Attribute graph clustering method based on tensor modeling and sequential game

Abstract

The invention discloses an attribute graph clustering method based on tensor modeling and sequential game, and relates to the technical field of data mining. The method comprises the following steps: constructing a third-order tensor representation of an attribute graph by combining a topology slice, a feature slice and a topology-feature transfer slice; the method comprises the following steps: defining a weighted modularity, introducing a regularization item to constrain class cluster scale distribution and an overlapping ratio between class clusters, thereby constructing a global clustering objective function, and performing optimization through a sequential game: taking the class clusters as game players, taking membership degrees of nodes to the class clusters as player strategies, defining player utility based on the global clustering objective function, and performing optimization through the sequential game. And iteratively optimizing node class cluster attribution according to a node influence sequence, and outputting a final clustering result when the player utility reaches Nash equilibrium. Through the method, heterogeneous information of modeling structures and attributes can be unified, a clustering objective function is optimized, a high-order structure and class cluster constraints are fused, stable clustering is achieved, the overall performance is improved, and a real class cluster mechanism is mined.

Description

Technical Field

[0001] The present invention relates to the technical field of data mining, and particularly relates to an attributed graph clustering method based on tensor modeling and sequential game. Background Art

[0002] With the wide application of complex data structures such as social networks, bioinformatics graphs, and recommendation systems, graph clustering, as a key technology for revealing potential structures and functional modules in graphs, has received extensive attention. Especially in an attributed graph, nodes not only have a topological structure but also carry rich attribute feature information. How to simultaneously fuse the structural connections and attribute features of nodes for high-quality clustering has become an important research direction.

[0003] The existing attributed graph clustering algorithms have the following problems to be solved urgently:

[0004] Insufficient handling of structural and attribute heterogeneity: Many existing algorithms generally assume that nodes belonging to the same cluster have better topological compactness and feature homogeneity. However, the topological structure and attribute features of nodes essentially belong to heterogeneous information sources. Forcing the fusion of the two may lead to biases in cluster division driven by structure or features, thus reducing the authenticity and interpretability of the clustering results.

[0005] Ineffective utilization of high-order structural information: Most clustering methods only focus on the low-order information of the graph, that is, they focus on the features and topology of clusters from the perspectives of nodes and edges, ignoring more complex graph structure units - motifs. Existing research has shown that motifs, as high-order connection patterns, can more accurately reflect the roles and functions of nodes in local structures and play an important role in revealing the mechanism of cluster formation. Although some existing work has tried to introduce motifs into graph clustering, most algorithms are difficult to adapt to the attributed graph structure and usually only support specific types of motifs, with insufficient adaptability and generality.

[0006] Problems of algorithm efficiency and scalability are prominent: Currently, some methods rely on global optimization models, which have high computational overhead when dealing with large-scale graph data and are difficult to meet the efficiency requirements in practical applications. Relatively speaking, algorithms based on local information modeling have lower computational complexity, but due to ignoring the overall structural features of the graph, they often cannot balance the clustering effect and global consistency, resulting in the need to improve the clustering performance. Summary of the Invention

[0007] Aiming at the problems in the prior art such as the difficulty in unifying structural and property heterogeneity, insufficient utilization of high-order structural information, and unstable clustering optimization, the present invention proposes an attribute graph clustering method based on tensor modeling and sequential game, which can unify the modeling of structural information, attribute features, and semantic transfer relationships, and at the same time has high expressive ability and computational efficiency to more accurately and stably mine the potential cluster structures in the graph.

[0008] The present invention realizes the above object through the following technical solutions:

[0009] An attribute graph clustering method based on tensor modeling and sequential game, the method comprising:

[0010] Combining topological slices, feature slices, and topological-feature transfer slices to construct a third-order tensor representation of the attribute graph to realize the joint representation learning of node structural information and attribute features;

[0011] The topological slice is used to quantify the topological association strength between nodes based on motif structures, the feature slice is used to measure the similarity of nodes in the attribute feature space, and the topological-feature transfer slice is used to calculate the probability strength of eigenvalue transfer between nodes based on common non-empty feature dimensions;

[0012] Based on the third-order tensor representation, define the weighted modularity, and introduce a regularization term to constrain the cluster size distribution and the overlap ratio between clusters, thereby constructing a global clustering objective function;

[0013] Optimize the clustering objective function through sequential game, including: regarding the clusters as game players, the membership degree of nodes to the clusters as player strategies, defining the player utility based on the global clustering objective function, and iteratively optimizing the cluster membership of nodes in the order of node influence, and outputting the final clustering result when the player utility reaches the Nash equilibrium.

[0014] As a preferred solution of the present invention, the topological slice is defined as a weighted adjacency matrix , where represents the topological association strength between node and node based on a predefined motif structure; is a set of real numbers, is the number of nodes in the attribute graph;

[0015] For any given node pair , if there is no connection relationship between node and node , then ;

[0016] Conversely, if node and node If there is an edge between them, then based on the predefined motif structure, calculate the topological correlation strength between the node pair as follows:

[0017] ;

[0018] In the formula, represents the number of motifs jointly participated by the node pair ; represents the number of motifs individually participated by the node ; represents the number of motifs individually participated by the node ;

[0019] As a preferred solution of the present invention, the feature slice is defined as a weighted feature matrix , where represents the similarity weight of the node and the node in terms of attribute features;

[0020] For any given node pair , the feature vectors corresponding to the node and the node are respectively denoted as and . According to whether the eigenvalues of the corresponding dimensions of the feature vectors and are empty, it is divided into three cases: both eigenvalues are not empty, only one eigenvalue is empty, and both eigenvalues are empty;

[0021] Then the calculation formula of the similarity weight is:

[0022] ;

[0023] In the formula, represents the feature similarity calculated when both eigenvalues are not empty. Among them, if the eigenvalues are boolean type {0,1}, the feature similarity is measured by the Jaccard index; if the eigenvalues are real-time values [0,1], the feature similarity is measured by the cosine similarity; represents the proportion of the dimensions with only one empty eigenvalue in the total number of feature dimensions; represents the proportion of the dimensions with both empty eigenvalues in the total number of feature dimensions.

[0024] As a preferred solution of the present invention, the topology-feature transfer slice is defined as a probability transfer matrix , where represents the probability of transitioning from the node to the node On the premise of the existence of a common non-empty feature dimension, the probability intensity of the eigenvalue transferring from node to node .

[0025] For any given pair of nodes , if there is the same dimension with non-empty eigenvalues between the eigenvectors and , then node can transfer to node through features, and the calculation formula of the probability intensity is:

[0026] ;

[0027] In the formula, respectively represent the dimensions with non-empty eigenvalues in the eigenvector , respectively represent the corresponding eigenvalues of the eigenvector in the dimension , respectively represent the corresponding eigenvalues of the eigenvector in the dimension .

[0028] As a preferred solution of the present invention, the weighted modularity is defined as:

[0029] ;

[0030] In the formula, represents the total number of clusters, represents the th cluster; , are nodes in the cluster ; represents any one of the weighted adjacency matrix , the weighted feature matrix and the probability transition matrix ; represents the association intensity between node and node and node from the corresponding weighted perspective; represents the weighted degree of node ;

[0031] The regularization term is defined as:

[0032] ;

[0033] In the formula, Indicates the number of motif structures in the cluster ; Indicates a given attribute graph; Indicates the th cluster;

[0034] Fuse the weighted modularity with the regularization term to obtain the global clustering objective function , expressed as:

[0035] ;

[0036] In the formula, Indicates the number of slices in the third-order tensor representation, ; , respectively correspond to the topological slice, the feature slice, and the topological-feature transfer slice; Indicates the weighted modularity from the perspective of slice , Indicates the regularization term from the perspective of slice .

[0037] As a preferred solution of the present invention, the game player is defined as , where Indicates the set of , , respectively indicate the 1st, 2nd, th game players, Indicates the set of all nodes in the attribute graph, Indicates that all clusters cover all nodes;

[0038] The player strategy is defined as , where Indicates the set of strategy spaces of , , respectively indicate the membership degrees of all nodes to the 1st, 2nd, th clusters, corresponding to the strategy spaces of the 1st, 2nd, th game players;

[0039] The player utility is defined as , where Indicates the set of utility functions of any utility function ​

[0040] ;

[0041] ;

[0042] Wherein, is the modularity score of the cluster under the given matrix ; is the modularity score of the cluster from the perspective of the slice ;

[0043] In each iteration, all clusters are traversed, and the change in the utility function after each candidate node joins any cluster is calculated , represents the iteration number. If , then add the candidate node to the corresponding cluster and update the cluster strategy;

[0044] After each iteration ends, calculate the global clustering objective function . If the difference between the global clustering objective function and the previous round is less than the preset threshold, or the maximum number of iterations is reached, then terminate the iteration;

[0045] When the iteration terminates, the player utilities of all clusters no longer increase, the clustering result converges to the Nash equilibrium state, and the final node cluster division is output.

[0046] A computer device, the device includes:

[0047] A memory for storing computer-executable instructions;

[0048] One or more processors for executing the computer-executable instructions to implement an attribute graph clustering method based on tensor modeling and sequential game as described above.

[0049] A computer-readable storage medium, on which a computer program is stored, and when the computer program is executed by a processor, it implements an attribute graph clustering method based on tensor modeling and sequential game as described above.

[0050] Compared with the prior art, the beneficial effects of the present invention are: by constructing three types of matrices, namely topological slices, feature slices, and topological-feature transfer slices, fusing the similarity relationships between nodes from three perspectives of structure, attribute, and semantic transfer, a unified third-order tensor representation is formed, effectively improving the expression ability of multi-dimensional heterogeneous relationships in the attribute graph.

[0051] Based on tensor modeling, an improved weighted modularity function is proposed to quantify the tightness of nodes within clusters from multiple perspectives. Meanwhile, a regularization term is introduced to jointly consider the scale distribution of clusters and the overlapping relationships between clusters, thereby avoiding the generation of overly large or highly overlapping clusters while preserving the true structural features.

[0052] Clusters are modeled as game players, and the membership degrees of nodes are used as the strategy space. The cluster utility function is optimized through an iterative game process. This process theoretically converges to a Nash equilibrium state, effectively overcoming the defects of unstable results and dependence on initial conditions in traditional clustering methods, and improving the convergence and interpretability of the model.

[0053] The method of the present invention has been systematically compared with existing clustering algorithms on multiple public datasets. The results show that the method has an average improvement of about 25% in mainstream evaluation indicators such as the F1 score. Not only is the overall clustering performance better, but it also reveals the internal structural characteristics of clusters of different scales in the attribute graph: small-scale clusters exhibit higher topological tightness and characteristic homogeneity consistency, while large-scale clusters more reflect the aggregation mechanism dominated by topology, further deepening the understanding of the true cluster structure. BRIEF DESCRIPTION OF THE DRAWINGS

[0054] To more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings required for the description of the embodiments will be briefly introduced below. Obviously, the drawings in the following description are only some embodiments of the present invention, and those of ordinary skill in the art can obtain other drawings based on these drawings without creative efforts. Among them: Figure 1 is the method flow chart of the present invention; Figure 2 is the schematic diagram of the tensor modeling and optimization process of the attribute graph in the embodiment of the present invention; Figure 3 is the schematic diagram of the pseudo-code process of the local optimization algorithm for attribute graph clustering based on sequential game in the embodiment of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0055] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the drawings of the embodiments of the present invention. Obviously, the described embodiments are some, but not all, embodiments of the present invention. All other embodiments obtained by those of ordinary skill in the art based on the described embodiments of the present invention fall within the protection scope of the present invention.

[0056] As Figure 1 shown, it is an embodiment of the present invention, and this embodiment provides an attribute graph clustering method based on tensor modeling and sequential game.

[0057] The attribute graph contains both the topological structure between nodes and the high-dimensional attributes of the nodes themselves. Traditional matrix methods are difficult to uniformly represent this heterogeneous information and cannot capture the high-order interaction relationships (such as community overlap or multi-node collaboration) commonly existing in the network. Tensor modeling naturally breaks through the dimensionality limitation of the matrix by introducing third-order or higher-order tensors. This invention focuses on the theme of attribute graph clustering. In tensor construction, different similarity measurement methods are proposed to describe the node features, graph topology, and the connection between them in the attribute graph; a new regularization term is proposed to further expand the modularity index; finally, different from common alternating optimization methods, a sequential game framework is proposed to optimize the objective function to achieve attribute graph clustering.

[0058] The overall implementation of the method in this embodiment can be divided into three stages: tensor construction of the attribute graph, modularity expansion based on tensors and regularization, and local optimization based on sequential games.

[0059] (1) Tensor construction of the attribute graph

[0060] The community structure in the attribute graph is determined by node features, graph topology, and the connection between them. Therefore, in this embodiment, a third-order tensor representation of the attribute graph is constructed by combining topological slices, feature slices, and topological-feature transfer slices to realize the joint representation learning of node structure information and attribute features, forming a three-dimensional data model covering network structure, attributes, and interaction relationships, thus solving the problem of the disconnection between topological structure and node features.

[0061] Among them, the topological slice is used to quantify the topological association strength between nodes based on the motif structure, the feature slice is used to measure the similarity of nodes in the attribute feature space, and the topological-feature transfer slice is used to calculate the probability strength of eigenvalue transfer between nodes based on common non-empty feature dimensions.

[0062] In one specific embodiment, the topological slice is defined as a weighted adjacency matrix , where represents the topological association strength between node and node based on the predefined motif structure; is the set of real numbers, is the number of nodes in the attribute graph;

[0063] For any given node pair , if there is no connection relationship between node and node , then ;

[0064] Conversely, if node and node If there is an edge between them, then based on the predefined motif structure, calculate the topological association strength between node pairs as follows:

[0065] …………………………………(1)

[0066] In the formula, represents the number of motifs jointly participated by node pair ; represents the number of motifs individually participated by node ; represents the number of motifs individually participated by node ; ranges from [0, 1], and the larger the value, the more motifs are formed by the given edge.

[0067] The feature slice is defined as the weighted feature matrix , where represents the similarity weight of nodes and node in terms of attribute features;

[0068] For any given node pair , the feature vectors of nodes and node are respectively denoted as and . According to whether the eigenvalues of the corresponding dimensions of feature vectors and are empty, it can be divided into three cases: both eigenvalues are not empty, only one eigenvalue is empty, and both eigenvalues are empty;

[0069] Then the calculation formula of the similarity weight is as follows:

[0070] …………………………………(2)

[0071] In the formula, represents the feature similarity calculated when both eigenvalues are not empty. Among them, if the eigenvalues are boolean type {0, 1}, the feature similarity is measured by the Jaccard index; if the eigenvalues are real-time values [0, 1], the feature similarity is measured by the cosine similarity; represents the proportion of the dimensions with only one empty eigenvalue in the total number of feature dimensions; represents the proportion of the dimensions with both empty eigenvalues in the total number of feature dimensions; ranges from [0, 1], and the larger the value, the more similar the features of the given node pair .

[0072] The topology - feature transfer slice is defined as a probability transfer matrix , where represents the probability intensity of the eigenvalue transferring from node to node on the premise that there is a common non - empty feature dimension between node and node ;

[0073] For any given pair of nodes , if there is the same dimension with non - empty eigenvalues between eigenvectors and , then node can transfer features to node , and the calculation formula of the probability intensity is:

[0074] …………………………(3)

[0075] In the formula, respectively represent the dimensions with non - empty eigenvalues in eigenvector , respectively represent the corresponding eigenvalues of eigenvector in dimension , respectively represent the corresponding eigenvalues of eigenvector in dimension ; The first part (left side of the multiplication sign) represents the probability of the common features transferred from node to node and node , and the second part (right side of the multiplication sign) represents the probability of the common features transferred from node and node transferred out to node . The value range of is [0, 1]. The larger the value, the higher the probability that node transfers features to node

[0076] (2) Modularity extension based on tensors and regularization

[0077] Based on the third - order tensor representation, a weighted modularity is defined, and a regularization term is introduced to constrain the class cluster size distribution and the overlap ratio between class clusters, thereby constructing a clustering objective function.

[0078] Modularity can not only be used as an evaluation index to measure the quality of the discovered class cluster structure, but also be used as an optimization objective to achieve clustering by maximizing. When modularity is used as an optimization objective, there is a so - called "resolution limit" problem. The present invention solves this problem through a weighting strategy.

[0079] In this embodiment, the optimization objective is improved based on tensor construction. All slices of the tensor are included in the calculation, enabling the objective function to capture both directly connected edges (low order) and complex motifs (high order) simultaneously. A regularization constraint is introduced - by means of a penalty term to limit the expansion of cluster sizes (such as preventing a single community from expanding infinitely) and suppress excessive overlap (such as preventing a user from belonging to too many groups simultaneously), ensuring that the clusters have both topological compactness and reasonable feature distribution.

[0080] In one of the embodiments, the weighted modularity is defined as:

[0081] …………………………………(4)

[0082] In the formula, represents the total number of clusters, represents the th cluster; , are nodes in the cluster ; represents any one of the weighted adjacency matrix , the weighted feature matrix and the probability transition matrix , that is, , and are respectively substituted into for evaluation to obtain the weighted modularity indicators , , and (that is, in the global clustering objective function ); represents the association strength (whether topological, feature, or transition probability) between node and node in the corresponding weighted perspective; represents the weighted degree of node , that is: represents the weighted sum of node and all other nodes in the graph in the matrix , used to reflect the magnitude of its overall connection or similarity or transfer ability; represents the Inside, the proportion of the total weight of the actual connections between all node pairs in the entire graph weight sum can be understood as the edge weight density actually existing within the cluster, which is used to measure the tightness within the cluster. The larger this value, the denser the connections within the cluster, the more compact the structure, and the better the clustering effect. It represents the expected weight proportion of the edges within the cluster when nodes are probabilistically connected only according to their degrees (or weights) under the unbiased random connection model; the difference between the two represents whether the current clustering is tighter than the random partition, and the larger the value, the more concentrated and meaningful the cluster.

[0083] It does not consider the high-order information in the graph. Although the weighting strategy can be directly adjusted based on various motifs, such adjustments weaken the definition of the low-order information in the graph. Therefore, in this embodiment, the regularization term is improved by and extended to tensor optimization. The regularization term is defined as:

[0084] …………………………(5)

[0085] In the formula, represents the number of motif structures in the cluster ; represents the given attributed graph; represents the number of motifs in the entire graph after removing the cluster ; represents the th cluster; represents the cluster and the number of overlapping nodes with other clusters;

[0086] By fusing the weighted modularity and the regularization term , the global clustering objective function is obtained, which is expressed as:

[0087] …………………………………………(6)

[0088] In the formula, represents the number of slices in the third-order tensor representation, indicating observing the network structure from multiple perspectives (such as topology, features, feature transfer). In this embodiment ; , that is correspond to the topology slice, the feature slice, and the topology-feature transfer slice respectively; represents the weighted modularity from the perspective of slice , represents the slice Regularization terms from a perspective (in each slice under which the clusters have different structural manifestations. Therefore, the regularization terms also independently calculate depending on the structural or semantic consistency under each slice).

[0089] By and to declare the low - order and high - order characteristics of the clusters. Among them, the regularization terms further consider the scale and overlap of the clusters to suppress the formation of ultra - large - scale clusters and excessive overlap between clusters. It is not difficult to find from the definition of the optimization objective that does not directly assume a strong correlation between the topology and features of the clusters, but simultaneously considers node features and graph topology through the tensor paradigm, and strengthens the topological characteristics of the clusters through regularization terms. On the basis of ensuring the tightness of the cluster topology, the distribution of features is made more reasonable.

[0090] (3) Local optimization based on sequential game

[0091] Optimize the clustering objective function through sequential game, including: regarding the clusters as game players, the membership degree of nodes to the clusters as the strategy space, defining the player utility based on the clustering objective function, iteratively optimizing the cluster membership of nodes in the order of node influence, and outputting the final clustering structure when the player utility reaches the Nash equilibrium.

[0092] The optimization models of clustering can be divided into two categories according to whether they depend on the overall information of the graph: global models and local models. Global models analyze the overall information of the graph to discover the cluster structure. As the scale of the graph expands, it becomes more and more difficult to obtain the overall information. At the same time, the optimization process faces bottlenecks in memory and computing resources. Local models focus on discovering the cluster structure in the local area of the graph or the node neighborhood, which makes the optimization more efficient. On this basis, local models based on game theory generally convert the optimization process into a non - cooperative game process, and then, based on various characteristics of non - cooperative games, discover the clusters more accurately.

[0093] Local models based on non - cooperative games include three elements: game players, player strategies, and player benefits. Existing research generally focuses on node - level games, regarding nodes as game players, the candidate clusters of nodes as player strategies, and determining the cluster membership of nodes by defining and optimizing the benefit objectives of players. However, such modeling does not consider the influence of nodes and also ignores the interaction between clusters. Extensive research has shown that the formation of clusters is dominated by high - influence nodes. Determining the cluster membership of these nodes first helps to more accurately grasp the distribution of clusters. In addition, by considering the interaction between clusters, it helps to clarify the interaction between clusters of different scales, thus more comprehensively understanding the internal mechanism of the graph.

[0094] Therefore, to solve the problem of optimization efficiency, this embodiment proposes a sequential game mechanism: each cluster is defined as an independent player (such as a "research team"), whose strategy is to dynamically adjust the node membership (recruit or remove members), and the utility function directly reuses the improved modularity objective in the second stage (the global clustering objective function ). The optimization process is carried out in the order of node influence: first, determine the ownership of the hub nodes (for example, the joining of a big V in the social network drives the followers), and then each cluster takes turns to adjust its strategy - when a cluster expands, the competing clusters will re-evaluate the node allocation (for example, when Team A recruits new members, Team B will re-optimize its own structure). By replacing the global calculation with multiple rounds of local games, the efficiency is improved while maintaining the clustering accuracy (the measured F1 score is significantly improved), and finally, a clustering result that better fits the real scenario is output.

[0095] In a specific embodiment of the present invention, the game players are defined as , where represents the set of , , respectively represent the 1st, 2nd, and th game players, that is, each cluster corresponds to a game player; represents the set of all nodes in the attribute graph, represents that all clusters cover all nodes;

[0096] The player strategy is defined as , where represents the set of strategy spaces of , , respectively represent the membership degrees of all nodes to the 1st, 2nd, and th clusters, corresponding to the strategy spaces of the 1st, 2nd, and th game players;

[0097] The player utility is defined as , where represents the set of utility functions of game players, and any utility function

[0098] ……………………………(7)

[0099] …………………………………………(8)

[0100] Based on the utility function, the benefits of player actions (cluster growth, overlap between clusters, cluster shrinkage) are measured. In the formula, is a cluster The modularity score under a given matrix such as , , ), is the local (by cluster) contribution term of the global , that is is the result of a cluster, is the result of all clusters, is for all summation; is the modularity score of cluster from the perspective of slice .

[0101] In each iteration, all clusters are traversed, and the change in the utility function after each candidate node joins any cluster is calculated , represents the number of iteration rounds. If , then the candidate node is added to the corresponding cluster and the cluster strategy is updated;

[0102] After each iteration ends, the global clustering objective function is calculated. If the difference between the global clustering objective function and the previous round is less than the preset threshold, or the maximum number of iterations is reached, then the iteration is terminated;

[0103] When the iteration terminates, the player utilities of all clusters no longer increase, and the clustering result converges to the Nash equilibrium state, and the final node cluster division is output.

[0104] As Figure 2 , Figure 3 shown, this is a specific embodiment of the present invention, Figure 2 which schematically shows the whole process from data input, tensor modeling to sequential game optimization output in dealing with the attribute graph clustering task in this embodiment. Specifically, it includes the following steps:

[0105] 1. Input stage: Given an attribute graph (including 9 nodes, represented by numbers 1-9, there are 13 edges between the nodes, each node has 3 features, and the value of each feature is 0 or 1), a motif (such as a triangle), and the number of clusters (such as 2).

[0106] 2. Attribute graph tensor construction:

[0107] (1) Construct a 9*9 weighted adjacency matrix (blue): First, initialize all matrix elements to 0; then, traverse the edge information between nodes and assign values to the matrix elements. For example, if there is an edge between node 1 and node 2, the element in the first row and second column of the matrix is assigned 1, and so on. Store the 13 edge information of 9 nodes in the matrix; finally, based on the given motif (triangle), adjust the values of the elements in the matrix according to formula (1). For example, there are 3 triangles containing node 1 (123, 124, 134), 3 triangles containing node 2 (213, 214, 234), and 2 triangles containing the edge between node 1 and 2 (123, 124). Therefore, the value of the element in the first row and second column of the matrix is adjusted to 2*2 / 3 + 3 = 0.67, and so on, to form a topological slice.

[0108] (2)Construct a 9*9 weighted feature matrix (red): Calculate the feature similarity between any two nodes based on formula (2) and assign values to the matrix elements. For example, the feature of node 1 is (100), and the feature of node 2 is (101). Then, based on the Jaccard coefficient, ValSum = 0.33, NulSum = 0.33, NonSum = 0.33. Therefore, the value of the element in the first row and second column of the matrix is 0.33 + 0.33 - 0.33 = 0.33, and so on, to form a feature slice.

[0109] (3)Construct a 9*9 probability transition matrix (green): Calculate the topological-feature transition probability between any two nodes based on formula (3). For example, the feature of node 1 is (100), and the feature of node 2 is (101). There is 1 common feature, and the total number of features of node 1 is 3. The probability from node 1 to the common feature is 1 / 3 = 0.33. Similarly, the probability from the common feature to node 2 is 1 / 3 = 0.33. Therefore, the value of the element in the first row and second column of the matrix is 0.33 * 0.33 = 0.11, and so on, to form a topological-feature transition slice.

[0110] Merge the three slices to obtain a tensor representation of the attribute graph with a dimension of 9*9*3, where the element in the (i, j, k) dimension represents the similarity between the i-th node and the j-th node in the k-th dimension.

[0111] 3. Sequential game clustering

[0112] (1)According to the given motif (triangle) and the number of clusters (2), initialize 2 clusters, and each cluster randomly contains 2 (the number of motif nodes - 1) nodes. For example, C1 = {1, 2}, C2 = {8, 9}.

[0113] (2) Based on formulas (7) and (8), calculate the utility of other nodes joining C1 and C2. For example, the utility of node 3 joining C1 is 0.24 and the utility of joining C2 is 0, so node 3 joins C1.

[0114] And so on, continuously iterate until the termination condition of the algorithm is met (the clusters no longer change, the number of iterations = 20 times).

[0115] 4. Output: The result of attribute graph clustering. For example, C1 = {1, 2, 3, 4, 5}, C2 = {6, 7, 8, 9}.

[0116] Figure 3 For Figure 2 the pseudo-code flowchart corresponding to the "sequential game clustering" step, which details the game-based iterative strategy of this embodiment for optimizing the tensor clustering objective function, aiming to locally optimize the community membership of nodes in the attribute network through the mechanism of sequential games.

[0117] First, the algorithm initializes the number of iterations , and constructs the tensor representation of the attribute network according to formulas (1)-(3). Subsequently, set a number of game players, and each player corresponds to a community. Initially, each community randomly joins a number of nodes to form the initial community set .

[0118] In the strategy space, each player has a strategy space of length , and its elements represent the membership degrees of each node belonging to the community . At initialization, set the overall benefit placeholder variable to 0; then, calculate the benefit of each community under the initial community partition according to formula (8) , and take their sum as the overall benefit of the first round of iteration . .

[0119] Subsequently, the algorithm enters the main loop, and the iteration conditions are that the benefit improvement amplitude is not less than 0.005 and the number of iterations does not exceed 20 times. In each round of iteration, first determine the update order of communities from large to small according to the current benefit , and then optimize each player in turn. Traverse all strategies within its strategy space , and recalculate the new benefit brought by the node joining the community If the new benefit is not lower than the old benefit (i.e., ), then update the current strategy .

[0120] After the strategy update is completed, summarize the new benefits of all communities to update the overall benefit , and at the same time increment the iteration count by 1. Then, further check each strategy . If the membership degree of a certain node to the community satisfies , that is, its membership degree exceeds the reciprocal of the number of neighbors, then officially incorporate this node into the community .

[0121] The entire process is repeated until the benefit improvement is less than the set threshold or the maximum number of iterations is reached, thereby outputting the community membership matrix of the nodes , achieving the final optimized result of community division, which is consistent with Figure 2 the clustering division on the right.

[0122] In the above embodiments, it can be implemented in whole or in part by software, hardware, firmware, or any other combination. When implemented using software, it can be implemented in whole or in part in the form of a computer program product, which includes one or more computer instructions. When loading and executing the computer program instructions on a computer, the processes or functions according to the present application are generated in whole or in part. The computer can be a general-purpose computer, a special-purpose computer, a computer network, or other programmable devices. The computer instructions can be stored in a computer-readable storage medium, or transmitted from one computer-readable storage medium to another.

[0123] In addition, in each embodiment of the present application, the functional units can be integrated into one processing module, or each unit can exist physically alone, or two or more units can be integrated into one module. The above integrated module can be implemented in the form of hardware or in the form of a software functional module. When the above integrated module is implemented in the form of a software functional module and sold or used as an independent product, it can also be stored in a computer-readable storage medium. This storage medium can be a read-only memory, a magnetic disk, or an optical disc, etc.

[0124] Therefore, an embodiment of the present invention provides a computer device, which includes: a memory for storing computer-executable instructions; and one or more processors for executing the computer-executable instructions to implement the steps of an attribute graph clustering method based on tensor modeling and sequential games.

[0125] An embodiment of the present invention further provides a computer-readable storage medium, on which a computer program is stored, and when the computer program is executed by a processor, the steps of an attribute graph clustering method based on tensor modeling and sequential game are implemented.

[0126] In summary, the present invention has achieved significant improvements in aspects such as clustering accuracy, model stability, and structural interpretability, and is applicable to structural mining tasks of complex attribute graphs such as social networks, biological graphs, and recommendation systems, having good theoretical value and broad application prospects.

[0127] The above is only the specific implementation manner of the present application, but the protection scope of the present application is not limited thereto. Any person skilled in the art within the technical scope disclosed by the present application can easily think of various changes or substitutions thereof, and these should all be covered within the protection scope of the present application. Therefore, the protection scope of the present application should be subject to the protection scope of the claims.

Claims

1. An attribute graph clustering method based on tensor modeling and sequential game, characterized in that The method includes: Constructing a third-order tensor representation of the attribute graph by combining topological slices, feature slices, and topological-feature transfer slices to achieve joint representation learning of node structure information and attribute features; The topological slice is used to quantify the topological association strength between nodes based on the motif structure, the feature slice is used to measure the similarity of nodes in the attribute feature space, and the topological-feature transfer slice is used to calculate the probability strength of eigenvalue transfer between nodes based on common non-empty feature dimensions; Based on the third-order tensor representation, a weighted modularity is defined, and a regularization term is introduced to constrain the class cluster size distribution and the overlap ratio between class clusters, thereby constructing a global clustering objective function; Optimizing the clustering objective function through sequential games, including: regarding the class clusters as game players, the membership of nodes to the class clusters as player strategies, defining player utilities based on the global clustering objective function, and iteratively optimizing the class cluster membership of nodes in the order of node influence. When the player utilities reach the Nash equilibrium, the final clustering result is output.

2. The attribute graph clustering method based on tensor modeling and sequential game according to claim 1, wherein The topological slice is defined as a weighted adjacency matrix , where represents the topological association strength between node and node based on the predefined motif structure; is a set of real numbers, is the number of nodes in the attributed graph; For any given pair of nodes , if there is no connection relationship between node and node , then ; Conversely, if there is an edge between node and node , then based on the predefined motif structure, calculate the topological correlation strength between node pairs . The formula is as follows: ； In the formula, represents the number of motifs jointly participated by node pairs; represents the number of motifs individually participated by node represents the number of motifs individually participated by node.

3. A method for attribute graph clustering based on tensor modeling and sequential game according to claim 2, characterized in that, The feature slice is defined as a weighted feature matrix , where represents the similarity weight of node and node on the attribute features; For any given pair of nodes , the feature vectors corresponding to node and node are respectively denoted as and . According to whether the eigenvalues corresponding to the feature vectors and in the corresponding dimensions are empty, it is divided into three cases: neither eigenvalue is empty, only one of the eigenvalues is empty, and both eigenvalues are empty; The similarity weight is calculated by the following formula: ； In the formula, represents the feature similarity calculated when all feature values are not empty. Among them, if the feature values are of boolean type {0, 1}, the Jaccard index is used to measure the feature similarity; if the feature values are real-time values [0, 1], the cosine similarity is used to measure the feature similarity; represents the proportion of the dimensions where only one party of the feature values is empty in the total number of feature dimensions; represents the proportion of the dimensions where all feature values are empty in the total number of feature dimensions.

4. A method for attribute graph clustering based on tensor modeling and sequential game according to claim 3, characterized in that, The topological-feature transfer slice is defined as a probability transfer matrix , where represents the probability intensity that, on the premise that there is a common non-empty feature dimension between node and node , the eigenvalue transfers from node to node . For any given pair of nodes , if there are the same dimensions with non-empty eigenvalues between the eigenvectors and , then the node can be transferred to the node through features, and the calculation formula of the probability intensity is as follows: ； wherein, respectively represent the eigenvectors in the dimensions where the eigenvalues are not empty, respectively represent the eigenvectors in the dimension of the corresponding eigenvalues, respectively represent the eigenvectors in the dimension of the corresponding eigenvalues.

5. A method for attribute graph clustering based on tensor modeling and sequential game according to claim 4, characterized in that The weighted modularity is defined as: ； In the formula, represents the total number of clusters, represents the -th cluster; , are nodes in the cluster ; represents any one of the weighted adjacency matrix , the weighted feature matrix and the probability transition matrix ; represents the association strength between the node in the matrix and the node and the node from the corresponding weighted perspective; represents the weighted degree of the node ; The regularization term is defined as: ； In the formula, represents the number of motif structures in the cluster; represents a given attribute graph; represents the th cluster; Combine the weighted modularity with the regularization term to obtain the global clustering objective function , expressed as: ； wherein, represents the number of slices in the third-order tensor representation, ; , correspond to the topological slice, the feature slice, and the topological-feature transfer slice respectively; represents the weighted modularity from the perspective of slice , represents the regularization term from the perspective of slice .

6. The attribute graph clustering method based on tensor modeling and sequential game according to claim 5, characterized in that The game players are defined as , where represents the set of game players, , , respectively represent the 1st, 2nd, th game players, represents the set of all nodes in the attribute graph, represents that all clusters cover all nodes; The player strategy is defined as , where denotes the set of strategy spaces of , , respectively denote the membership degrees of all nodes to the 1st, 2nd, th clusters, corresponding to the strategy spaces of the 1st, 2nd, th players; The player utility is defined as , where represents the set of utility functions of game players, and any utility function ； ； In the formula, is the modularity score of the cluster under the given matrix ; is the modularity score of the cluster from the perspective of the slice ; In each iteration, traverse all clusters and calculate the change in the utility function after each candidate node joins any cluster , denotes the iteration number. If , then add the candidate node to the corresponding cluster and update the cluster strategy; After each iteration, calculate the global clustering objective function , if the difference between the global clustering objective function and that of the previous round is less than the preset threshold, or the maximum number of iterations is reached, then terminate the iteration; When the iteration terminates, the player utilities of all class clusters no longer increase, and the clustering result converges to the Nash equilibrium state, and the final node class cluster division is output.

7. A computer device, characterized in that, The device includes: A memory for storing computer-executable instructions; One or more processors for executing the computer-executable instructions to implement an attribute graph clustering method based on tensor modeling and sequential games as described in any one of claims 1-6.

8. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by the processor, it implements an attribute graph clustering method based on tensor modeling and sequential games as described in any one of claims 1 to 6.

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