High-order network propagation dynamics classification method based on node weighted clique degree and deep learning

Abstract

The invention discloses a high-order network propagation dynamics classification method based on node weighted clique degree and deep learning, and belongs to the technical field of computers, and the method specifically comprises the steps: data collection and preprocessing: constructing a hypergraph and network representation thereof from real data; establishment of propagation dynamic models: respectively constructing the propagation dynamic models on the graph network, the simple complex and the hypergraph, and describing dynamic evolution laws under different network structures; feature processing, including three sub-tasks of structural feature calculation, dynamic feature processing and feature merging and sample division; and inputting the features into a model for model training, and performing prediction and effect evaluation on the trained model. According to the method, a real system structure is described by adopting a hypergraph, a pure complex and other high-order networks, so that the defect of insufficient consideration of a node multi-element interaction mode in an existing network propagation dynamic process is overcome.

Description

Technical Field

[0001] This invention belongs to the field of computer technology, specifically relating to a high-order network propagation dynamics classification method based on node weighted clique degree and deep learning. Background Technology

[0002] Network propagation dynamics is an important research direction in network science, aiming to characterize the dynamic behavior and evolution of nodes in a system. Research in this field is widely applied to the analysis and solution of problems such as disease transmission and control, and information transmission and diffusion analysis. Specifically, in disease transmission and control, propagation dynamics models simulate the spread and infection process of viruses and other transmission media among individuals to study the spread speed and scope of epidemics in networks. In information diffusion, propagation dynamics models can effectively simulate individuals' information adoption behavior, thereby revealing the degree of information diffusion and evolutionary trends. However, in many practical scenarios, it is often necessary to analyze the diffusion mechanisms of specific scenarios to determine which propagation dynamics model should be used to abstractly model the propagation evolution process under specific conditions, thus providing a basis for formulating guiding strategies for further diffusion or suppression of the propagation process. This type of problem is called the propagation dynamics model classification and discrimination problem, the core objective of which is to identify the most appropriate propagation dynamics mechanism from observable propagation sequences.

[0003] In the real world, propagation between entities typically relies on the connections between them. The most basic form of connection can be modeled using traditional graph structures, where nodes have only simple one-to-one relationships, and propagation occurs solely through direct "point-to-point" contact. However, many real-world systems contain higher-order network structures that incorporate group interactions. For example, on online social platforms, users often post information within a circle of friends comprised of multiple users. In such contexts, traditional graph structures cannot effectively describe the crucial role of group interactions in the propagation process.

[0004] To more comprehensively consider the diversity of dynamic patterns resulting from the diversity of node interactions in a network, this invention introduces a propagation dynamics model that can characterize the propagation dynamics process in higher-order network structures such as hypergraphs and simplex complexes, based on graph network dynamics. Summary of the Invention

[0005] To overcome the shortcomings of existing technologies, this invention aims to provide a high-order network propagation dynamics classification method based on node weighted clique degree and deep learning. By combining the proposed topological features such as node weighted clique degree and node infection order features with a deep learning model, a method is designed that can effectively classify propagation dynamics mechanisms, thus solving the problem of discriminating propagation dynamics processes in real systems.

[0006] The technical problem solved by this invention can be achieved through the following specific technical solutions:

[0007] The aforementioned high-order network propagation dynamics classification method based on node weighted clique degree and deep learning includes the following steps:

[0008] Step 1: Data acquisition and preprocessing, including constructing hypergraphs and their network representations from real data;

[0009] Step 2: Establishment of propagation dynamics models. Propagation dynamics models are constructed on graph networks, simplex complexes, and hypergraphs respectively to characterize the dynamic evolution laws under different network structures.

[0010] Step 3, Feature Processing, includes three sub-tasks: structural feature calculation, dynamic feature processing, and feature merging and sample partitioning.

[0011] Step 4: Input the features into the model for training, and then use the trained model to make predictions and evaluate its performance.

[0012] Furthermore, in step 1, the specific content of constructing the hypergraph from real data is as follows: collecting relevant data, constructing the hypergraph based on the association between nodes and hyperedges, denoted as... Where V = [v1, v2, ..., v N [] represents the set of N nodes in the hypergraph, where N = |V| represents the size of the network; M = |E| represents the total number of hyperedges in the network, and correspondingly, E = [e1, e2, ..., e] M [] represents the set of superedges consisting of M superedges in the hypergraph; superedge e j A hyperedge is a subset of the node set V, containing two or more nodes. The total number of nodes contained therein is defined as the size of the hyperedge.

[0013] Furthermore, in step 1, based on the hypergraph constructed using real data, the hypergraph needs to be generated. The two corresponding network representations: simple complex network HeTu Network

[0014] SuperMap The corresponding simplicial complex network representation is obtained through the following steps: First, all nodes in the original hypergraph are retained in the new network, i.e., the node set is V; second, the hyperedges of size k+1 in the original hypergraph are represented using k-simplexes, where k represents the order of the simplicium, and a k-simplex is defined as an edge structure containing k+1 nodes; third, all hyperedges are mapped to several simpliciums of different orders, and these simpliciums form a set U, which, together with the node set V, constitutes the simplicial complex network.

[0015] SuperMap The steps for mapping to a graph network are as follows: First, preserve the original hypergraph. First, all nodes are represented as the node set V of the new network; second, for each pair of node relationships, whether an edge is established between the nodes depends on their position in the network. First, determine if there are any shared hyperedges between the nodes. If so, establish an edge connecting these node pairs in the new network. Second, add the established edges between the node pairs to the new network, ensuring that no edges are repeated. Thus, a graph network corresponding to the original hypergraph is formed. The construction is then complete, including, This represents the set of all connected edges in the mapped graph network.

[0016] Furthermore, in step 2, the propagation dynamics on the graph network adopts the "Susceptible-Infected" (SI) propagation dynamics model. In this model, node states are divided into two categories: susceptible (S) and infected (I). Nodes in the susceptible (S) state will be transformed into the infected state by the propagation of infected (I) nodes. In the SI model acting on the graph network, for each propagation time step, nodes in the susceptible state will transform their state into the infected state by contacting their connected infected nodes at a propagation rate β. As time steps are iterated, the number of infected nodes in the system will gradually increase over time until all nodes in the system become infected.

[0017] Furthermore, in step 2, within the simplex network, in the k-simplex where a susceptible node resides, an individual is not only independently affected by the spread of infected individuals through simple links, but also by the collective effect generated by several groups of infected nodes. Therefore, the probability of an individual being infected can be characterized by the propagation rate of several simplexes composed of infected nodes within the individual's neighborhood; that is, the propagation rate resulting from a node independently being adjacent to a k-simplex composed of infected nodes is β. k That is, when k=1, the propagation degenerates into a simple link propagation mode that only propagates through edges.

[0018] Furthermore, in step 2, regarding the propagation process on the hypergraph, susceptible nodes are propagated by infected nodes in their subordinate hyperedges. Specifically, the propagation probability of a susceptible node in each adjacent hyperedge is related to the number of infected nodes in that hyperedge, i.e., the propagation rate is... Where λ and ξ are adjustable parameters characterizing the linearity and nonlinearity of the propagation, respectively. Indicates the superedge e j (e j The number of infected nodes contained in (∈E).

[0019] Furthermore, in step 3, the structural feature calculation includes two parts: calculating the node degree of nodes in the network and calculating the clustering coefficient of nodes. The node degree of a node is defined as the total number of unique nodes contained in the hyperedges to which the node belongs (excluding the node itself). The node degree of node i is denoted as D(i). The calculation of node degree centrality depends on the association matrix I constructed on the hypergraph. This association matrix is ​​constructed through the membership relationships between nodes and their neighbors in the hypergraph. Specifically, when node i is contained in hyperedge e... j At that time, I ij The value is 1 if the condition is met, and 0 otherwise. Based on the constructed incidence matrix, the adjacency matrix A between nodes in the hypergraph can be calculated.

[0020]

[0021] in, The matrix is ​​a diagonal matrix whose elements Let represent the number of hyperedges that node i belongs to in the hypergraph. Then, calculate the degree centrality value for each node:

[0022]

[0023] in, It is matrix A ij The binary matrix, that is, for matrix A ij For each element, elements with values ​​greater than 0 are assigned the value 1, otherwise they are assigned the value 0;

[0024] Next, computing networks The clustering coefficient (CC) of each node reflects the connectivity between its neighbors and is defined as follows:

[0025]

[0026] In the formula, This represents the number of edges that connect node i to its neighboring nodes.

[0027] Finally, computational networks The weighted clique degree (CD) of each node in the graph is defined as the sum of the products of the number of cliques formed by node v and its adjacent nodes, and the number of nodes within each clique. A clique is a complete subgraph, meaning that every pair of nodes in the subgraph is connected by an edge. Therefore, the clique degree CD(i) of node i can be mathematically expressed as:

[0028]

[0029] in, Let ε be the set of neighboring nodes of node i, that is, the set of nodes directly connected to node i through edges. Let κ be a subset of the set of neighboring nodes of node i. Let κ∪{i} be a set of nodes containing node i and its neighboring node subset κ. It must be a clique, denoted as δ(κ∪{i}), that is, all nodes in the set κ∪{i} must be pairwise connected. In addition, ε κ∪{i} Let |δ(κ∪{i})| represent the number of cliques adjacent to node i, and let |δ(κ∪{i})| represent the total number of nodes contained in the clique.

[0030] Furthermore, in step 3, the dynamic feature processing records the changes in the node degree, clustering coefficient, and weighted clique degree of the node based on the infection order of the node in the network.

[0031] In SuperGraph In the process, firstly, a node is randomly selected from the node set V as the initial infected node; then, in the hypergraph... The propagation dynamics process on the hypergraph is executed, where the propagation parameters are set as follows: and Corresponding to the propagation process under each set of λ and ξ parameter configurations, the time t at which each node i is infected is recorded during the process. i This continues until the propagation process ends, with the end time T being the moment the propagation process finally reaches a steady state. Subsequently, for the propagation dynamics process executed under each set of propagation parameters, nodes are arranged according to their infection time t. i Sort the nodes from smallest to largest to obtain the node sequence. and the infection time series corresponding to the node sequence Based on this, according to Calculate the node degree, clustering coefficient, and weighted clique degree of the corresponding node in the sequence to obtain a node degree sequence. A sequence of node clustering coefficients and a node weighted clique degree sequence

[0032] Repeat the above process R times to obtain R infection time series for each set of transmission parameters. The node degree sequence corresponding to R infection sequences Clustering coefficient sequence of nodes corresponding to R infection sequences and the node clique degree sequences corresponding to the R infection sequences In simple complex HeTu Network The propagation processes corresponding to step 2 are executed respectively, wherein the parameters of the propagation dynamics process on the simplex complex are configured as follows: and The parameters of the propagation dynamics process on the diagram are configured as follows: After performing the propagation process on the simple complex, node infection sequence feature sets were obtained under each set of propagation parameters. Node degree sequence group Node clustering coefficient sequence group and node clique degree sequence group and in graph networks In the middle, the node infection sequence group under each set of propagation parameters Node degree sequence group Node clustering coefficient sequence group and node clique degree sequence group

[0033] Furthermore, in step 3, feature merging and sample partitioning include two sub-tasks: feature merging and sample partitioning. For the feature merging process, taking a hypergraph as an example, based on the hypergraph... In λ=λ m (1≤m≤L λ ) and ξ=ξ n (1≤n≤L ξ The data obtained is used to infect the r-th (1≤r≤R) node with the sequence. Node degree corresponding to infection sequence Clustering coefficient features of nodes By merging, a unified feature sequence is obtained. Right now Since 1 ≤ r ≤ R, the above steps yield R merged feature sequences, which form a feature matrix. Then, the feature matrices under different parameters λ and ξ are merged row by row to obtain a merged feature matrix. Right now:

[0034]

[0035] Correspondingly, in simple complex HeTu Network The corresponding feature matrix is ​​obtained from the above. and Furthermore, label vectors representing the actual categories of dynamic processes on hypergraphs, simplexes, and graph networks are processed, i.e. in, and These are vectors composed of values ​​0, 1, and 2, respectively. Next, the feature matrix... and tags Both sets were divided in a 7:3 ratio to form training and test sets.

[0036] Furthermore, in step 4, the model training relies on a Multilayer Perceptron (MLP) model. This neural network model is used to train the classification task, including the following:

[0037] First, an MLP model is built, consisting of an input layer, three hidden layers, and an output layer. The input layer receives feature data, and the number of neurons in the input layer matches the feature dimension of the training set. The number of neurons in the three hidden layers is twice, once, and half the number of neurons in the input layer, respectively, and the ReLU (Rectified Linear Unit) activation function is used, i.e., ReLU(x) = max(0,x), to enhance the model's non-linear expressive power. The output layer has three neurons, corresponding to the prediction units of the hypergraph, simplex, and general graph network dynamics, respectively, and uses the Softmax activation function, i.e.:

[0038]

[0039] Among them, z α For the unnormalized output of the α-th class, the Softmax function can convert the network output into a probability distribution for multi-class classification tasks;

[0040] Subsequently, the constructed training set is input into the defined MLP model, and the model is trained using the Stochastic Gradient Descent (SGD) optimization algorithm. During training, the Cross Entropy Loss Function is selected as the loss function to measure the error between the model's prediction and the true label. The learning rate (lr) is dynamically adjusted during training; that is, the trend of the loss function value is monitored. If the rate of loss descent slows down or oscillates drastically, the learning rate is appropriately reduced to avoid getting trapped in local minima or gradient oscillations. Conversely, if the loss descent is stable, the learning rate can be appropriately increased to accelerate the training process.

[0041] Ultimately, the trained classifier r has the ability to effectively distinguish the propagation dynamics processes on different topologies (including graphs, simplexes, and hypergraphs). This classifier can predict the propagation category based on the structural features and dynamic evolution patterns of the input data.

[0042] In the prediction phase, the test set is input into the trained classifier model v for prediction. For each sample in the test set, after calculation by the classifier ψ, the three neurons in the output layer generate three probability weight values, namely... and These correspond to the propagation dynamics processes on hypergraphs, simplexes, and graph networks, respectively. Then, the three probability weights for each sample are compared, and the class corresponding to the highest probability is selected as the predicted class label for that sample, i.e.:

[0043]

[0044] in, This represents the final prediction result of the classifier model, summing the prediction results for all samples in the test set. In preparation for evaluating the effectiveness of the model.

[0045] Model performance evaluation comprises two sub-tasks: calculating model evaluation metrics and comparing algorithms. First, multiple evaluation metrics for the proposed model are calculated to comprehensively measure its classification performance. These metrics include: (i) Accuracy: measuring the proportion of correctly classified samples; (ii) Recall: evaluating the model's ability to identify samples of a specific class, defined as the proportion of correctly classified samples of that class out of the total number of samples in that class; (iii) Precision: measuring the proportion of correctly predicted samples of a given class; and (iv) F1-score: the harmonic mean of the model's precision and recall, balancing accuracy and coverage. Second, to further validate the model's effectiveness, traditional machine learning methods (e.g., Random Forest (RF) and Support Vector Machine (SVM)) are selected as benchmarks. Classification tasks are performed using both methods, and the above four evaluation metrics are calculated to obtain baseline performance. Finally, the evaluation metrics of the proposed model are compared with those based on Random Forest and SVM classifiers.

[0046] By analyzing the classification performance of each model on different topologies and its recognition ability on different propagation dynamics categories, the advantages of the proposed model in terms of classification accuracy, generalization ability and robustness are evaluated, thereby verifying its applicability and superiority in the classification task of propagation dynamics in complex networks.

[0047] Compared with the prior art, the present invention has the following advantages:

[0048] (1) This invention uses high-order networks such as hypergraphs and simplex complexes to characterize the high-order features of individuals in real interactions. It can identify the type of network dynamics process based on the topological features and propagation laws of nodes in the network, thus making up for the shortcomings of existing propagation dynamics models in considering high-order interactive behaviors.

[0049] (2) This invention, combined with a deep learning framework, explores a high-order network propagation dynamics classification method based on node weighted clique degree. At the micro level, it incorporates high-order topological features (e.g., weighted clique degree) in addition to considering the low-order topological features of nodes; at the macro level, it establishes a mapping law between node dynamics features and node structural features through deep learning methods, exhibiting better generalization ability. This method also provides a novel approach to reconstructing the propagation dynamics process in high-order systems, further offering more comprehensive solutions to practical problems such as the positive guidance of public opinion and the precise prevention and control of diseases. Attached Figure Description

[0050] Figure 1 This is a flowchart of the method of the present invention. Detailed Implementation

[0051] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0052] The purpose of this invention is to propose a high-order network propagation dynamics classification method. This method comprehensively considers topological features such as node weighted clique degree, node degree, and node clustering coefficient, as well as dynamic features such as node infection order, to accurately identify the appropriate propagation dynamics model from a given propagation sequence. Furthermore, this method maintains good generalization ability in systems of different sizes.

[0053] like Figure 1 As shown, the high-order network propagation dynamics classification method based on node weighted clique degree and deep learning provided by this invention is carried out according to the following steps:

[0054] Step 1: Data acquisition and preprocessing, including constructing hypergraphs and their network representations from real data.

[0055] A hypergraph is constructed from relevant data based on the nodes and the interactions between them (i.e., the relationships between nodes and hyperedges), denoted as . Where V = {v1, v2, ..., v} N Let E = {e1, e2, ..., e} represent the set of N nodes in the hypergraph. Correspondingly, E = {e1, e2, ..., e} M Let} represent the set of superedges consisting of M hyperedges in the hypergraph. Based on the hypergraph... Mapping this hypergraph yields two network representations of the hypergraph: the simplex complex. HeTu Network The specific steps are as follows:

[0056] ① In the simple complex network, all nodes in the original hypergraph are retained, i.e., the node set is V.

[0057] ② Represent the hyperedges of size k+1 in the original hypergraph using a k-simplex, where k represents the order of the simplex. A k-simplex is defined as an edge structure containing k+1 nodes.

[0058] ③ Map all hyperedges onto simplexes, form a set U of all simplexes, and together with the node set V, form a simplex complex network.

[0059] ④ Retain the original supernetwork in the graph network. All nodes are represented as the node set V of the graph network.

[0060] ⑤ For each pair of node relationships, whether an edge is established between the nodes depends on their... If there is a common superedge, then establish an edge between the node pairs.

[0061] ⑥ Add the edges between the established node pairs to the graph network, ensuring that the edges are not repeated. Thus, a graph network corresponding to the original hypergraph is formed. The construction is then complete, including, This represents the set of all connected edges in the mapped graph network.

[0062] The second step is to establish a propagation dynamics model. Propagation dynamics models are constructed on graph networks, simplex complexes, and hypergraphs to characterize the dynamic evolution under different network structures.

[0063] This invention employs the Susceptible-Infected (SI) propagation model to simulate the propagation dynamics of nodes on a hypergraph. Specifically, during propagation, a node at any given moment can only be in either a susceptible state (S) or an infected state (I). A susceptible (S) node will be influenced by an infected (I) node with probability β, thus changing its state to infected. At each time step, for the propagation dynamics on the graph network, a susceptible node will be independently influenced by the propagation of each infected node within its neighborhood with probability β; for the propagation dynamics on a simple complex, a susceptible node will be independently influenced by the propagation of infected nodes within its neighborhood with probability β, and simultaneously, the node will be influenced by the propagation of infected nodes within its neighborhood with probability β. k It is affected by the propagation of a k-simplex consisting of any k+1 nodes within its neighborhood; for the hypergraph, at each time step, a susceptible node in the hypergraph will... The probability is affected by the adjacent hyperedge e j The joint propagation effect of infected nodes in the hyperedge is considered, where λ and ξ represent the linear and nonlinear effects of infected nodes in the hyperedge, respectively. Indicates the superedge e j (ej The number of infected nodes contained in (∈E). The specific execution steps are as follows:

[0064] ① Execute the propagation dynamics process on the hypergraph. First, randomly select a node from the node set V of the hypergraph as the initial infected node, and begin the propagation dynamics process on the hypergraph as described above. For each set of λ and ξ parameter configurations, record the infection time t for each node i during the process. i This continues until the propagation process ends, with the end time T being the moment when the propagation process finally reaches a steady state. The propagation parameters are set as follows: as well as Under each set of propagation parameters, the above three propagation dynamics processes are repeated R times. For each propagation process under each set of parameters, the nodes are arranged according to their infection time t. i Sort the nodes in ascending order to obtain a sequence of R nodes. and R and their corresponding infection time series

[0065] ② Execute the propagation dynamics process on the simplex complex. Randomly select a node from the node set V of the simplex complex as the initial infected node to begin the propagation dynamics process on the simplex complex. Similarly, corresponding to each set β and β... k The propagation process under parameter configuration records the infection time t of each node i during the process. i This continues until the time step reaches the steady-state time T. The parameters are configured as follows: For each set of propagation parameters, we obtain R node sequences. and R infection time series

[0066] ③ Execute the propagation dynamics process on the graph network. Randomly select a node from the node set V of the graph network as the initial infected node and begin the propagation process. The parameters of the propagation dynamics process on the graph are configured as follows: For each set of propagation parameters, we obtain R node sequences. and R infection time series

[0067] Step 3: Calculate structural features, which involves calculating the topological properties of each node. For each node, calculate its degree, clustering coefficient, and clique degree.

[0068] ① Calculate the degree of each node i in the network, i.e. in, Matrix I is an incidence matrix representing the membership relationships between nodes and their neighbors in a hypergraph. When node i is contained within hyperedge e... j When, element I ijThe value is 1 if it is 1, otherwise it is 0; in addition, It is matrix A ij The binary matrix, that is, for matrix A ij For each element in the array, elements with values ​​greater than 0 are assigned the value 1, otherwise they are assigned the value 0.

[0069] ② Calculate the node clustering coefficient for each node i in the network, i.e., in, Represents network The number of edges between the neighboring nodes of node i.

[0070] ③ Calculate the weighted clique degree of each node j in the network, i.e., in, Let ε be the set of neighboring nodes of node i, i.e., the set of nodes adjacent to node i. Let κ be a subset of the set of neighboring nodes of node i. Let κ∪{i} be a set of nodes containing node i and its neighboring node subset κ, which must be a clique (denoted as δ(κ∪{i})), meaning that all nodes in this subset must be fully connected. κ∪{i} Let |δ(κ∪{i})| represent the number of cliques adjacent to node i, and let |δ(κ∪{i})| represent the total number of nodes in that clique.

[0071] Step 4: Feature Processing. Based on the node sequence sorted by infection time obtained in Step 2. And the node topological features calculated in the third step, to obtain the sequence. The corresponding node degree sequence, node clustering coefficient sequence, and node weighted clique degree sequence, the element values ​​in these sequences are the sequence values. The topological feature value of the corresponding node.

[0072] ① Hypergraph For example, after the second and third steps, the propagation process for each propagation parameter yields a sequential sequence of infection times for R nodes. And R sequences of nodes and their corresponding nodes For each node sequence Calculate the node degree and node clustering coefficient of the corresponding node in the sequence to obtain a node degree sequence. and node clustering coefficient sequence Finally, the node degree sequences corresponding to the R infection sequences are obtained. Clustering coefficient sequence of nodes corresponding to R infection sequences and the node clique degree sequences corresponding to the R infection sequences Similarly, for the dynamic processes on simplex complexes and graph networks, we obtain the following for simplex complexes: Node infection sequence feature group Node degree sequence group Node clustering coefficient sequence group and node clique degree sequence group and in graph networks In the middle, the node infection sequence group under each set of propagation parameters Node degree sequence group Node clustering coefficient sequence group and node clique degree sequence group

[0073] ② Merge the features corresponding to each propagation process in a specific network. For example, for a hypergraph... In λ=λ m (1≤m≤L λ ) and ξ=ξ n (1≤n≤L ξ The data obtained is used to infect the r-th (1≤r≤R) node with the sequence. Node degree corresponding to infection sequence Clustering coefficient features of nodes By merging, a unified feature sequence is obtained. Right now Since 1 ≤ r ≤ R, the above steps can yield R merged feature sequences, which form a feature matrix. Then, the feature matrices under different parameters λ and ξ are merged row by row to obtain a merged feature matrix. Right now:

[0074]

[0075] Correspondingly, in simple complex HeTu Network The corresponding feature matrix is ​​obtained from the above. and Furthermore, it processes the actual label vectors representing dynamic processes on hypergraphs, simplexes, and graph networks, i.e. in, and These are R-dimensional vectors composed of values ​​0, 1, and 2; and the samples (i.e., feature matrices) and tags The datasets were divided into training and testing sets in a 7:3 ratio.

[0076] Step 5: Model training.

[0077] ① Build an MLP model consisting of an input layer, three hidden layers, and one output layer. The input layer receives feature data, with the number of neurons matching the feature dimension of the training set. The three hidden layers have 2, 1, and 1 / 2 times the number of neurons in the input layer, respectively, and use the ReLU (Rectified Linear Unit) activation function (ReLU(x) = max(0,x)) to enhance the model's non-linear expressive power. The output layer consists of three neurons, corresponding to the prediction units of a hypergraph, a simplex, and a general graph network dynamics, respectively, and uses the Softmax activation function. Among them, z α This is the unnormalized output for the α-th category.

[0078] ② The constructed training set is input into the constructed MLP model, and training is performed using the Stochastic Gradient Descent (SGD) optimization algorithm and the Cross Entropy Loss Function. During training, the learning rate (lr) is dynamically adjusted. Specifically, if the rate of loss descent slows down or oscillates violently, the learning rate is appropriately reduced; conversely, if the loss descent is stable, the learning rate can be appropriately increased to accelerate the training process. Finally, a classifier ψ is trained that can effectively distinguish propagation dynamics processes on graphs, simplicities, and hypergraphs.

[0079] ③ Input the test set into the trained classifier model ψ for prediction. For each sample in the test set, after calculation by the classifier ψ, the three neurons in the output layer generate three probability weight values, i.e. and These correspond to the propagation dynamics processes on hypergraphs, simplexes, and graph networks, respectively. Then, the three probability weights for each sample are compared, and the class corresponding to the highest probability is selected as the predicted class label for that sample, i.e.: in, This represents the final prediction result of the classifier model. Finally, the prediction results for all samples in the test set are statistically analyzed. In preparation for evaluating the effectiveness of the model.

[0080] Step 6: Evaluate the model's predictive ability using model evaluation metrics and compare its performance with other algorithms. Calculate the accuracy, recall, precision, and F1-score of the proposed model to quantify its predictive performance. Furthermore, select traditional machine learning methods such as random forest and support vector machine as benchmarks. Perform classification tasks using these two methods respectively, and calculate the aforementioned evaluation metrics to assess baseline performance. Finally, compare the evaluation metrics of the proposed model with those based on random forest and SVM classifiers. By analyzing the classification performance of each model on different topologies and its recognition ability across different propagation dynamics categories, evaluate the proposed model's classification accuracy and generalization ability.

[0081] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A high-order network propagation dynamics classification method based on node weighted clique degree and deep learning, characterized in that, Includes the following steps: Step 1: Data acquisition and preprocessing, including constructing hypergraphs and their network representations from real data; Step 2: Establishment of propagation dynamics models. Propagation dynamics models are constructed on graph networks, simplex complexes, and hypergraphs respectively to characterize the dynamic evolution laws under different network structures. Step 3, Feature Processing, includes three sub-tasks: structural feature calculation, dynamic feature processing, and feature merging and sample partitioning. Step 4: Input the features into the model for training, and then use the trained model to make predictions and evaluate its performance.

2. The high-order network propagation dynamics classification method based on node weighted clique degree and deep learning according to claim 1, characterized in that, In step 1, the specific content of constructing a hypergraph from real data is as follows: collect data, construct a hypergraph based on the association between nodes and hyperedges, denoted as . Where V = [v1, v2, ..., v N [] represents the set of N nodes in the hypergraph, where N = |V| represents the size of the network; M = |E| represents the total number of hyperedges in the network, and correspondingly, E = [e1, e2, ..., e] M [] represents the set of superedges consisting of M superedges in the hypergraph; superedge e j A hyperedge is a subset of the node set V, containing two or more nodes. The total number of nodes contained therein is defined as the size of the hyperedge.

3. The high-order network propagation dynamics classification method based on node weighted clique degree and deep learning according to claim 2, characterized in that, In step 1, based on the hypergraph constructed using real data, the hypergraph needs to be generated. The two corresponding network representations: simple complex network HeTu Network SuperMap The corresponding simplicial complex network representation is obtained through the following steps: First, all nodes in the original hypergraph are retained in the new network, i.e., the node set is V; second, the hyperedges of size k+1 in the original hypergraph are represented using k-simplexes, where k represents the order of the simplicium, and a k-simplex is defined as an edge structure containing k+1 nodes; third, all hyperedges are mapped to several simpliciums of different orders, and these simpliciums form a set U, which, together with the node set V, constitutes the simplicial complex network. SuperMap The steps for mapping to a graph network are as follows: First, preserve the original hypergraph. First, all nodes are represented as the node set V of the new network; second, for each pair of node relationships, whether an edge is established between the nodes depends on their position in the network. First, determine if there are any shared hyperedges between the nodes. If so, establish an edge connecting these node pairs in the new network. Second, add the established edges between the node pairs to the new network, ensuring that no edges are repeated. Thus, a graph network corresponding to the original hypergraph is formed. The construction is then complete, including, This represents the set of all connected edges in the mapped graph network.

4. The high-order network propagation dynamics classification method based on node weighted clique degree and deep learning according to claim 1, characterized in that, In step 2, the propagation dynamics on the graph network are established using a "susceptible-infected" propagation dynamics model. In this model, node states are divided into two categories: susceptible state S and infected state I. Nodes in susceptible state S will be propagated by infected state I nodes and change their state to infected state. In the SI model applied to the graph network, for each propagation time step, nodes in the susceptible state will change their state to infected state by contacting their connected infected state nodes with a propagation rate β. As time steps are iterated, the number of infected individuals in the system will gradually increase over time until all nodes in the system become infected.

5. The high-order network propagation dynamics classification method based on node weighted clique degree and deep learning according to claim 4, characterized in that, In step 2, within the simplex network, the probability of an individual being infected within the k-simula of a susceptible node is characterized by the propagation rate of several simplexes composed of infected nodes in the individual's neighborhood. That is, the propagation rate resulting from a node independently being adjacent to a k-simula composed of infected nodes is β. k That is, when k=1, the propagation degenerates into a simple link propagation mode that only propagates through edges.

6. The high-order network propagation dynamics classification method based on node weighted clique degree and deep learning according to claim 5, characterized in that, In step 2, for the propagation process on the hypergraph, the probability of a susceptible node being propagated along each adjacent hyperedge is related to the number of infected nodes in that hyperedge; that is, the propagation rate is... Where λ and ξ are adjustable parameters characterizing the linearity and nonlinearity of the propagation, respectively. Indicates the superedge e j (e j The number of infected nodes contained in (∈E).

7. The high-order network propagation dynamics classification method based on node weighted clique degree and deep learning according to claim 1, characterized in that, In step 3, the structural feature calculation includes two parts: calculating the node degree of nodes in the network and calculating the clustering coefficient of nodes. The node degree of a node is defined as the total number of unique nodes contained in the hyperedges to which the node belongs. The node degree of node i is denoted as D(i). The calculation of node degree centrality depends on the association matrix I constructed on the hypergraph. This association matrix is ​​constructed through the membership relationships between nodes and their neighbors in the hypergraph. Specifically, when node i is contained in hyperedge e... j At that time, I ij The value is 1 if the condition is met, and 0 otherwise. Based on the constructed incidence matrix, the adjacency matrix A between nodes in the hypergraph can be calculated. in, The matrix is ​​a diagonal matrix whose elements Let represent the number of hyperedges that node i belongs to in the hypergraph. Then, calculate the degree centrality value for each node: in, It is matrix A ij The binary matrix, that is, for matrix A ij For each element, elements with values ​​greater than 0 are assigned the value 1, otherwise they are assigned the value 0; Next, computing networks The clustering coefficient of each node, which reflects the connectivity between the node's neighbors, is defined as: In the formula, This represents the number of edges that connect node i to its neighboring nodes. Finally, computational networks The weighted clique degree of each node in the algorithm, where the clique degree CD(i) of node i can be mathematically represented as: in, Let κ∪{i} be the set of neighboring nodes of node i, that is, the set of nodes directly connected to node i through edges. Let k be a subset of the set of neighboring nodes. κ∪{i} is a set of nodes containing node i and its neighboring subset κ, and it must be a clique, denoted as δ(κ∪{i}), meaning that all nodes in the set κ∪{i} must be pairwise connected. Furthermore, ε... κ∪{i} Let |δ(κ∪{i})| represent the number of cliques adjacent to node i, and let |δ(κ∪{i})| represent the total number of nodes contained in the clique.

8. The high-order network propagation dynamics classification method based on node weighted clique degree and deep learning according to claim 7, characterized in that, In step 3, the dynamic feature processing records the changes in the node degree, clustering coefficient, and weighted clique degree of a node based on its infection order in the network, within the hypergraph. In the process, firstly, a node is randomly selected from the node set V as the initial infected node; then, in the hypergraph... The propagation dynamics process on the hypergraph is executed, where the propagation parameters are set as follows: and Corresponding to the propagation process under each set of λ and ξ parameter configurations, the time t at which each node i is infected is recorded during the process. i This continues until the propagation process ends, with the end time T being the moment the propagation process finally reaches a steady state. Subsequently, for the propagation dynamics process executed under each set of propagation parameters, nodes are arranged according to their infection time t. i Sort the nodes from smallest to largest to obtain the node sequence. and the infection time series corresponding to the node sequence Based on this, according to Calculate the node degree, clustering coefficient, and weighted clique degree of the corresponding node in the sequence to obtain a node degree sequence. A sequence of node clustering coefficients and a node weighted clique degree sequence Repeat the above process R times to obtain R infection time series for each set of transmission parameters. The node degree sequence corresponding to R infection sequences Clustering coefficient sequence of nodes corresponding to R infection sequences and the node clique degree sequences corresponding to the R infection sequences In simple complex HeTu Network The propagation processes corresponding to step 2 are executed respectively, wherein the parameters of the propagation dynamics process on the simplex complex are configured as follows: and The parameters of the propagation dynamics process on the diagram are configured as follows: After performing the propagation process on the simple complex, node infection sequence feature sets were obtained under each set of propagation parameters. Node degree sequence group Node clustering coefficient sequence group and node clique degree sequence group and in graph networks In the middle, the node infection sequence group under each set of propagation parameters Node degree sequence group Node clustering coefficient sequence group and node clique degree sequence group 9. The high-order network propagation dynamics classification method based on node weighted clique degree and deep learning according to claim 8, characterized in that, In step 3, feature merging and sample partitioning include two sub-tasks: feature merging and sample partitioning. For the feature merging process, taking a hypergraph as an example, based on the hypergraph... In λ=λ m (1≤m≤L λ ) and ξ=ξ n (1≤n≤L ξ The data obtained is used to infect the r-th (1≤r≤R) node with the sequence. Node degree corresponding to infection sequence Clustering coefficient features of nodes By merging, a unified feature sequence is obtained. Right now Since 1 ≤ r ≤ R, the above steps yield R merged feature sequences, which form a feature matrix. Then, the feature matrices under different parameters λ and ξ are merged row by row to obtain a merged feature matrix. Right now: Correspondingly, in simple complex HeTu Network The corresponding feature matrix is ​​obtained from the above. and Furthermore, label vectors representing the actual categories of dynamic processes on hypergraphs, simplexes, and graph networks are processed, i.e. in, and These are vectors composed of values ​​0, 1, and 2, respectively. Next, the feature matrix... and tags Both sets were divided in a 7:3 ratio to form training and test sets.

10. The high-order network propagation dynamics classification method based on node weighted clique degree and deep learning according to claim 9, characterized in that, In step 4 Includes the following: First, an MLP model is built, consisting of an input layer, three hidden layers, and one output layer. The input layer receives feature data, and the number of neurons in the input layer matches the feature dimension of the training set. The number of neurons in the three hidden layers is twice, once, and half the number of neurons in the input layer, respectively, and they employ the ReLU activation function, i.e., ReLU(x) = max(0,x), to enhance the model's non-linear expressive power. The output layer consists of three neurons, corresponding to the prediction units of a hypergraph, a simplex, and a general graph network dynamics, respectively, and employs the Softmax activation function, i.e.: Among them, z α For the unnormalized output of the α-th class, the Softmax function can convert the network output into a probability distribution for multi-class classification tasks; Subsequently, the constructed training set is input into the defined MLP model, and the model is trained using the stochastic gradient descent optimization algorithm. During the training process, the cross-entropy loss function is selected to measure the error between the model's prediction results and the true labels. The learning rate lr is dynamically adjusted during the training process, that is, the changing trend of the loss function value is monitored. Ultimately, the trained classifier S has the ability to effectively distinguish the propagation dynamics processes on different topologies. This classifier can predict the propagation category based on the structural features and dynamic evolution patterns of the input data. In the prediction phase, the test set is input into the trained classifier model ψ for prediction. For each sample in the test set, after calculation by the classifier ψ, the three neurons in the output layer generate three probability weight values, namely... and These correspond to the propagation dynamics processes on hypergraphs, simplexes, and graph networks, respectively. Then, the three probability weights for each sample are compared, and the class corresponding to the highest probability is selected as the predicted class label for that sample, i.e.: in, This represents the final prediction result of the classifier model, summing the prediction results for all samples in the test set. In preparation for evaluating the effectiveness of the model.

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