Graph neural network-based crowd-sourcing high-order network critical threshold prediction method and device

Abstract

The invention belongs to the technical field of network analysis, and particularly relates to a crowd-sourcing high-order network critical threshold prediction method and device based on a graph neural network. The method comprises the following steps: integrating local structure features of nodes, position codes of positions of the nodes in a whole graph and parameters influencing the global state of a system, and providing comprehensive and three-dimensional cognition on the network for a model; in a network information processing level, a local and global mixed GNN architecture is adopted, and a GIN module and a global self-attention module are arranged in parallel: the former is used for accurately capturing a local neighborhood structure of a node, and the latter is used for capturing a long-distance dependency relationship in a whole graph range; by fusing the information of the two dimensions, the model can understand how the local structure affects the global dynamics more deeply. The problems that the prior art depends on dynamic evolution data, and critical threshold prediction cannot be accurately carried out on the high-order network are solved.

Description

Technical Field

[0001] This invention belongs to the technical field of swarm intelligence systems and network analysis, and more specifically, relates to a method and apparatus for predicting critical thresholds in high-order swarm intelligence networks based on graph neural networks. Background Technology

[0002] In swarm intelligence systems, when external conditions change slowly, the system may suddenly transition from a stable state to another state with drastically different characteristics—a phenomenon known as a "critical transition." Critical transitions often lead to severe consequences, such as the loss of coordination among drone swarms, sudden outbreaks of disease, and the large-scale spread of malicious information. Therefore, developing analytical methods capable of accurately identifying and predicting these transitions is crucial for preventing or mitigating their adverse impacts on ecosystems and human society. Currently, the mainstream technology for predicting critical transitions utilizes deep learning to analyze system evolution data.

[0003] The Chinese paper "Early predictor for the onset of critical transitions in networked dynamical systems" (Physical Review X, 2024, 14(3), 031009) proposes an early prediction method for critical transitions based on graph neural networks (GNNs) and gated recurrent units (GRUs). This method requires acquiring time-series data of the network system during its evolution, i.e., collecting the state information of each node in the network at multiple consecutive time points to form a time-ordered network snapshot. Subsequently, the GNN is used to process the network snapshot at each time point, aggregating information along the edges of the network to extract spatial topological features. These features are then sequentially input into the GRU for dynamic pattern analysis, thereby achieving the prediction of critical transitions.

[0004] In summary, firstly, the core module of existing technical solutions, GRU, heavily relies on complete dynamic evolution time-series data, making it unsuitable for common scenarios where only static network snapshots are available. Secondly, the core characteristic of swarm intelligence systems lies in the fact that their complex collective behaviors (such as propagation and synchronization) are typically driven by the collaborative effects between multiple individuals, rather than simple pairwise interactions. Higher-order networks are the key framework for mathematical abstraction and research of such systems. Existing GNN modules focus on aggregating information along edges, effectively modeling only pairwise interactions between nodes, but lack sufficient ability to represent higher-order interactions that drive system phase transitions. This limits their ability to be effectively used in higher-order networks for critical threshold prediction, thus restricting the model's prediction accuracy and scope of application. Summary of the Invention

[0005] The present invention aims to overcome at least one of the defects of the prior art and provide a critical threshold prediction method for high-order swarm intelligence networks based on graph neural networks, so as to solve the problem that the prior art relies on dynamic evolution data and cannot accurately predict the critical threshold of high-order networks.

[0006] The detailed technical solution of this invention is as follows: A method for predicting critical thresholds in a high-order swarm intelligence network based on graph neural networks, the method comprising: S1. Obtain the network to be processed and divide the dataset. Extract node features and global features from the input network, standardize the data, and calculate the position code.

[0007] S2. Map the obtained node features, global features, and positional codes to a high-dimensional space to form independent high-dimensional feature embeddings; then, concatenate and fuse these high-dimensional feature embeddings to form a comprehensive feature representation Z. S3. Input the comprehensive feature representation Z into the core processing module consisting of four stacked GPS layers. Update the state of the node itself using the graph isomorphic network module and the global self-attention module respectively. Then, information is fused by element-wise addition to obtain an enhanced representation. Finally, the final node embedding matrix is ​​obtained. S4. The final node embedding matrix is ​​aggregated into a graph-level representation vector, and then threshold prediction is performed to finally output the predicted critical threshold.

[0008] Furthermore, the node features include node degree, the average number of 2-simplexes in which the node participates, and the clustering coefficient; the global features include the initial infection ratio and higher-order infection parameters. The position encoding includes: obtaining the Laplacian eigenvector position encoding (LapPE) by calculating the eigenvectors of the graph Laplacian matrix; and obtaining the random walk position encoding (RWPE) by simulating a random walk process on the graph. The LapPE reflects the global and spectral information of the nodes, and the RWPE reflects the local diffusion characteristics of the nodes.

[0009] This invention addresses the limitations of traditional node features in describing the location information of complex structures by supplementing the structural location information of nodes.

[0010] Furthermore, S2 specifically includes: S21. Pass the node feature matrix X through a linear layer to form the node feature embedding X′; S22. The LapPE and RWPE positional encoding matrices are each processed through independent batch normalization and linear layers to form two different positional feature embeddings. and ; S23. The global feature ρ0 is a discrete value, and λΔ is a continuous value. Therefore, ρ0 is input into the embedding layer and λΔ is input into the linear layer for processing, resulting in two low-dimensional dense one-dimensional feature vectors h(1) and h(2). In order for each node to perceive these global feature information, the two vectors are broadcast to form two global feature embeddings. , ; S24. Embed the above features into X′. and , By concatenating along the feature dimensions, we obtain the comprehensive feature representation Z.

[0011] Furthermore, the core processing module includes four stacked GPS layers, each GPS layer including: a parallel-connected graph isomorphic network module and a global self-attention module, as well as a multilayer perceptron; First, the comprehensive feature representation Z is input in parallel into the graph isomorphic network module and the global self-attention module; The graph isomorphic network module uses a message passing mechanism to update the state of a node by collecting neighbor information along the edges of the network, based on a specific edge list. The global self-attention module uses a self-attention mechanism to calculate the attention score between any node in the network and all nodes in the graph. Then, based on the attention score, it performs a weighted summation of the features of all nodes to update the node's own state. The results processed by the two modules are fused by adding them element by element to form an enhanced representation that takes into account both local and global information, thus obtaining the output of this layer of GPS. After continuous processing through four GPS layers, the final node embedding matrix is ​​obtained.

[0012] Furthermore, the graph-level representation aggregation specifically includes: inputting the final node embedding matrix, where each row of the matrix represents a high-dimensional node embedding vector, and using a global summing pooling layer to sum and aggregate the embedding vectors of all nodes to finally form a graph-level representation vector zG; The threshold prediction specifically includes: inputting the graph-level representation vector zG into a prediction head composed of a multilayer perceptron, and finally outputting the predicted critical threshold.

[0013] Based on the above processing, the present invention can effectively handle complex dynamic systems driven by higher-order interactions and exhibiting hysteresis loops and asymmetric thresholds, thus overcoming the limitations of such problems.

[0014] In another aspect of the invention, an apparatus is provided for implementing a high-order network critical threshold prediction method based on graph neural networks, the apparatus comprising: At least one processor; and The memory stores instructions that, when executed by the at least one processor, cause the at least one processor to perform a swarm intelligence high-order network critical threshold prediction method based on a graph neural network as described above.

[0015] In another aspect of the invention, a computer-readable storage medium is also provided, which stores executable instructions that, when executed, cause the processor to perform a critical threshold prediction method for a high-order swarm intelligence network based on a graph neural network as described above.

[0016] Compared with the prior art, the beneficial effects of the present invention are as follows: (1) The present invention provides a method and device for predicting critical thresholds of a high-order swarm intelligence network based on graph neural network. It designs a multi-scale feature fusion mechanism, which integrates the local structural features of nodes, the position encoding of nodes in the whole graph, and the parameters that affect the global state of the system, providing the model with a comprehensive and three-dimensional understanding of the network.

[0017] (2) The present invention provides a method and device for predicting critical thresholds of a high-order swarm intelligence network based on graph neural networks. At the network information processing level, a hybrid GNN architecture that takes into account both local and global aspects is adopted, and a GIN module and a global self-attention module are set up in parallel: the former is used to accurately capture the local neighborhood structure of nodes, while the latter is used to capture long-distance dependencies in the entire graph. By fusing information from these two dimensions, the model can more deeply understand how local structures affect global dynamics. Attached Figure Description

[0018] Figure 1 This is an overall flowchart of the method described in this invention.

[0019] Figure 2 This is a framework diagram of the algorithm model described in Embodiment 1 of the present invention.

[0020] Figure 3 This is a prediction result diagram of Embodiment 1 of the present invention.

[0021] Figure 4 This is a robustness verification diagram of the hidden node features in Embodiment 1 of the present invention.

[0022] Figure 5 This is a robustness verification diagram for randomly adding or deleting some connecting edges in Embodiment 1 of the present invention. Detailed Implementation

[0023] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0024] It should be noted that the following detailed description is exemplary and is intended to provide further explanation of the present invention. Unless otherwise specified, all technical and scientific terms used herein have the same meaning as commonly understood by those of ordinary skill in the technical field to which the present invention belongs.

[0025] It should be noted that the terms used herein are only for describing specific embodiments and are not intended to limit the exemplary embodiments according to the present invention. As used herein, unless the context clearly dictates otherwise, the singular form is also intended to include the plural form. In addition, it should be understood that when the terms "comprise" and / or "include" are used in this specification, they indicate the presence of features, steps, operations, devices, components, and / or combinations thereof.

[0026] In the case of no conflict, the embodiments in the present invention and the features in the embodiments can be combined with each other.

[0027] Embodiment 1 In the present invention, on random networks, scale-free networks and community networks, the high-order network propagation and critical threshold situations are simulated based on the principle of the Simplicial Contagion Model (SCM). In this embodiment, taking the random network as an example, by setting different parameters, the diversity of the generated network is achieved. In this example, the present invention sets the total number of network nodes to 1600, the average degree of the entire network to 20, the average number of triangles to 6, and sets the initial infection probability ρ0 to 0.02, λ Δ to 1.5. For each set of parameters, multiple simulations are performed to obtain the critical threshold . The random network data is input into the trained algorithm model to obtain the predicted critical threshold .

[0028] Refer Figure 1 , this embodiment provides a method for predicting the critical threshold of group intelligence high-order networks based on a graph neural network, and the method includes: S1. Data acquisition and its preprocessing, and the preprocessing includes dataset division, feature extraction, data standardization, and position encoding.

[0029] The specific content of S1 includes: Data acquisition: In the present invention, for random networks, community networks and scale-free networks, respectively according to the corresponding network generation equations, by adjusting relevant parameters, network samples with various different structural features are generated, thereby constituting a complete dataset. This embodiment is described by taking the random network as an example.

[0030] Dataset partitioning: Obtain the network to be processed to form a dataset and partition it randomly into a training set, validation set, and test set in a 7:1:2 ratio. The training set is used to train the model, i.e., the algorithm model that executes this prediction method; the validation set is used to adjust hyperparameters; and the test set is used to test the training effect of the model.

[0031] Feature extraction: For each static network structure in the dataset, its node features and global features are extracted first, and positional encoding is calculated. This embodiment uses a random network as an example, setting the total number of network nodes N to 1600, and the average degree of the entire network... <k>The number is 20, and the average number of triangles is k. Δ It is 6.

[0032] Among them, node features are obtained by calculating the structural attributes of each node, including: node degree, the average number of 2-simplexes (i.e., the average number of triangles) in which the node participates, and the clustering coefficient; Among them, global features: global parameters that affect network dynamics are used as inputs for global features, including: the initial infection ratio ρ0 and the higher-order infection parameter λ. Δ The initial infection probability ρ0 is set to 0.02, and λ... Δ It is 1.5; Data standardization: For continuous features, i.e., node features and higher-order transmission parameters λ Δ Z-score standardization is used to improve the stability and convergence speed of subsequent model training. For discrete features, i.e., the initial infection probability ρ0, no standardization is performed. Instead, a dictionary mapping table is constructed to convert it into integer indices so that the model can embed it as a class feature.

[0033] Position encoding, or position feature: To supplement the structural position information of nodes and overcome the limitations of traditional node features (such as node degree) in describing the position information of complex structures, this invention obtains the Laplacian eigenvector position encoding (LapPE) by calculating the eigenvectors of the graph Laplacian matrix, and obtains the random walk position encoding (RWPE) by simulating a random walk process on the graph. LapPE reflects the global and spectral information of the nodes, while RWPE reflects the local diffusion characteristics of the nodes.

[0034] S2, Feature Embedding and Fusion; The algorithm model of this prediction method maps the extracted features—node features, global features, and positional encoding—into a high-dimensional space, forming independent high-dimensional feature embeddings. These feature embeddings are then concatenated and fused to form a unified comprehensive feature representation Z, which serves as the input to subsequent processing layers. The specific process is as follows: Figure 2 As shown, it includes: S21. Pass the node feature matrix X through a linear layer to form the node feature embedding X′. Each row of the node feature matrix X represents a feature vector of a node. Specifically, X is an N*d matrix, where N is the number of nodes and d=3, representing the three node attributes mentioned above. In other words, each row here represents a node feature vector. S22. The LapPE and RWPE positional encoding matrices are each processed through independent batch normalization and linear layers to form two different positional feature embeddings. and .

[0035] S23, The global feature ρ0 is a discrete value, λ Δ Since ρ0 is a continuous value, λ is input into the embedding layer. Δ The input is processed by a linear layer to obtain two low-dimensional dense one-dimensional feature vectors h. (1) and h (2) To ensure that every node can perceive this global feature information, broadcasting these two vectors means creating a copy for each node in the network, forming two global feature embeddings. , .

[0036] S24. Embed the above features into X′. and , By concatenating along the feature dimensions, we obtain the comprehensive feature representation Z.

[0037] S3, Core Information Processing, i.e., the GPS layer; The comprehensive feature representation Z is input into a core processing module consisting of four stacked GPS layers. The input to the first GPS layer is Z, while the input to each subsequent GPS layer comes from the output of the layer above. Within each GPS layer, the input features are fed in parallel into the graph isomorphic network module GIN and the global self-attention module.

[0038] The GIN module uses the message passing mechanism MPNN, which updates the state of a node by collecting neighbor information along the edges in the network, based on a specific edge list.

[0039] The global self-attention module uses a self-attention mechanism to calculate the attention score between any node in the network and all nodes in the graph. Then, based on the attention score, it performs a weighted summation of the features of all nodes to update the node's own state.

[0040] The results processed by the two modules are fused by adding them element by element. Then, a multilayer perceptron (MLP) module is used for deep, non-linear processing to form an enhanced representation that takes into account both local and global information, thus obtaining the output of this layer's GPS. After continuous processing through four GPS layers, the final node embedding matrix is ​​obtained.

[0041] S4. Graph-level representation and prediction output: Graph-level representation aggregation: The input is the final node embedding matrix, where each row represents a high-dimensional node embedding vector. For graph-level prediction, a global add pooling layer is used to aggregate the embedding vectors of all nodes, ultimately forming a graph-level representation vector z. G .

[0042] Threshold prediction: Representing the graph-level vector z G The input is fed into a prediction head (i.e., an MLP module) composed of a multilayer perceptron (MLP), and the final output is the predicted critical threshold.

[0043] In the random network of this embodiment, the obtained critical threshold has clear application value. It identifies key points where dynamic processes such as information dissemination or disease spread on the network undergo abrupt changes, providing a quantifiable intervention target. In terms of network security, to prevent the explosive spread of misinformation on the network, administrators can take measures to control the actual effective infection rate below the predicted critical threshold, thereby effectively preventing large-scale cascading. In terms of disease control, by blocking transmission, the probability of infection can be controlled, making the spread unsustainable and ultimately allowing the disease to naturally disappear within the population.

[0044] Compared with existing technologies, the technical solution proposed in this invention has significant performance advantages. Existing technologies must rely on complete dynamic evolution data, while this invention can make predictions using only the static topology of the network. This greatly saves the cost of data acquisition and processing in real-world scenarios where data acquisition is limited.

[0045] Regarding prediction accuracy, this invention has undergone extensive experimental verification. This invention uses three metrics to evaluate the training accuracy of the model: mean squared error (MSE), mean absolute error (MAE), and coefficient of determination (R²). MSE evaluates the average of the squares of the model's prediction errors and is more sensitive to results with large errors; MAE directly reflects how much the predicted result deviates from the true value on average, providing an intuitive reflection of the average prediction deviation; R² evaluates the model's fit to the data, with lower MSE and MAE being better, and a higher R² being better.

[0046] Figure 3 The left figure shows the comparison between the true value and the predicted value of the model on the test set. The red line indicates that the true value and the predicted value are exactly the same, without any deviation. As can be seen, the prediction results of this invention have a low deviation, which reflects high prediction accuracy.

[0047] Figure 3 The right figure shows the training loss and validation loss of the model during training. It can be seen that both the training loss and validation loss decrease to low values ​​during training, reflecting the excellent fit of the model on the dataset. Specific test results on the test set are shown in Table 1. The model's performance scores in mean squared error (MSE), mean absolute error (MAE), and coefficient of determination (R²) are significantly better than existing techniques in all network types, including random networks, community networks, and scale-free networks.

[0048] Table 1

[0049] In terms of robustness, this invention also performs excellently. This invention simulates real-world scenarios where feature information is incomplete, such as sensor failure or the inability to observe certain node information, and measurement errors or random noise in the network structure itself. For scenario one, this invention randomly selects a certain proportion of nodes and sets their node features to zero, i.e., hides them, for simulation testing. The results for scenario one are as follows: Figure 4 This compares the true and predicted values ​​at concealment ratios of 5% (a), 15% (b), and 30% (c). Table 2 shows the results for Scenario 1, including the mean squared error (MSE), mean absolute error (MAE), and coefficient of determination (R²) scores for concealment ratios of 5%, 10%, 15%, 20%, 25%, and 30%. Table 2

[0050] For scenario two, this invention perturbs the network structure of the test set by randomly removing or adding a certain proportion of edges to disrupt the network's topology. The comparison results between the ground truth and predicted values ​​for scenario two are as follows: Figure 5 The true and predicted values ​​are compared in three cases: adding 10% (a), deleting 10% (b), and deleting first and then adding 10% (c). The performance scores for scenario two are shown in Table 3, which shows the mean squared error (MSE), mean absolute error (MAE), and coefficient of determination (R²) scores in the three cases: adding 10%, deleting 10%, and deleting first and then adding 10%.

[0051] Table 3

[0052] Experimental results show that even under conditions where some node features are missing or the network structure is significantly perturbed, the model can still maintain stable and high-performance prediction capabilities, proving that the present invention remains reliable in complex environments with incomplete information or noise.

[0053] Furthermore, when tested on more complex real-world network data, this invention also demonstrates excellent generalization ability. In summary, this invention offers innovative implementation, accurate and effective prediction, strong robustness and generalization ability, meeting the application requirements for high-precision critical threshold prediction in complex network domains.

[0054] Example 2 This embodiment provides an apparatus for implementing a high-order network-based critical threshold prediction method based on graph neural networks. The apparatus includes: At least one processor; and The memory stores instructions that, when executed by the at least one processor, cause the at least one processor to perform a swarm intelligence high-order network critical threshold prediction method based on a graph neural network as described above.

[0055] In this embodiment, electronic devices include, but are not limited to: personal computers, server computers, workstations, desktop computers, laptop computers, notebook computers, mobile computing devices, smartphones, tablet computers, cellular phones, personal digital assistants (PDAs), handheld devices, messaging devices, wearable computing devices, consumer electronic devices, etc.

[0056] Example 3 This embodiment also provides a computer-readable storage medium storing executable instructions, which, when executed, cause the processor to perform a critical threshold prediction method based on a graph neural network for high-order crowd intelligence networks as described above.

[0057] Specifically, a system or apparatus equipped with a readable storage medium may be provided, on which software program code implementing the functions of any of the embodiments described above is stored, and the computer or processor of the system or apparatus can read and execute the instructions stored in the readable storage medium.

[0058] In this case, the program code itself, which can be read from the readable medium, can perform the functions of any of the above embodiments, and therefore the computer-readable code and the readable storage medium storing the computer-readable code constitute a part of this specification.

[0059] Examples of readable storage media include floppy disks, hard disks, magneto-optical disks, optical disks (such as CD-ROM, CD-R, CD-RW, DVD-ROM, DVD-RAM, DVD-RW, DVD-RW), magnetic tapes, non-volatile memory cards, and ROMs. Alternatively, program code can be downloaded from a server computer or the cloud via a communication network.

[0060] Those skilled in the art will understand that embodiments of the present invention can be provided as methods, systems, or computer program products. Therefore, the present invention can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present invention can take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0061] Obviously, the above embodiments of the present invention are merely examples for clearly illustrating the technical solutions of the present invention, and are not intended to limit the specific implementation of the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the claims of the present invention should be included within the protection scope of the claims of the present invention.< / k>

Claims

1. A method for predicting critical thresholds in a high-order swarm intelligence network based on graph neural networks, characterized in that, The method includes: S1. Obtain the network to be processed and divide the dataset. Extract node features and global features from the input network, standardize the data, and calculate the position code. S2. Map the obtained node features, global features, and positional codes to a high-dimensional space to form independent high-dimensional feature embeddings; then, concatenate and fuse these high-dimensional feature embeddings to form a comprehensive feature representation. S3. Input the comprehensive feature representation into the core processing module consisting of four stacked GPS layers. Update the state of the node itself using the graph isomorphic network module and the global self-attention module respectively. Then, perform information fusion by element-wise addition to obtain an enhanced representation. Finally, obtain the final node embedding matrix. S4. The final node embedding matrix is ​​aggregated into a graph-level representation vector, and then threshold prediction is performed to finally output the predicted critical threshold.

2. The critical threshold prediction method for a high-order swarm intelligence network based on graph neural networks according to claim 1, characterized in that, The node features include: node degree, the average number of 2-simplexes in which the node participates, and clustering coefficient; The global features include: the initial infection rate and higher-order infection parameters; The position encoding includes: obtaining the Laplacian eigenvector position encoding LapPE by calculating the eigenvectors of the graph Laplacian matrix; and obtaining the random walk position encoding RWPE by simulating a random walk process on the graph.

3. The critical threshold prediction method for a high-order swarm intelligence network based on graph neural networks according to claim 2, characterized in that, S2 specifically includes: S21. Pass the node feature matrix X through a linear layer to form the node feature embedding X′; S22. The LapPE and RWPE positional encoding matrices are each processed through independent batch normalization and linear layers to form two different positional feature embeddings. and ; S23. The global feature ρ0 is a discrete value, and λΔ is a continuous value. Therefore, ρ0 is input into the embedding layer and λΔ is input into the linear layer for processing, resulting in two low-dimensional dense one-dimensional feature vectors h(1) and h(2). Then, a broadcast operation is performed on these two vectors to form two global feature embeddings. , ; S24. Embed the above features into X′. and , By concatenating along the feature dimensions, we obtain the comprehensive feature representation Z.

4. The critical threshold prediction method for a high-order swarm intelligence network based on graph neural networks according to claim 3, characterized in that, The core processing module includes four stacked GPS layers, each GPS layer including: a parallel-connected graph isomorphic network module and a global self-attention module, as well as a multilayer perceptron; First, the comprehensive feature representation Z is input in parallel into the graph isomorphic network module and the global self-attention module; The graph isomorphic network module uses a message passing mechanism to update the state of a node by collecting neighbor information along the edges of the network, based on a specific edge list. The global self-attention module uses a self-attention mechanism to calculate the attention score between any node in the network and all nodes in the graph. Then, based on the attention score, it performs a weighted summation of the features of all nodes to update the node's own state. The results processed by the two modules are fused by adding them element by element to form an enhanced representation that takes into account both local and global information, thus obtaining the output of this layer of GPS. After continuous processing through four GPS layers, the final node embedding matrix is ​​obtained.

5. The critical threshold prediction method for a high-order swarm intelligence network based on graph neural networks according to claim 4, characterized in that, The graph-level representation aggregation specifically includes: inputting the final node embedding matrix, where each row of the matrix represents a high-dimensional node embedding vector; using a global summing pooling layer to sum and aggregate the embedding vectors of all nodes, ultimately forming a graph-level representation vector zG; The threshold prediction specifically includes: inputting the graph-level representation vector zG into a prediction head composed of a multilayer perceptron, and finally outputting the predicted critical threshold.

6. The critical threshold prediction method for a high-order swarm intelligence network based on graph neural networks according to claim 2, characterized in that, The data standardization specifically refers to: Z-score standardization is used for continuous features; Instead of standardizing discrete features, we convert them into integer indices by constructing a dictionary mapping table.

7. An apparatus for implementing a high-order network-based critical threshold prediction method using graph neural networks, characterized in that, The device includes: processor; A memory on which computer programs that can run on the processor are stored; When the computer program is executed by the processor, it implements the steps of the critical threshold prediction method for a high-order swarm intelligence network based on graph neural networks as described in any one of claims 1 to 6.

8. A computer-readable storage medium storing a computer program, characterized in that, When the computer program is executed by a processor, it implements the steps of the method as described in any one of claims 1 to 6.

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