Method for enhancing robustness of multilayer network based on representation learning

Abstract

The invention discloses a method for enhancing robustness of a multilayer network based on representation learning, which comprises the following steps of: constructing initial features, and comprehensively extracting local information and global information of the multilayer network by capturing local structure features and node pair relation features so as to provide high-quality input for subsequent steps. Uniform node representation is generated through intra-layer feature learning, inter-layer feature mapping and an inter-layer attention mechanism, and the modeling capability of a complex coupling relation is enhanced. And calculating a node selection probability by using a multi-head attention mechanism, selecting nodes and adding connecting edges according to the node selection probability, completing an action decision, and updating a network state.

Description

Technical Field

[0001] The present invention relates to the field of network enhancement technologies, and particularly to a method for enhancing the robustness of multi-layer networks based on representation learning. Background Art

[0002] With the rapid development of information technology, the structure of complex networks increasingly exhibits multi-layer characteristics. For example, traffic networks, social networks, and communication networks, etc., often have complex coupling relationships between layers. How to enhance the robustness of multi-layer networks to cope with problems such as node failures or line short circuits has become one of the core challenges in the field of network enhancement.

[0003] Traditional network enhancement methods are mostly based on heuristic rules or expert experience. For example, the connectivity of single-layer networks is improved by adding redundant edges or optimizing key nodes. However, these methods have significant limitations when dealing with multi-layer networks: on the one hand, the heterogeneity of multi-layer networks leads to large differences in the topological structures and functions of different layers, making it difficult to optimize them through unified rules; on the other hand, the inter-layer dynamic coupling effects (such as cascading failures) are not fully considered, resulting in insufficient robustness of enhancement strategies in complex scenarios. In addition, existing methods usually rely on static analysis, lack adaptability to the dynamic evolution of networks, and are difficult to achieve the globally optimal enhancement effect under a limited edge budget.

[0004] Graph representation learning provides a data-driven solution for network analysis by mapping nodes and edges to a low-dimensional vector space. Existing technologies (such as graph neural networks, graph convolutional networks, etc.) can effectively capture the local structures and global features of single-layer networks. However, in multi-layer networks, the interaction relationships and cross-layer influences of nodes in each layer are highly complex, and it is difficult for traditional methods to effectively fuse inter-layer information. For example, single-layer graph representations cannot reflect the dynamic role differences of nodes in different layers, and static cross-layer aggregation strategies (such as fixed-weight fusion) ignore the time-varying characteristics of inter-layer relationships, resulting in the generated node representations lacking accurate characterization of the global coupling effect.

[0005] Reinforcement learning has been gradually applied to the field of network optimization due to its advantages in dynamic decision-making problems. Existing research mainly learns the optimal node or edge operation strategies through the interaction between agents and the environment. However, in multi-layer network scenarios, reinforcement learning faces two major challenges: firstly, the complexity of the state space (such as multi-layer topology, cross-layer coupling) makes it difficult for traditional policy networks to efficiently represent network states; secondly, the design of the reward function fails to fully consider key indicators such as the cascading failure mechanism and the maximum common connected component (MCCC) of multi-layer networks, resulting in a deviation between the goal of policy optimization and the actual robustness requirements. In addition, existing methods lack the ability to adaptively adjust the inter-layer dynamic weights in the action space design, which limits the generalization and scalability of the strategies.

[0006] The existing technologies have deficiencies in network enhancement, graph representation learning, and reinforcement learning: traditional network enhancement methods are difficult to handle multi-layer heterogeneity and dynamic coupling; graph representation learning lacks fine-grained modeling of cross-layer interactions; reinforcement learning strategies are inefficient in complex state spaces and reward function design. Summary of the Invention

[0007] To solve the limitations and defects of the existing technologies, the present invention provides a method for enhancing the robustness of multi-layer networks based on representation learning, including:

[0008] Construct initial features, and comprehensively extract the local information and global information of the multi-layer network by capturing local structure features and node pair relationship features;

[0009] Perform intra-layer representation on nodes through a multi-layer perceptron and a graph neural network, dynamically fuse node features between different layers using an inter-layer attention mechanism, and generate unified node representations through intra-layer feature learning, inter-layer feature mapping, and inter-layer attention mechanism;

[0010] Use a multi-head attention mechanism to calculate node selection probabilities, select nodes and add edges according to the node selection probabilities, complete action decisions, and update the network state.

[0011] Optionally, it further includes:

[0012] Design a reward function according to the cascade failure mechanism and the largest common connected component, and use the reward function to evaluate the robustness of the multi-layer network.

[0013] Optionally, it further includes:

[0014] The expression for obtaining the local structure feature is as follows:

[0015]

[0016] Wherein, represents the local structure feature of node i in the l-th layer, f is a feature extraction function, is the degree centrality of node i in the l-th layer, is the average neighbor degree of node i in the l-th layer, is the clustering coefficient of node i in the l-th layer;

[0017] The expression for obtaining the node pair relationship feature is as follows:

[0018]

[0019] Wherein, represents the node pair relationship feature between node i and node j in the l-th layer, g is a feature extraction function, is the physical distance between node i and node j in the l-th layer, is the product of the degrees of node i and node j in the l-th layer, is the algebraic distance between node i and node j in the l-th layer, is the Jaccard coefficient between node i and node j in the l-th layer;

[0020] Form a comprehensive node feature based on the local structural features and the node pair relationship features Normalize the comprehensive node feature X to obtain the feature matrix of each layer of the network:

[0021]

[0022] where X (l) is the feature matrix of the l-th layer, with dimension n×d (l) , n is the number of nodes, d (l) is the feature dimension of each layer.

[0023] Optionally, it further includes:

[0024] Generate the initial feature vector representation of the nodes in the multi-layer network according to the multi-layer perceptron, and the expression is as follows:

[0025]

[0026] where, is the initial feature vector representation of node i in the l-th layer, is the feature matrix of node i in the l-th layer, W0 is the trainable weight matrix shared by the multi-layer network, ReLU is the activation function for introducing non-linearity, and Norm is the normalization operation for adjusting the scale of the features;

[0027] Use the graph neural network to capture the relationships between nodes and the global structure of the graph, and the expression is as follows:

[0028]

[0029] where, represents the features of the nodes in the multi-layer network during the information propagation process of the graph neural network in the (k-1)-th layer, and the initial k represents the information of the k-hop neighbors to be captured, represents all the neighbor nodes of node i in the l-th layer of the multi-layer network, represents the features of the neighbor node u of node i, Agg is the aggregator for aggregating the information of neighbor nodes, and are the learnable weight matrices of the k-th layer graph neural network for adjusting the contribution degrees of the node's own features and the neighbor node's features in the feature update, Tanh is the activation function for introducing non-linearity, and Norm is the normalization operation for adjusting the scale of the features;

[0030] Embed nodes from different layers into the same space, eliminate the representation differences of embeddings in each layer of the multi-layer network, and enable the interaction and fusion of node features under unified semantics and unified scale. The expression is as follows:

[0031]

[0032] Among them, h i (l) is the initial feature vector representation of node i in the l-th layer, W3 is the trainable weight matrix, b1 is the bias, and Tanh is the activation function, which is used to introduce non-linear characteristics.

[0033] Optionally, it also includes:

[0034] Through the inter-layer attention mechanism, flexibly adjust the information propagation and aggregation strategy according to the changes of node characteristics and network structure, and adaptively adjust the fusion weight of inter-layer features for each node. The expression is as follows:

[0035]

[0036] Among them, is the inter-layer attention weight of node i from layer n to layer m, W4 is the trainable weight matrix, h i (m) is the initial feature vector representation of node i in the m-th layer, h i (n) is the initial feature vector representation of node i in the n-th layer, h i (l) is the initial feature vector representation of node i in the l-th layer, b2 is the bias, represents the Hadamard product, function is the activation function, which is used to introduce non-linear characteristics;

[0037] Combine the intra-layer representation and inter-layer influence weight of the node to generate the final node embedding of the multi-layer network, and at the same time obtain the global representation of each layer graph in the multi-layer network according to the node embedding. The expression is as follows:

[0038]

[0039]

[0040] Among them, represents the final feature representation of node i in the l-th layer, is the feature vector representation of node i in the l-th layer after unified mapping, represents the weighted sum of features of node i except the l-th layer, represents the feature representation of the l-th layer network, and Pooling is the function that aggregates the features of all nodes in the l-th layer.

[0041] Optionally, it further includes:

[0042] Taking the state vector as a query, taking all the representations of the nodes in the l-th layer as keys and values, calculating the correlation between the nodes and the state space, and obtaining the global perspective vector g (l) , and the expression is as follows:

[0043]

[0044] where g (l) is the global perspective vector, MHA represents the multi-head attention mechanism, is the set of nodes in the currently selected layer l, is the representation of the current graph state, is the representation of the currently selected node. When no node is selected When a node has been selected

[0045] Taking the global perspective vector g (l) and transforming and mapping it into a query vector q (l) , and linearly mapping the nodes in the l-th layer to generate key vectors , and the expression is as follows:

[0046] q (l) = W5g (l)

[0047]

[0048] where q (l) is the query vector, g (l) is the global perspective vector, is the key vector, W5 and W6 are trainable weight matrices, is the final node representation in each layer of the network.

[0049] Optionally, it further includes:

[0050] Calculating the dot product of the query vector q (l) and the key vector , normalizing the dot product, measuring the matching degree between the current state and the node (l) and obtaining the selection score of each node , and the expression is as follows:

[0051]

[0052] where C is a control coefficient, and q (l) represents the query vector of the l-th layer. Denote the key vector of the $l$-th layer, $d$ represents the query vector dimension, and Tanh is the activation function, which is used to introduce non-linearity;

[0053] Use the Softmax function to convert the selected score into the probability distribution of node selection. Select the node with the highest probability according to the probability distribution to complete the action decision, update the network state and policy learning. The expression is as follows:

[0054]

[0055] where, represents the selection probability of node $i$ in the $l$-th layer, is the selected score of node $i$ in the $l$-th layer, and exp is the exponential function, represents the exponential sum of the selected scores of all nodes.

[0056] The present invention has the following beneficial effects:

[0057] The present invention provides a method for enhancing the robustness of a multi-layer network based on representation learning, including: constructing initial features, comprehensively extracting local information and global information of the multi-layer network by capturing local structural features and node pair relationship features, and providing high-quality input for subsequent steps. Through intra-layer feature learning, inter-layer feature mapping and inter-layer attention mechanism, generate unified node representations, and enhance the modeling ability for complex coupling relationships. Use the multi-head attention mechanism to calculate the node selection probability, and perform operations of selecting nodes and adding edges according to the node selection probability to complete the action decision and update the network state. BRIEF DESCRIPTION OF THE DRAWINGS

[0058] Figure 1 It is a schematic diagram of a multi-layer network representation framework for fusing inter-layer influences provided in Embodiment 1 of the present invention.

[0059] Figure 2 It is a schematic diagram of the multi-layer network enhancement process of reinforcement learning provided in Embodiment 1 of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0060] To enable those skilled in the art to better understand the technical solutions of the present invention, the method for enhancing the robustness of a multi-layer network based on representation learning provided by the present invention will be described in detail below with reference to the accompanying drawings.

[0061] Embodiment 1

[0062] This embodiment provides a method for enhancing the robustness of multi-layer networks based on representation learning. By fusing multi-dimensional node features, considering the dynamic influence between layers, and designing a reward function suitable for evaluating the robustness of multi-layer networks, the limitations of traditional network enhancement methods in multi-layer networks are solved. The method includes: a multi-layer network representation framework that fuses inter-layer influence, and policy learning and robustness evaluation. First, construct the initial node features, and extract local structural features and node pair relationship features. Then, perform intra-layer representation of nodes through multi-layer perceptrons and graph neural networks, and use the inter-layer attention mechanism to dynamically fuse node features between different layers. Finally, calculate the selection probability of nodes using multi-head attention based on the obtained node fusion features. Then, the agent learns the optimal policy through interaction with the environment, selects nodes and adds edges to enhance the network robustness. The state space includes the node features of the entire network. The agent operates according to the node selection probability, and the goal is to maximize the improvement of network robustness. To accurately design the reward function to evaluate robustness, a cascade failure mechanism and the largest common connected component are introduced to measure the impact of node failures on the network. This method overcomes the problems of heterogeneity and dynamic coupling in traditional methods when dealing with multi-layer networks, has better dynamic adaptability and global optimization ability, and can effectively improve the robustness of multi-layer networks.

[0063] This embodiment proposes a collaborative framework for multi-layer network representation learning and robustness enhancement based on reinforcement learning through the cascade failure mechanism and multi-dimensional feature extraction, aiming to break through the above technical bottlenecks and achieve the unity of dynamic optimization and global robustness improvement of multi-layer networks.

[0064] This embodiment uses representation learning and reinforcement learning to solve the problem of multi-layer network enhancement. By fusing multi-dimensional node features and considering the dynamic influence weights between layers, the model can understand the network from both local and global perspectives, generate high-quality node representations, guide policy optimization, introduce the cascade failure mechanism of the system, and effectively evaluate the robustness of multi-layer networks through the largest common connected component, better solving the problem of enhancing the robustness of multi-layer networks. This method includes two parts: a multi-layer network representation framework that fuses inter-layer influence, and policy learning and robustness evaluation.

[0065] 1.1 Construction of Initial Node Features

[0066] Figure 1 This is a schematic diagram of the multi-layer network representation framework that fuses inter-layer influence provided by Embodiment 1 of the present invention. In this embodiment, the multi-layer network is defined as graph G=(G (1) ,G (2) ,...,G (|L|) )=(V,E), where |L| represents the number of layers of the network. Each layer of the network shares nodes, and V (1) =V (2) =…=V (|L|)= V, where the node set V = {v1, v2…, v N}, the relationship set of nodes within each layer of the network, where E (l) represents the edge set of the l-th layer network.

[0067] Although traditional heuristic algorithms are simple and easy to implement, their performance is limited by expert experience. In multi-layer networks, the heterogeneity and scale differences of network structures make it difficult for traditional methods to adapt to networks in different scenarios. The feature extraction module absorbs the advantages of heuristic algorithms while overcoming their limitations. By capturing local and global information, it helps the model understand the essence of the network from a higher dimension. At the same time, the decision-making effect of the reinforcement learning method depends on high-quality state and action representations. By extracting key features, the decision-making accuracy of the model for selecting nodes and adding edges can be improved. This module includes the following two types of features:

[0068] Local structure features, a) Centrality metrics: including degree centrality (the number of direct connections of a node) and average neighbor degree (the average degree of neighbor nodes), used to measure the importance of a node. b) Clustering coefficient: describes whether there is a triangular closed loop in the neighborhood of a node, used to reflect local connectivity. The specific formulas are as follows:

[0069]

[0070] where represents the local comprehensive feature of node i in the l-th layer.

[0071] Node pair relationship features, a) Node pair distance: represents the physical or topological distance between two nodes. b) Product of degrees: the product of the degrees of the two nodes in a node pair, used to capture the interaction relationship between high-degree and low-degree nodes. c) Algebraic distance: based on the eigenvectors of the Laplacian matrix, reflecting the potential connection strength between node pairs. d) Jaccard coefficient: measures the similarity of the neighbors of a node pair, reflecting the closeness of the relationship between node pairs. The specific formulas are as follows:

[0072]

[0073] where represents the relationship feature between node i and node j in the l-th layer.

[0074] In this embodiment, the local features and node pair relationship features of each layer are combined into the comprehensive feature of the node Then the comprehensive feature X is normalized, and finally the feature matrix of each layer of the network is obtained:

[0075]

[0076] where X (l)is the feature matrix of the l-th layer, with dimensions n×d (l) , where n is the number of nodes, and d (l) is the feature dimension of each layer.

[0077] This module aims to comprehensively capture the local and global information of the network, thereby enhancing the generalization and decision-making capabilities of the model. By identifying the key elements in the network, it can effectively guide the direction of policy optimization. Mining the local information of nodes provides a microscopic perspective for network representation, and the method of measuring the potential interaction and correlation strength between nodes enables the model to formulate more comprehensive optimization strategies from a global perspective while paying attention to local details, helping the model to more accurately learn the strategy of adding edges. These features complement each other, providing higher generalization ability and optimization effect for solving the network enhancement problem.

[0078] 1.2 Multi-layer Network Representation Module

[0079] 1.2.1 Intra-node Layer Representation

[0080] In a multi-layer network, first, it passes through a multi-layer perceptron layer (MLP) to generate the initial feature vector representation of the nodes in the multi-layer network. The specific formula is as follows:

[0081]

[0082] where is the feature matrix of node i in the l-th layer obtained by the feature extraction module, is the feature matrix of node i in the l-th layer, W0 is the trainable weight matrix shared by the multi-layer network, ReLU is the activation function, introducing non-linearity, and Norm is the normalization operation, adjusting the scale of the features and optimizing the learning process of the model.

[0083] Then, a graph neural network can be used to represent the topological structure of the entire graph. Nodes update their own representations by exchanging information with their neighbor nodes, thereby realizing the transmission of information on the graph. This mechanism enables the graph neural network to capture complex relationships between nodes and the global structure of the graph. The specific formula is as follows:

[0084]

[0085] where represents the features of the nodes in the multi-layer network during the information propagation process of the GNN in the (k - 1)-th layer. The initial k represents the information of k-hop neighbors to be captured, represents all the neighbor nodes of node i in the l-th layer of the multi-layer network, represents the features of the neighbor node u of node i, Agg represents the aggregator, and by aggregating the information of neighbor nodes, it generates a representation for the node related to its local structure and neighborhood features, and is the learnable weight matrix of the k-th layer GNN, which adjusts the contribution degrees of the node's own features and the neighbor nodes' features in feature update. Tanh is the activation function, introducing non-linearity. Norm is the normalization operation, adjusting the scale of features and optimizing the learning process of the model.

[0086] 1.2.2 Node Feature Mapping

[0087] In a multi-layer network, nodes in different layers influence each other. To quantify the cross-layer influence and address the heterogeneity of node representations within a layer, by unifying the node embeddings of different layers into a shared embedding space, it is possible to effectively eliminate the representational differences of the embeddings in each layer of the multi-layer network, enabling the interaction and fusion of node features under a unified semantics and scale, while ensuring the consistency of the global optimization objective. The specific formula is as follows:

[0088]

[0089] where W3 is the trainable weight matrix, b1 is the bias, and the parameters are shared among nodes in each layer to ensure the consistency of cross-layer mapping. Tanh is the activation function, introducing non-linearity.

[0090] 1.2.3 Inter-layer Feature Fusion of Nodes

[0091] Traditional aggregation algorithms usually use fixed cross-layer weights to quantify the inter-layer influence. Although this method is convenient for calculation, it ignores the dynamics of the inter-layer node relationships, especially the variability of the weights of different cross-layer connections of the same node. The fixed-weight approach is difficult to comprehensively characterize the high-order node interaction relationships in complex networks, limiting the model's ability to express the dynamic details of the network structure. To address this problem and enhance the model's encoding effect, a node-level attention mechanism is designed. This mechanism can dynamically calculate custom attention weights for each cross-layer node relationship, thereby capturing richer high-order interaction patterns. By introducing the inter-layer attention mechanism, the model can flexibly adjust the information propagation and aggregation strategies according to the changes in node characteristics and network structure, and adaptively adjust the fusion weights of inter-layer features for each node. The specific formula is as follows:

[0092]

[0093] where is the calculated inter-layer attention weight of node i from layer n to layer m. W4 is the trainable weight matrix, b2 is the bias, represents the Hadamard product, function is the activation function, introducing non-linearity. It should be noted that α i is asymmetric, that is is not equal to This indicates that a node in one layer may be important to another layer, but vice versa may not be the case.

[0094] Finally, by combining the intra-layer representation and the inter-layer influence weights of the nodes, the final node embeddings of the multi-layer network are generated. At the same time, the global representation of each layer of the graph in the multi-layer network can be obtained from the node embeddings in this embodiment. The specific formula is as follows:

[0095]

[0096]

[0097] where Pooling is a function that aggregates the features of all nodes in the l-th layer, which can be the mean, maximum, or attention-weighted aggregation.

[0098] 1.3 Node Selection Module

[0099] In reinforcement learning, the agent learns the optimal policy by interacting with the environment. In the multi-layer network problem, the environmental state not only includes the node information of a single layer, but also covers the dynamic relationship of node interactions between layers. This module uses the and node feature representations as the state space. The agent uses the multi-head attention mechanism to calculate the selection probability of the nodes to determine the probability distribution of the next action. Each action corresponds to selecting a node in layer l of the network, thereby completing the addition of edges.

[0100] To fuse the current state with the node features of the specified layer l, this embodiment uses the multi-head attention mechanism (MHA) to calculate a global perspective vector g (l) , which contains the global graph representation and the local information of the currently selected node, and is the overall characterization of the environmental state by the reinforcement learning agent. By using the state vector as the query, and all the node representations in the l-th layer as the keys and values, the correlation between each node and the state space is calculated, thereby extracting the vector g (l) that can reflect the global information within the current layer l. This step can dynamically focus on the most useful part of the information in the state space and eliminate those nodes that do not meet the conditions or are irrelevant (for example, by setting the attention scores of unselectable nodes to -∞). The specific formula is as follows:

[0101]

[0102]

[0103] where g (l) is the global perspective vector, MHA represents the multi-head attention mechanism, is the set of nodes in the currently selected layer l, is the representation of the current graph state, is the representation of the currently selected node. When no node is selected If there are selected nodes

[0104] Then, after obtaining the global perspective vector g (l) Next, it is transformed and mapped into the query vector q (l) At the same time, a linear mapping is performed on each node in the l-th layer to generate the key vector The specific formula is as follows:

[0105] q (l) = W5g (l)

[0106]

[0107] where both W5 and W6 are trainable weight matrices is the final node representation in each layer of the network

[0108] Then, this embodiment uses the obtained query vector q (l) and the key vector By calculating the dot product of q (l) and and normalizing it, the matching degree between the current state and the node is measured, and then the selection score of each node is obtained The specific formula is as follows:

[0109]

[0110] where C is the hyperparameter that controls the exploration and exploitation of the agent in reinforcement learning, d represents the dimension of the query vector, Tanh is the activation function, introducing non-linear characteristics. Optional condition control: For nodes that do not meet the conditions (such as those that have been selected or are not selectable due to constraint conditions), their scores are set to -∞ to ensure that the probability after Softmax is zero, so that they are not sampled

[0111] Finally, the selection scores are converted into the probability distribution of node selection through the Softmax function. The reinforcement learning agent samples or selects the node with the highest probability according to this distribution, thus completing the action decision, and then updating the network state and policy learning. The specific formula is as follows:

[0112]

[0113] 2. Policy learning and robustness evaluation

[0114] In the multi-layer network enhancement problem, reinforcement learning is used to solve the multi-layer network enhancement problem based on the above representation model. The state is defined as the vector representation of the entire network and nodes. The action is to add edges according to the node selection probability, and the reward is the improvement of the network robustness after taking the action. The following is a detailed elaboration.

[0115] Figure 2 This is a schematic diagram of the multi-layer network enhancement process of reinforcement learning provided in the first embodiment of the present invention. For a given edge budget edges, the decision-making process of the agent in reinforcement learning requires 2*edges steps. In each step, the agent observes the state of the entire network G, selects a node in the l-th layer of the network G, and then executes an action to change the network state. The detailed process is as follows: for 0 ≤ t ≤ edges - 1, the state s 2t is a tuple where is obtained by adding the decision result of the agent to the graph G, is the node selected by the agent in each step, 1 ≤ l ≤ |L|, and l represents the l-th layer of the selected node in the graph (the l in the 2t-th step and the 2t + 1-th step is the same), represents that there is no edge to be added initially. When the agent observes the state s 2t it selects a node in the l-th layer and executes an action to update the state of the graph to At this time, the graph structure does not change, that is, G 2t+1 = G 2t . Then the agent continues to select the next node in the l-th layer. At this time, the operation of adding an edge is executed, and the edge is added to the graph G 2t+1 to obtain the updated graph state The agent continuously repeats the above process within the edge budget. Finally, the reward it can obtain is the improvement of the robustness of the graph G 2*edges relative to the original graph G.

[0116] The goal of the agent's learning is to maximize the expected cumulative reward of the policy. The specific formula is as follows:

[0117]

[0118] where the policy is parameterized as π θ (a|s), which represents the probability of selecting action a in state s, and θ is the parameter of the model. τ is the trajectory generated by the policy π θ , R(τ) represents the improvement of the robustness of the graph G 2*edges relative to the original graph G.

[0119] To better evaluate the robustness of multi-layer networks, this embodiment introduces the cascading failure mechanism (Cascading Failure) here. That is, in a multi-layer network, there is a coupling relationship between layers, which means that the failure of nodes in one layer may cause the failure of nodes in other layers that depend on these nodes. In addition, this impact may recursively expand, thereby causing a chain reaction of failure sequences. At the same time, this embodiment needs to consider the maximum common connected component (MCCC) in the multi-layer network, which refers to the largest set of nodes that exist simultaneously in multiple network layers and are connected within these layers. That is, any two nodes can reach each other not only in a single layer but also across different layers. Through the above analysis, it is possible to better evaluate which nodes will have a profound and extensive impact on the entire network, so as to formulate more effective learning strategies. Therefore, the definition of robustness is the minimum number of nodes that must be deleted to make the MCCC of graph G equal to 1 under the cascading failure mechanism, where represents the strategy for deleting nodes, CF(·) represents the cascading failure mechanism, q is the sequence of selected nodes to be deleted, and the specific formula is as follows:

[0120]

[0121] This embodiment proposes a robustness enhancement method based on representation learning, aiming to address the challenges of multi-layer networks in terms of heterogeneity, dynamic coupling, and global optimization. By introducing the cascading failure mechanism and the maximum common connected component (MCCC), the model can effectively evaluate the robustness of multi-layer networks. The reinforcement learning agent learns the optimal node or edge operation strategy through interaction with the environment. The state is defined as the entire network, the action is to select a node to add an edge, and the reward is the improvement of network robustness.

[0122] The core modules of the model include initial feature construction, multi-layer network representation module, and node selection module. Initial feature construction comprehensively extracts the local and global information of the network by capturing local structure features (such as degree centrality, clustering coefficient) and node pair relationship features (such as node pair distance, Jaccard coefficient), providing high-quality input for subsequent modules. The multi-layer network representation module generates unified node representations through intra-layer feature learning, inter-layer feature mapping, and inter-layer attention mechanisms, enhancing the modeling ability for complex coupling relationships. The node selection module then calculates the node selection probability using the multi-head attention mechanism, guiding the agent to select nodes and add edge operations, completing action decisions and updating the network state.

[0123] Overall, through the collaboration of reinforcement learning and graph representation learning, this embodiment solves the limitations of traditional methods in multi-layer network enhancement, and has dynamic adaptability and global optimization capabilities. In the future, this method is expected to be widely applied in complex network scenarios such as transportation, social, and communication, further improving the robustness and adaptability of multi-layer networks, and providing innovative solutions for the field of network optimization.

[0124] It can be understood that the above embodiments are merely exemplary embodiments adopted to illustrate the principles of the present invention, but the present invention is not limited thereto. For those of ordinary skill in the art, various modifications and improvements can be made without departing from the spirit and essence of the present invention, and these modifications and improvements are also considered within the protection scope of the present invention.

Claims

1. A method for enhancing the robustness of a multi-layer network based on representation learning, characterized in that, Including: Construct initial features, and comprehensively extract local and global information of the multi-layer network by capturing local structural features and node pair relationship features; Perform intra-layer representation of nodes through a multi-layer perceptron and a graph neural network, dynamically fuse node features between different layers using an inter-layer attention mechanism, and generate unified node representations through intra-layer feature learning, inter-layer feature mapping, and inter-layer attention mechanism; Use a multi-head attention mechanism to calculate node selection probabilities, select nodes and add edges according to the node selection probabilities, complete action decisions, and update the network state.

2. The method for enhancing the robustness of a multi-layer network based on representation learning according to claim 1, wherein Also including: Design a reward function according to the cascade failure mechanism and the largest common connected component, and use the reward function to evaluate the robustness of the multi-layer network.

3. The method for enhancing the robustness of a multi-layer network based on representation learning according to claim 2, wherein Also including: The expression for obtaining the local structural feature is as follows: Among them, represents the local structural feature of node i in the l-th layer, f is the feature extraction function, is the degree centrality of node i in the l-th layer, is the average neighbor degree of node i in the l-th layer, is the clustering coefficient of node i in the l-th layer; The expression for obtaining the node pair relationship feature is as follows: Among them, represents the node pair relationship feature between node i and node j in the l-th layer, and g is the feature extraction function. is the physical distance between node i and node j in the l-th layer. is the product of the degrees of node i and node j in the l-th layer. is the algebraic distance between node i and node j in the l-th layer. is the Jaccard coefficient between node i and node j in the l-th layer. Form a comprehensive node feature based on the local structural feature and the node pair relationship feature Normalize the comprehensive node feature X to obtain the feature matrix of each layer of the network: Among them, X (l) is the feature matrix of the l-th layer, with a dimension of n×d (l) , where n is the number of nodes, and d (l) is the feature dimension of each layer.

4. The method for enhancing the robustness of a multi-layer network based on representation learning according to claim 3, wherein Also including: Generate an initial feature vector representation of the nodes in the multi-layer network according to the multi-layer perceptron, and the expression is as follows: Among them, is the initial feature vector representation of node i in the l-th layer, is the feature matrix of node i in the l-th layer, W0 is the trainable weight matrix shared by the multi-layer network, ReLU is the activation function used to introduce non-linearity, and Norm is the normalization operation used to adjust the scale of the features; Use a graph neural network to capture the relationships between nodes and the global structure of the graph, and the expression is as follows: Among them, represents the features of multi - layer network nodes during the information propagation process of the (k - 1)-th layer graph neural network. The initial k represents the information of k - hop neighbors to be captured. represents all neighbor nodes of node i in the l - th layer of the multi - layer network. represents the features of neighbor node u of node i. Agg represents an aggregator used to aggregate the information of neighbor nodes. and are learnable weight matrices of the k - th layer graph neural network, used to adjust the contribution degrees of the node's own features and neighbor node features in feature update. Tanh is an activation function used to introduce non - linear characteristics, and Norm is a normalization operation used to adjust the scale of features. Embed nodes in different layers into the same space, eliminate the representation differences of the embeddings in each layer of the multi-layer network, and enable the interaction and fusion of node features under unified semantics and unified scales, and the expression is as follows: Among them, h i (l) is the initial feature vector representation of node i in the l-th layer, W3 is a trainable weight matrix, b1 is a bias, and Tanh is an activation function used to introduce non-linearity.

5. The method for enhancing the robustness of a multi-layer network based on representation learning according to claim 4, wherein Also including: Through the inter-layer attention mechanism, flexibly adjust the information propagation and aggregation strategies according to the changes in node characteristics and network structure, and adaptively adjust the fusion weights of inter-layer features for each node, and the expression is as follows: Among them, is the inter-layer attention weight from layer n to layer m of node i, W4 is the trainable weight matrix, h i (m) is the initial feature vector representation of node i in the m-th layer, h i (n) is the initial feature vector representation of node i in the n-th layer, h i (l) is the initial feature vector representation of node i in the l-th layer, b2 is the bias, represents the Hadamard product, and function is the activation function used to introduce non-linearity; Combine the intra-layer representation of nodes and the inter-layer influence weights to generate the final node embeddings of the multi-layer network, and at the same time obtain the global representation of each layer of the graph in the multi-layer network according to the node embeddings, and the expression is as follows: Among them, represents the final feature representation of node i in the l-th layer, is the feature vector representation of node i in the l-th layer after unified mapping, represents the weighted sum of the features of node i except for the l-th layer, represents the feature representation of the l-th layer network, and Pooling is a function that aggregates the features of all nodes in the l-th layer.

6. The method for enhancing the robustness of a multi-layer network based on representation learning according to claim 5, wherein Also including: Take the state vector as a query, take all the representations of the nodes in the l-th layer as keys and values, calculate the correlation between the nodes and the state space, and obtain the global perspective vector g (l) , and the expression is as follows: Among them, g (l) is the global perspective vector, MHA represents the multi-head attention mechanism, is the set of nodes of the currently selected layer l, is the representation of the current graph state, is the representation of the currently selected node. When no node is selected When a node has been selected Transform the global perspective vector g (l) into a query vector q (l) , and perform a linear mapping on the nodes of the l-th layer to generate key vectors The expression is as follows: Among them, q (l) is the query vector, g (l) is the global perspective vector, is the key vector, W5 and W6 are trainable weight matrices, is the final node representation in each layer of the network.

7. The method for enhancing the robustness of a multi-layer network based on representation learning according to claim 6, wherein Also including: According to the query vector q (l) and the key vector calculate the dot product of the query vector q (l) and the key vector normalize the dot product to measure the matching degree between the current state and the node and obtain the selection score of each node The expression is as follows: Among them, C is the control coefficient, q (l) represents the query vector of the l-th layer, represents the key vector of the l-th layer, d represents the dimension of the query vector, and Tanh is the activation function used to introduce non-linearity; Use the Softmax function to convert the selected score into the probability distribution of node selection. Select the node with the highest probability according to the probability distribution to complete the action decision, update the network state and policy learning. The expression is as follows: Among them, represents the selection probability of node i in the l-th layer, is the selection score of node i in the l-th layer, and exp is the exponential function, represents the exponential sum of the selection scores of all nodes.

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