0.05; (2) If the current label k does not belong to then generate a random number. When the random number is greater than the perturbation factor η i , select the largest as the k'-th element of u i , and all other elements are 0; otherwise, randomly select one as the k'-th element of u i , and all other elements are 0.

4. A community discovery method based on fusing node features and network topology with a triangular structure according to claim 1, characterized in that: The local learning framework includes a local learning framework for overlapping community discovery. For overlapping community discovery, since each node can belong to multiple communities, the local learning framework for overlapping community discovery introduces a label set to save the community labels of node υ i at the t-th round. The specific stages are as follows: Phase 1: Compromise utility update, given the perturbation factor η i , if the generated random number is greater than η i , then select the largest element from the membership vector u i of υ i as Otherwise, randomly select an element as Suppose represents the compromise utility of node υ i with label k in the t-th round. Then, given a label k′ of a neighbor node, υ i will update the compromise utility of node υ i with label k′ in the (t + 1)-th round based on the utility gain obtained by changing the current label and the compromise utility in the t-th round: where α ∈ [0, 1] is a weighting factor; Stage 2: Update the candidate label set. Based on the compromise utility updated in stage 1, each node updates its own candidate label set: Phase 3: Overlapping community label update, node υ i Update its membership vector u i and the label set as follows: (1) If the largest element is selected in Phase 1, then η i decays by 0.

05. When only has k, u i remains unchanged. Taking as Conversely, randomly select several k from to form For each k″ in, select as the k″-th element of u i and all other elements are 0; (2) If the element is randomly selected in Phase 1, then randomly select several k′ and k from and to form For each k″ in, if it comes from then select as the k″-th element of u i and if it comes from then select as the k″-th element of u i and all other elements are 0.

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