Two-stage influence maximization method based on graph neural network
By employing a two-stage approach using graph neural networks, combining a global scorer and a local selector model, the problem of efficient and accurate identification of key nodes in complex networks is solved. This approach achieves a combination of influential nodes with low redundancy and high synergy, thereby expanding the scope of information dissemination.
Patent Information
- Application Number
- CN202511134575.7
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-08-14
- Publication Date
- 2025-11-18
AI Technical Summary
Existing technologies struggle to efficiently and accurately identify key, dispersed nodes with complementary propagation capabilities in complex networks, leading to issues of redundancy in propagation range and seed selection.
A two-stage influence maximization method based on graph neural networks is adopted. By combining SIR diffusion scores, 3D bridge scores and composite features through a global scorer and a local selector model, a combination of influence nodes with low redundancy and high synergy is selected.
It enables efficient and accurate identification of key nodes with dispersed topologies and complementary influences in large-scale networks, thereby maximizing the reach of information dissemination.
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Figure CN120974672A_ABST
Abstract
Description
TECHNICAL FIELD
[0001] The application belongs to the technical field of complex network information propagation and influence diffusion modeling, and particularly relates to a two-stage influence maximization method based on a graph neural network. BACKGROUND
[0002] Influence maximization (IM) is a core problem in network science, which aims to find a small group of key "seed nodes" in the network to trigger the maximum range of information propagation, and has great application value in viral marketing, public opinion monitoring, disease prevention and control, etc. However, the existing technology faces the contradiction between efficiency, accuracy and redundancy elimination when solving this problem.
[0003] The contradiction between efficiency and accuracy: traditional methods can be roughly divided into two categories. One is the local heuristic algorithm represented by degree centrality, which has fast calculation speed (close to linear time), but lacks a global perspective, so the identified node influence is often inaccurate. The other is the global algorithm represented by betweenness centrality or greedy strategy-based algorithm, which has good effect, but usually needs to calculate the shortest path of all node pairs in the network or perform a large number of Monte Carlo simulations, with extremely high computational complexity (such as O(n*m) or higher), which is almost impossible to implement on large-scale networks (such as millions of nodes).
[0004] Influence overlap problem (redundancy problem): Even some better-performing algorithms often select nodes that are highly clustered in network topology when selecting Top-K influence nodes. These nodes have high influence themselves, but their influence ranges overlap a lot, resulting in the "1+1<2" effect, and the final propagation effect of the entire seed set is greatly reduced. How to select a group of nodes that have high influence, are dispersed in structure, and can form complementary propagation is a big difficulty in the existing technology.
[0005] Limitations of learning methods: In recent years, methods based on machine learning and graph neural networks (GNN) have shown potential, but still have many shortcomings. For example, many methods need expensive global features (such as network disassembly scores) as labels, and the computational cost is still high; or rely on training on the target network, lack of inductive capability, i.e. cannot directly reason quickly on a completely new, unseen network; in addition, most models are still simple regression or ranking models, and have not solved the influence overlap problem from the mechanism.
[0006] Therefore, there is an urgent need for a new influence node identification method that can meet the following four ideal standards: (1) induction, which can directly predict new networks; (2) comprehensiveness, which can perceive the global and local characteristics of nodes; (3) low redundancy, which can suppress the overlap of influence between seed nodes; (4) high efficiency, which can process large-scale networks. SUMMARY
[0007] The core problem to be solved by the present application is to efficiently and accurately identify a group of key influence nodes that are dispersed and have complementary propagation capabilities in a complex network, thereby balancing the maximization of propagation range and the non-redundancy of seed selection. Traditional methods are difficult to achieve both efficiency and accuracy, and have not effectively avoided the overlap of influence between high-impact nodes. Therefore, the present application proposes a two-stage influence maximization method based on graph neural network to coordinate the two links of global influence prediction and local fine selection, and improve the performance of key node discovery in large-scale networks.
[0008] The present application provides a two-stage influence maximization method based on graph neural network, comprising the following steps:
[0009] S1. Input the target network, extract the local structure features of each node of the target network and normalize them;
[0010] S2. Construct a global scorer model based on a graph attention network GAT, input the normalized features of step S1 into the global scorer model, and regress to obtain SIR diffusion scores and 3d bridge scores, and then obtain a comprehensive score by weighting according to a preset weight;
[0011] The SIR diffusion score is used to predict the information propagation potential of the node under the SIR model;
[0012] The 3d bridge score is used to evaluate the "structural bridge" role played by the node in the network;
[0013] S3. Select the top M nodes in descending order of the comprehensive score to form a candidate node set;
[0014] S4. Extract the candidate subgraph and the composite feature;
[0015] Extract the candidate subgraph based on the nodes and node relationships in the candidate node set, and construct a composite feature containing the comprehensive score, global degree, local degree and degree ratio for each node based on the candidate subgraph;
[0016] S5. Construct a local selector model based on a graph attention network GAT, input the composite feature constructed in step S4 into the local selector model, and the local selector model outputs the probability of the node being selected as a seed;
[0017] S6. Select the top K nodes in descending order of the probability as the final seed set to achieve influence maximization.
[0018] Further, in the step S1, the local structural features of each node of the target network include the degree of the node, the number of two-hop neighbors, and the average neighbor degree.
[0019] Further, in the step S2, the 3D-Bridging score, i.e., the three-dimensional bridging centrality score, is the product of the weighted community entropy, the betweenness centrality, and the structural vulnerability, and the calculation expression is as follows:
[0020] C 3DB (v)=WCE(v)×CB(v)×SF(v)
[0021]
[0022]
[0023] In the formula, C 3DB (v) represents the three-dimensional bridging centrality of node v; WCE(v) represents the weighted community entropy of node v; CB(v) represents the betweenness centrality of node v; SF(v) represents the structural vulnerability of node v; represents the proportion of nodes belonging to community c in the neighbors of node v, and is normalized and weighted by the overall size of the community; n c (v)=|N(v)∩C c | is the number of nodes belonging to community c in the neighbors of node v; |C c | is the total number of nodes of community c in the entire graph; c′ represents a community in the network; n c′ (v) represents the number of nodes belonging to community c′ in the neighbors of node v; |C c′ | represents the total number of nodes of community c′; σ st (v) represents the number of shortest paths from node s to node t; C(v) is the local clustering coefficient of node v, defined as the ratio of the number of connection edges between the neighbors of node v to the maximum possible number of connection edges.
[0024] Further, the global scorer model in the step S2 includes the following two parallel regression tasks: predicting the SIR model propagation score of the node; predicting the 3D bridging centrality score of the node;
[0025]
[0026] In the formula, represents the predicted value of the SIR influence score of node i by the global scorer model, i.e., the SIR diffusion score; represents the predicted value of the 3D-Bridging score of node i by the global scorer model, i.e., the 3D bridging score; denotes the transpose of the weight parameters of a multi-layer perceptron, MLP, in the SIR diffusion score prediction task; MLP s (h i′ ) denotes a specific multi-layer perceptron network for SIR diffusion score prediction with input being the feature vector h i′ ; b s denotes the bias parameter in the SIR diffusion score prediction task; denotes the transpose of the weight parameters of a multi-layer perceptron, MLP, in the 3D bridge centrality score prediction task; MLP b (h i′ ) denotes a specific multi-layer perceptron network for 3D bridge centrality score prediction with input being the feature vector h i′ ; b b denotes the bias parameter in the 3D bridge centrality score prediction task;
[0027] The loss functions of the two regression tasks of the global scorer model are as follows:
[0028]
[0029] where L sir denotes the loss function of SIR diffusion score; L bri denotes the loss function of 3d bridge score; N denotes the total number of nodes;
[0030] The loss weighting fusion obtains the final multi-task joint loss function L total ;
[0031] L total = L sir + λ·L bri
[0032] where λ denotes a hyperparameter.
[0033] Further, the local selector model in the step S5 introduces a composite loss function for guiding the local selector model to achieve dynamic balance between influence and structural diversity, a degree penalty for inhibiting nodes with sparse connections, and a diversity constraint for punishing node combinations that are too clustered in topology.
[0034] where the expression of the composite loss function L local is:
[0035] L local = L BCE + λ degree ·L degree + λ diversity ·L diversity
[0036] where L BCE is the binary cross-entropy loss; λ degree is the hyper-parameter of the degree correlation loss; L degree is the degree correlation loss; λ diversity is the hyper-parameter of the diversity loss; L diversity is the diversity loss;
[0037]
[0038] where LND i is the local neighborhood degree of node i; C represents the candidate node set in the second stage; |C| represents the number of nodes in set C; Norm(LND i ) represents the normalization processing of the local neighborhood degree; and a represents the related parameter. represents the predicted score of node i.
[0039]
[0040] where f penalty (·) represents the piecewise penalty function. represents the expectation of all node pairs (u, v) in the Top-K pred set, that is, the average is calculated; and dist(u, v) represents the distance between nodes u and v.
[0041] Further, the piecewise penalty function can depict the structural redundancy degree of two predicted nodes according to the shortest path distance between any two predicted nodes; the expression of the piecewise penalty function f penalty (d) is:
[0042]
[0043] where d represents the shortest path distance between any two predicted nodes.
[0044] Further, the local selector model in the step S5 is also provided with a diversity-aware greedy algorithm for generating higher-quality training labels.
[0045] v * = argmax v∈C、S [(1-λ dis )·Influence(v)+λ dis ·min s∈S dist(v, s)]
[0046] where v * represents the optimal seed node selected in this round; C represents the candidate node set; S represents the selected seed node set; λ disdenotes the diversity weight, Influence(v) denotes the individual influence of node v, and dist(v,s) denotes the topological distance between node v and the selected node s.
[0047] Advantages:
[0048] The application discloses a two-stage influence maximization method based on a graph neural network, and proposes a "global rough screening-local fine selection" framework. In the first stage, a global score model is constructed, the propagation potential of a node and 3D bridge centrality proposed by the application are jointly learned through a graph attention network, and a candidate node set is screened out with near-linear time complexity. In the second stage, a diversity-aware local selector model is designed, and the final seed node set is selected by using the global information in the first stage to fine rearrange the candidate subgraph, so that the final seed node set is topologically dispersed and has complementary influence. The application can efficiently and accurately identify the low-redundancy and high-collaborative-effect influence node combination, and has inductive ability.
[0049] Other advantages, objects, and features of the application will be set forth in part in the following specification taken in conjunction with the accompanying drawings, and in part will become apparent to those skilled in the art from a consideration of the following specification and drawings, or can be learned from the practice of the application. The advantages and other objects of the application can be realized and attained by the instrumentalities set forth in the following specification. BRIEF DESCRIPTION OF DRAWINGS
[0050] Figure 1 A flowchart of the two-stage influence maximization method based on the graph neural network of the application;
[0051] Figure 2 A network structure diagram for explaining that the 3D bridge centrality can effectively identify the real bridge node in the embodiment of the application. DETAILED DESCRIPTION
[0052] In order to make the technical solutions, advantages and purposes of the application clearer, the technical solutions of the embodiments of the application will be described clearly and completely below in conjunction with the drawings of the embodiments of the application. Obviously, the described embodiments are part of the embodiments of the application, rather than all the embodiments. Based on the described embodiments of the application, all other embodiments obtained by those skilled in the art without creative labor fall within the protection scope of the present application.
[0053] As shown in Figure 1 The application provides a two-stage influence maximization method based on a graph neural network, which comprises the following steps:
[0054] S1. Input the target network, extract the local structure features of each node of the target network and normalize them;
[0055] Local structural features of target network nodes include the degree of the node, the number of two-hop neighbors, and the average neighbor degree;
[0056] S2. A global scorer model is constructed based on a graph attention network (GAT), and the normalized features from step S1 are input into the global scorer model to regress SIR diffusion scores and 3d bridge scores, and then a comprehensive score is obtained by weighting according to a preset weight;
[0057] The global scorer model learns a deep feature representation for each node that can fuse the neighborhood information of the node through a message passing mechanism. Given a graph G=(V,E), each node vi has an initial feature vector GAT first maps the initial feature vector of each node vi to a low-dimensional space through a linear transformation :
[0058] h i ′=Wh i
[0059] Subsequently, for each pair of connected nodes (i,j)∈E, the attention weight of node i to neighbor node j is calculated:
[0060] e ij =LeakyReLU(a[h i ′||h′ j ])
[0061] where is a learnable attention weight vector, and || represents a feature concatenation operation. Then, the neighbor attention coefficients are normalized by softmax:
[0062]
[0063] Finally, the representation of node i is updated by aggregating neighbor information:
[0064]
[0065] where σ(·) represents an activation function, and N(i) is the neighbor set of node i. GAT usually adopts a multi-head attention mechanism, that is, K independent attention channels are executed in parallel, and are spliced or averaged in the output layer:
[0066]
[0067] where ij (k) represents the attention weight from neighbor node v j to target node v i , and W(k) is a learnable weight matrix of the kth layer.
[0068] The global scorer computes two key scores for each node in parallel based on the learned node representations, through a multi-task learning framework. SIR spread score, which is used to predict the information propagation potential of a node under the SIR (Susceptible-Infectious-Recovered) model. 3D bridge score, which is used to assess the "structural bridge" role a node plays in the network.
[0069] 3D bridge score is a key technology of the invention, aiming to solve the limitations of traditional centrality indicators. Referring to Figure 2 , the figure shows an example network containing two communities. Among them, node 8 is located inside the left community, but due to its connection to many (such as connections 0, 1, 2, 3), traditional betweenness centrality may mistakenly give it a high score, identifying it as a "betweenness-identified" core node. However, the real structural bridge in the network is node 9, which is the only channel connecting the left and right communities, and its importance is self-evident. The 3D bridge centrality proposed by the invention can accurately give node 9 a high score by comprehensively evaluating the cross-community connectivity (such as weighted community entropy), path intermediacy, and local structural vulnerability of the node, while identifying node 8 as only an "intra-community core node", thus achieving more accurate bridge role identification.
[0070] The 3D bridge score in step S2 is the three-dimensional bridge centrality score, which is the product of weighted community entropy, betweenness centrality, and structural vulnerability. The calculation expression is as follows:
[0071] C 3DB (v) = WCE(v) x CB(v) x SF(v)
[0072] Where WCE measures the uniformity of a node's neighbors' distribution in different communities, and is weighted by the relative size of the community. The more different communities and the larger the size of the community in which a node's neighbors are distributed, the higher the WCE value, indicating the stronger cross-community connectivity. The specific definition is as follows:
[0073]
[0074] CB(v) represents the betweenness centrality of node v, which describes the extent to which a node is located on the shortest path between other nodes in the network. Nodes with high betweenness centrality are often bottlenecks or intermediaries of information propagation, and play an important role in controlling network information flow, traffic, or virus spread. The formal definition is as follows:
[0075]
[0076] Structural Fragility (SF) measures the sparsity of internal connections among the neighbors of a node, i.e., whether the node is connected to a loosely structured and fragile local network structure. It is defined as:
[0077]
[0078] where C 3DB (v) represents the 3D bridging centrality of node v; WCE(v) represents the weighted community entropy of node v; CB(v) represents the betweenness centrality of node v; SF(v) represents the structural fragility of node v; represents the proportion of nodes in community c among the neighbors of node v, and is normalized by the overall size of the community; n c (v) = |N(v)∩C c | is the number of nodes in community c among the neighbors of node v; |C c | is the total number of nodes in community c in the entire graph; c' represents a community in the network; n c′ (v) represents the number of nodes in community c' among the neighbors of node v; |C c′ | represents the total number of nodes in community c'; σ st (v) represents the number of shortest paths from node s to node t; C(v) is the local clustering coefficient of node v, defined as the ratio of the number of connecting edges between the neighbors of node v to the maximum possible number of connecting edges. The fewer the connections between the neighbors of a node, the lower the clustering coefficient, and the more "fragile" the structure; therefore, the smaller C(v) is, the larger SF(v) is. This index reflects the potential structural damage to the neighborhood of node v after removing node v.
[0079] The global scorer model in step S2 includes the following two parallel regression tasks: predicting the SIR model propagation score of the node; predicting the 3D bridging centrality score of the node;
[0080]
[0081] where, represents the predicted value of the SIR influence score of node i by the global scorer model, i.e., the SIR diffusion score; represents the predicted value of the 3D-Bridging score of node i by the global scorer model, i.e., the 3d bridge score; represents the weight parameter transpose of the multi-layer perceptron MLP in the SIR diffusion score prediction task; MLP s (h i′ ) represents a specific multi-layer perceptron network used for SIR diffusion score prediction, whose input is the feature vector h i′ of node i; b sdenotes the bias parameter in the SIR diffusion score prediction task; denotes the weight parameter transpose of the multi-layer perception (MLP) in the 3D bridge centrality score prediction task; MLP b (h i′ ) denotes a specific multi-layer perception network for 3D bridge centrality score prediction, the input of which is the feature vector h i′ ; b b denotes the bias parameter in the 3D bridge centrality score prediction task;
[0082] The loss functions of the two regression tasks of the global scorer model are as shown in the following formula, which will jointly act in the subsequent multi-task joint optimization framework to realize the collaborative learning of the evaluation of node influence and bridging.
[0083]
[0084] In the formula, L sir denotes the loss function of the SIR diffusion score; L bri denotes the loss function of the 3D bridge score; N denotes the total number of nodes;
[0085] In order to realize the collaborative optimization of the two tasks, the above losses are weighted and fused to obtain the final multi-task joint loss function L total : L total = L sir + λ·L bri .
[0086] Wherein, the hyperparameter λ∈R + is used to adjust the relative weight between the two tasks, and the optimal value thereof is usually determined through cross-validation or grid search.
[0087] S3. Select the top M nodes in descending order of the comprehensive score to form a candidate node set;
[0088] According to the comprehensive score, all nodes are sorted, and the top M nodes with the highest score are selected to form a candidate node set (pool) C for use in the next stage. Wherein, M is a preset constant to ensure that the size of the candidate node set is much smaller than the number of original network nodes.
[0089] S4. Extract the candidate subgraph and the composite feature;
[0090] Based on the nodes and node relationships in the candidate node set, a candidate subgraph is extracted, and based on the candidate subgraph, a composite feature including the comprehensive score, global degree, local degree, and degree ratio is constructed for each node;
[0091] S5. Construct a local selector model based on a graph attention network (GAT), input the composite feature constructed in step S4 into the local selector model, and the local selector model outputs the probability of a node being selected as a seed.
[0092] The local selector model selects the final K seed node set S by reordering the candidate node subgraph * For each node v i in the candidate subgraph, the local selector model outputs a logit value, which is mapped by the Sigmoid function to generate the predicted probability of the node being selected into the Top-K set , whose supervised label is a binary variable y i ∈{0,1}.
[0093] The local selector model in step S5 introduces a composite loss function, as well as two structured regularization terms of degree penalty and diversity constraint, the composite loss function is used to guide the local selector model to achieve dynamic balance between influence and structural diversity, the degree penalty is used to suppress nodes with sparse connections, and the diversity constraint is used to punish node combinations that are too clustered in topology;
[0094] Wherein, the expression of the composite loss function L local is:
[0095] L local =L BCE +λ degree ·L degree +λ diversity ·L diversity
[0096] In the formula, L BCE is the binary cross-entropy loss; λ degree is the hyperparameter of the degree-related loss; L degree is the degree-related loss; λ diversity is the hyperparameter of the diversity loss; L diversity is the diversity loss
[0097] Under the guidance of multi-task learning and diversity-aware labels, the model is encouraged to preferentially select key nodes with bridging characteristics and topological dispersion. However, this design may introduce new potential bias: the model may mistakenly judge some nodes that are extremely sparse in themselves and their neighborhoods as “bridge nodes”, although such nodes have dispersion in structure, they often lack sufficient connection ability and local influence, and it is difficult to form effective individual spreading capability in the early stage of information propagation. Therefore, we introduce a structure-aware regularization term L degree , which aims to punish nodes with high predicted scores but in extremely sparse areas.
[0098]
[0099] In the formula, LNDi LNDiis the local neighborhood degree of node i, defined as the sum of the node's own degree and its first-order neighbor's degree; C represents the candidate node set in the second stage; |C| represents the number of nodes in set C; Norm(LND i ) represents the normalization processing of the local neighborhood degree; a represents a related parameter; represents the predicted score of node i;
[0100]
[0101] In the formula, f penalty (·) represents a piecewise penalty function; represents the expectation of all node pairs (u, v) in the Top-K pred set, that is, the average is calculated; dist(u, v) represents the distance between nodes u and v.
[0102] Among them, the piecewise penalty function can depict the structural redundancy degree of two predicted nodes according to the shortest path distance between any two predicted nodes; the piecewise penalty function f penalty (d) is expressed as a table:
[0103]
[0104] In the formula, d represents the shortest path distance between any two predicted nodes.
[0105] In order to solve the influence overlap problem from the root, we do not use the traditional greedy algorithm which only focuses on marginal gain to generate training labels. Because such "ground truth" labels are topologically aggregated themselves, which will give the model a wrong guide. Instead, we design and implement a diversity-aware greedy algorithm to generate higher quality training labels. This algorithm draws on the idea of maximizing marginal relevance (MMR) in the field of information retrieval. When selecting each seed node, the algorithm not only considers the individual influence (relevance) of the candidate node, but also considers the topological distance (diversity) between it and the selected seed set
[0106] v * = argmax v∈C、S [(1- l dis )·Influence(v) + l dis ·min s∈S dist(v, s)]
[0107] In the formula, v * represents the optimal seed node selected in this round; C represents the candidate node set; S represents the selected seed node set; l disdenotes the diversity weight; Influence(v) denotes the individual influence of node v; dist(v, s) denotes the topological distance between node v and the selected node s.
[0108] S6. Select the top K nodes in descending order of probability as the final seed set, and realize influence maximization.
[0109] Local selector model output Then select the top K nodes in descending order as the final seed S * , and the propagation scale as the result output. It can run on a general CPU+GPU platform, and an 8GB graphics memory can process a graph with a scale of ten million edges; Python3.10, PyTorch2.1, and PyG2.5. The training model and the inference process can be containerized and deployed.
[0110]
[0111] In the formula, S * denotes the final seed set, which is obtained by Top-K selection; σ is a Sigmoid function; means that all candidate nodes v i in the candidate subgraph C local , select the top K nodes with the highest scores; GNN (·) is LocalSelector, a structure-aware local scorer, which receives the candidate subgraph G C and the node features x i as input, and outputs a logit value for each candidate node v i , which represents the potential confidence of the node being selected into the final seed set. G C represents the candidate subgraph, x i is the feature of node v i in the candidate subgraph.
[0112] The present application can be widely applied in the fields of viral marketing, public opinion monitoring, power grid resilience analysis, and biomolecular target screening, etc.
[0113] It is declared that the above embodiments are only used to illustrate the technical solutions of the present application and not to limit it. Although the present application has been described in detail with reference to the preferred embodiments, it should be understood by those skilled in the art that the technical solutions of the present application can be modified or replaced equivalently without departing from the purpose and scope of the present application, and they should be covered in the scope of the claims of the present application.
Claims
1. A two-stage influence maximization method based on graph neural networks, characterized in that, Includes the following steps: S1. Input the target network, extract the local structural features of each node in the target network and normalize them; S2. Construct a global scorer model based on the graph attention network GAT, input the normalized features from step S1 into the global scorer model, regress to obtain the SIR diffusion score and 3d bridge score, and then weight them according to preset weights to obtain the comprehensive score. SIR diffusion score is used to predict the information propagation potential of a node under the SIR model; 3D bridge scores are used to evaluate the "structural bridge" role that nodes play in the network. S3. Select the top M nodes in descending order of comprehensive score to form a candidate node set; S4. Extract candidate subgraphs and composite features; Candidate subgraphs are extracted based on the nodes and node relationships in the candidate node set. For each node, a composite feature including comprehensive score, global degree, local degree, and degree ratio is constructed based on the candidate subgraphs. S5. Construct a local selector model based on the graph attention network GAT. Input the composite features constructed in step S4 into the local selector model. The local selector model outputs the probability that a node is selected as a seed. S6. Select the top K nodes in descending order of probability as the final seed set to maximize influence.
2. The two-stage influence maximization method based on graph neural networks according to claim 1, characterized in that: In step S1, the local structural features of each node in the target network include the node's degree, the number of two-hop neighbors, and the average neighbor degree.
3. The two-stage influence maximization method based on graph neural networks according to claim 2, characterized in that, The 3D bridge score in step S2 is the three-dimensional bridging centrality score. The three-dimensional bridging centrality is the product of weighted community entropy, betweenness centrality, and structural vulnerability, and its calculation expression is as follows: C 3DB (v)=WCE(v)×CB(v)×SF(v) In the formula, C 3DB WCE(v) represents the three-dimensional bridging centrality of node v; WCE(v) represents the weighted community entropy of node v; CB(v) represents the betweenness centrality of node v; SF(v) represents the structural fragility of node v. This represents the percentage of nodes belonging to community c among the neighbors of node v, multiplied by the overall size of that community and then weighted normally; n c (v)=|N(v)∩C c | represents the number of neighbors of node v who belong to community c; |C c | represents the total number of nodes in community c in the entire graph; c′ represents a community in the network; n c′ (v) represents the number of nodes belonging to community c′ among the neighbors of node v; |C c′ | represents the total number of nodes in community c′; σ st (v) represents the number of shortest paths from node s to node t; C(v) is the local clustering coefficient of node v, defined as the ratio of the number of connecting edges between the neighbors of node v to the maximum possible number of connecting edges.
4. The two-stage influence maximization method based on graph neural networks according to claim 3, characterized in that, The global scorer model in step S2 includes the following two parallel regression tasks: predicting the SIR model propagation score of the node; and predicting the 3D bridging centrality score of the node. In the formula, This represents the predicted SIR influence score of node i by the global scorer model, i.e., the SIR diffusion score. This represents the predicted 3D-Bridging score for node i by the global scorer model, i.e., the 3D bridge score. This represents the transpose of the weight parameters of the Multilayer Perceptron (MLP) in the SIR diffusion score prediction task; MLP s (h i′ ) denotes a specific multilayer perceptron network used for SIR diffusion score prediction, whose input is the feature vector h of node i. i′ b s This represents the bias parameter in the SIR diffusion score prediction task; This represents the transpose of the weight parameters of a multilayer perceptron (MLP) in a 3D bridging centrality score prediction task; MLP b (h i′ ) represents a specific multilayer perceptron network used for 3D bridging centrality score prediction, whose input is the feature vector h of node i. i′ b b This represents the bias parameter in the 3D bridging centrality score prediction task. The loss functions for the two regression tasks of the global scorer model are as follows: In the formula, L sir The loss function representing the SIR diffusion score; L bri The loss function represents the score of the 3D bridge; N represents the total number of nodes; Loss weighting and fusion yields the final multi-task joint loss function L. total ; L total L sir +λ·L bri In the formula, λ represents the hyperparameter.
5. The two-stage influence maximization method based on graph neural networks according to claim 3, characterized in that, The local selector model in step S5 introduces a composite loss function, as well as two structured regularization terms: degree penalty and diversity constraint. The composite loss function is used to guide the local selector model to achieve a dynamic balance between influence and structural diversity. The degree penalty is used to suppress sparsely connected nodes, and the diversity constraint is used to penalize node combinations that are too clustered in the topology. Among them, the composite loss function L local The expression is: L local =L BCE +λ degree ·L degree +λ diversity ·L diversity In the formula, L BCE For binary cross-entropy loss; λ degree L is the hyperparameter of the degree-dependent loss. degree For degree-related loss; λ diversity L is a hyperparameter for diversity loss. diversity For the loss of diversity; In the formula, LND i It represents the local neighborhood degree of node i; C represents the candidate node set for the second stage; |C| represents the number of nodes in set C; Norm(LND) i ) indicates that the local neighborhood degree is normalized; α represents the relevant parameters; This represents the prediction score for node i; In the formula, f penalty (·) denotes the piecewise penalty function; Indicates Top-K pred The expected value of all nodes (u,v) in the set is calculated, i.e., the average value is calculated; dist(u,v) represents the distance between nodes u and v.
6. The two-stage influence maximization method based on graph neural networks according to claim 3, characterized in that: The piecewise penalty function can characterize the structural redundancy of two predicted nodes based on the shortest path distance between any two predicted nodes; the piecewise penalty function f penalty (d) The expression for table (d) is: In the formula, d represents the shortest path distance between any two predicted nodes.
7. The two-stage influence maximization method based on graph neural networks according to claim 3, characterized in that: The local selector model in step S5 also includes a diversity-aware greedy algorithm to generate higher-quality training labels. in * =argmax v∈C、S [(1-λ dis )·Influence(v)+λ dis ·min s∈S dist(v,s)] In the formula, v * C represents the optimal seed node selected in this round; S represents the set of candidate nodes; λ represents the set of selected seed nodes; dis The diversity weight is represented by Influence(v); the individual influence of node v is represented by dist(v,s); and the topological distance between node v and the selected node s is represented by dist(v,s).