A method for enhancing robustness of a multi-layer network based on representation learning

Abstract

The application discloses a method for enhancing robustness of a multi-layer network based on representation learning, comprising the following steps: constructing initial features, capturing local structure features and node pair relationship features, and comprehensively extracting local information and global information of the multi-layer network to provide high-quality input for subsequent steps; generating unified node representation through intra-layer feature learning, inter-layer feature mapping and inter-layer attention mechanism, and enhancing modeling capability for complex coupling relationship; calculating node selection probability by using a multi-head attention mechanism, and performing node selection and edge addition operation according to the node selection probability to complete action decision and update network state.

Description

Technical Field

[0001] This invention relates to the field of network enhancement technology, and in particular to a method for enhancing the robustness of multilayer networks based on representation learning. Background Technology

[0002] With the rapid development of information technology, the structure of complex networks is increasingly characterized by multi-layering, such as transportation networks, social networks, and communication networks, where complex coupling relationships often exist between different layers. Enhancing the robustness of multi-layered networks to address issues such as node failures or short circuits has become one of the core challenges in the field of network enhancement.

[0003] Traditional network augmentation methods are mostly based on heuristic rules or expert experience, such as improving the connectivity of a single-layer network 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 significant differences in the topology and function of different layers, making it difficult to optimize using unified rules; on the other hand, dynamic coupling effects between layers (such as cascading failures) are not fully considered, resulting in insufficient robustness of augmentation strategies in complex scenarios. Furthermore, existing methods typically rely on static analysis, lacking adaptability to the dynamic evolution of networks, and struggle to achieve globally optimal augmentation results with limited edge budgets.

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

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

[0006] Existing technologies have shortcomings in network augmentation, graph representation learning, and reinforcement learning: traditional network augmentation methods have difficulty handling multi-layer heterogeneity and dynamic coupling; graph representation learning lacks fine-grained modeling of cross-layer interactions; and reinforcement learning strategies are inefficient in designing complex state spaces and reward functions. Summary of the Invention

[0007] To address the limitations and shortcomings of existing technologies, this invention provides a method for enhancing the robustness of multilayer networks based on representation learning, comprising:

[0008] Initial features are constructed, and local and global information of the multi-layer network is comprehensively extracted by capturing local structural features and node pair relationship features.

[0009] Nodes are represented within layers using multilayer perceptrons and graph neural networks. Node features from different layers are dynamically fused using interlayer attention mechanisms. A unified node representation is generated through intralayer feature learning, interlayer feature mapping, and interlayer attention mechanisms.

[0010] The node selection probability is calculated using a multi-head attention mechanism. Based on the node selection probability, nodes are selected and edges are added to complete the action decision and update the network state.

[0011] Optional, also includes:

[0012] A reward function is designed based on the cascading failure mechanism and the maximum common connectivity component, and the robustness of the multilayer network is evaluated using the reward function.

[0013] Optional, also includes:

[0014] The expression for obtaining the local structural features is as follows:

[0015]

[0016] in, Let f represent the local structural features of node i in the l-th layer, and f be the feature extraction function. It is the degree centrality of node i in the l-th layer. It is the average neighbor degree of node i in the l-th layer. It is the clustering coefficient of node i in the l-th layer;

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

[0018]

[0019] in, Let g represent the node pair relationship features between node i and node j in the l-th layer, and g be the feature extraction function. It is the physical distance between node i and node j in the l-th layer. It is the degree product of node i and node j at level l. It is the algebraic distance between node i and node j at level l. is the Jaccard coefficient of node i and node j in the l-th layer;

[0020] A comprehensive node feature is formed based on the local structural features and the node pair relationship features. The node comprehensive feature X is normalized to obtain the feature matrix of each network layer:

[0021]

[0022] Among them, X (l) It is the feature matrix of the l-th layer, with dimensions n×d. (l) n is the number of nodes, d (l) The feature dimension for each layer.

[0023] Optional, also includes:

[0024] The initial feature vector representation of the multilayer network nodes is generated based on the multilayer perceptron, as expressed below:

[0025]

[0026] in, It is the initial feature vector representation of node i in the l-th layer. W is the feature matrix of node i in layer l, W0 is the trainable weight matrix shared by multiple layers of the network, ReLU is the activation function used to introduce non-linear characteristics, and Norm is the normalization operation used to adjust the scale of the features.

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

[0028]

[0029] in, This represents the characteristics of multiple network nodes during information propagation in the (k-1)th layer of a graph neural network, initially... k represents the information of the k-hop neighbor that needs to be captured. This represents all neighboring nodes of node i in layer l of a multi-layer network. Let Agg represent the characteristics of node i's neighbor node u, and let Agg represent the aggregator, which is used to aggregate information about neighboring nodes. and is the learnable weight matrix of the k-th layer graph neural network, used to adjust the contribution of the node's own features and the features of its neighboring nodes in feature updates. Tanh is the activation function used to introduce non-linear characteristics, and Norm is the normalization operation used to adjust the scale of the features.

[0030] By embedding nodes from different layers into the same space, the representational differences between layers in a multi-layer network are eliminated, enabling node features to interact and merge under unified semantics and scale. The expression is as follows:

[0031]

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

[0033] Optional, also includes:

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

[0035]

[0036] in, W4 is the inter-layer attention weight of node i from layer n to layer m, and h is the trainable weight matrix. i (m) h is the initial feature vector representation of node i in the m-th layer. i (n) h is the initial feature vector representation of node i in the nth layer. i (l) b1 is the initial feature vector representation of node i in the l-th layer, and b2 is the bias. This represents the Hadamard product, where function is the activation function used to introduce nonlinear characteristics.

[0037] By combining the intra-layer representations and inter-layer influence weights of nodes, the final node embeddings of the multi-layer network are generated. Simultaneously, the global representation of the graph in each layer of the multi-layer network is obtained based on the node embeddings, as shown in the following expression:

[0038]

[0039]

[0040] in, Let i represent the final feature representation of node i in the l-th layer. It is the feature vector representation of node i in the l-th layer after unified mapping. This represents the weighted summation of features from the l-th layer, excluding node i. The feature representation of the l-th layer network is given by Pooling, which is a function that aggregates the features of all nodes in the l-th layer.

[0041] Optional, also includes:

[0042] The state vector As a query, all nodes at level l are represented as keys and values. The correlation between the nodes and the state space is calculated to obtain the global view vector g. (l) The expression is as follows:

[0043]

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

[0045] The global view vector g (l) Transform the mapping to the query vector q (l) Generate key vectors by performing linear mapping on the nodes of the l-th layer. The expression is as follows:

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

[0047]

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

[0049] Optional, also includes:

[0050] Based on query vector q (l) and key vector Calculate query vector q (l) With key vector The dot product is normalized to measure the relationship between the current state and the node. The degree of matching between nodes is used to obtain the selection score for each node. The expression is as follows:

[0051]

[0052] Where C is the control coefficient, q (l) This represents the query vector at level l. Let d represent the key vector of the l-th layer, d represent the query vector dimension, and Tanh is the activation function used to introduce non-linear characteristics.

[0053] The score will be selected using the Softmax function. The process is transformed into a probability distribution for node selection. Based on this probability distribution, the node with the highest probability is selected to complete the action decision, update the network state and policy learning, and the expression is as follows:

[0054]

[0055] in, Let i represent the probability of selecting node i in the l-th layer. Let exp be the score for selecting node i in the l-th layer, and let exp be an exponential function. This represents the sum of the scores of all selected nodes.

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

[0057] This invention provides a method for enhancing the robustness of multilayer networks based on representation learning, comprising: constructing initial features; comprehensively extracting local and global information of the multilayer network by capturing local structural features and node pair relationship features, providing high-quality input for subsequent steps; generating unified node representations through intra-layer feature learning, inter-layer feature mapping, and inter-layer attention mechanisms, enhancing the ability to model complex coupling relationships; calculating node selection probabilities using a multi-head attention mechanism; and performing node selection and edge addition operations based on the node selection probabilities to complete action decisions and update the network state. Attached Figure Description

[0058] Figure 1 This is a schematic diagram of a multilayer network representation framework for integrating interlayer influences provided in Embodiment 1 of the present invention.

[0059] Figure 2 This is a schematic diagram of the reinforcement learning multi-layer network enhancement process provided in Embodiment 1 of the present invention. Detailed Implementation

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

[0061] Example 1

[0062] This embodiment provides a method for enhancing the robustness of multilayer 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 multilayer networks, it overcomes the limitations of traditional network enhancement methods in multilayer networks. The method includes: a multilayer network representation framework that fuses inter-layer influences, policy learning, and robustness evaluation. First, initial node features are constructed, extracting local structural features and node pair relationship features. Then, nodes are represented intra-layer using a multilayer perceptron and graph neural network. Inter-layer attention mechanisms are used to dynamically fuse node features from different layers. Finally, multi-head attention is used to calculate the node selection probability based on the obtained fused node features. Next, the agent learns the optimal policy through interaction with the environment, selecting nodes and adding edges to enhance network robustness. The state space includes the node features of the entire network. The agent operates according to the node selection probability, aiming to maximize the network's robustness improvement. To accurately design the reward function to evaluate robustness, a cascading failure mechanism and the maximum common connectivity component are introduced to measure the impact of node failures on the network. This method overcomes the problems of heterogeneity and dynamic coupling in multilayer networks that traditional methods cannot solve. It has better dynamic adaptability and global optimization capabilities, and can effectively improve the robustness of multilayer networks.

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

[0064] This embodiment utilizes representation learning and reinforcement learning to address the enhancement problem of multilayer networks. 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, generating high-quality node representations to guide policy optimization. It introduces a cascading failure mechanism and effectively evaluates the robustness of multilayer networks through the maximum common connectivity component, thus better addressing the robustness enhancement problem of multilayer networks. This method consists of two parts: a multilayer network representation framework that fuses inter-layer influences, policy learning, and robustness evaluation.

[0065] 1.1 Initial Feature Construction of Nodes

[0066] Figure 1 This is a schematic diagram of a multilayer network representation framework for integrating interlayer influences provided in Embodiment 1 of the present invention. In this embodiment, the multilayer network is defined as graph G = (G (1) G (2) ,...,G (|L|) ) = (V, E), where |L| represents the number of layers in the network, and each layer shares nodes. (1) =V (2) =…=V (|L|)=V, where the set of nodes V = {v1, v2, ..., v N}, the set of relationships between nodes in each layer of the network. Where E (l) Let represent the set of edges in the l-th layer network.

[0067] While traditional heuristic algorithms are simple and easy to implement, their performance is limited by expert experience. In multi-layer networks, the heterogeneity of network structure and differences in scale make traditional methods difficult to adapt to different network scenarios. The feature extraction module absorbs the strengths 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. Furthermore, the decision-making performance of reinforcement learning methods relies on high-quality state and action representations. By extracting key features, the model's accuracy in selecting nodes and adding edges can be improved. This module includes the following two types of features:

[0068] Local structural features include: a) Centrality indices: including degree centrality (number of direct connections to a node) and average neighbor degree (average degree of neighboring nodes), used to measure the importance of a node. b) Clustering coefficient: describing whether a triangular closed loop exists in the neighborhood of a node, reflecting local connectivity. The specific formulas are as follows:

[0069]

[0070] in This represents the local integrated feature of node i at layer l.

[0071] Node pair relationship features include: a) Node pair distance: Represents the physical or topological distance between two nodes. b) Degree product: The product of the degrees of the two nodes in a node pair, used to capture the interaction relationship between nodes of different degrees. 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 between a node pair's neighbors, reflecting the closeness of the relationship between the node pairs. The specific formulas are as follows:

[0072]

[0073] in This represents the relationship characteristics between node i and node j at level l.

[0074] This embodiment combines the local features of each layer and the node-to-node relationship features into a comprehensive feature of the node. Then, the combined features X are normalized to obtain the feature matrix of each network layer:

[0075]

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

[0077] This module aims to comprehensively capture both local and global information of the network, thereby enhancing the model's generalization and decision-making capabilities. By identifying key elements in the network, it can effectively guide the direction of policy optimization. Mining local information from nodes provides a micro-perspective for network representation, and measuring the strength of potential interactions and connections between nodes allows the model to focus on local details while also formulating more comprehensive optimization strategies from a global perspective. This helps the model learn more accurately the strategies for adding edges. These features complement each other, providing higher generalization capabilities and optimization effects for solving network augmentation problems.

[0078] 1.2 Multilayer Network Representation Module

[0079] 1.2.1 Intra-node representation

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

[0081]

[0082] in It is the feature matrix of node i in the l-th layer obtained by the feature extraction module. W is the feature matrix of node i in layer l, W0 is the trainable weight matrix shared by multiple layers of the network, ReLU is the activation function that introduces non-linearity, and Norm is the normalization operation that adjusts the scale of the features and optimizes the learning process of the model.

[0083] Then, a graph neural network can be used to represent the topology of the entire graph. Nodes update their representations by exchanging information with their neighbors, thus enabling information transfer across the graph. This mechanism allows graph neural networks to capture complex inter-node relationships and the global structure of the graph. The specific formula is as follows:

[0084]

[0085] in This represents the characteristics of multi-layer network nodes during the information propagation process in the (k-1)th layer of the GNN, initially... k represents the information of the k-hop neighbor that needs to be captured. This represents all neighboring nodes of node i in the l-th layer of a multi-layer network. Let Agg represent the features of node i's neighbor node u, and let Agg represent the aggregator, which generates a representation for a node related to its local structure and neighborhood features by aggregating information from neighboring nodes. and It is the learnable weight matrix of the k-th layer GNN, which adjusts the contribution of the node's own features and the features of its neighboring nodes in feature updates. Tanh is the activation function, which introduces non-linearity. Norm is the normalization operation, which adjusts the scale of the features and optimizes the model's learning process.

[0086] 1.2.2 Node Feature Mapping

[0087] In multi-layer networks, nodes in different layers influence each other. To quantify cross-layer influence and address the heterogeneity of node representations within a layer, unifying the embeddings of nodes from different layers into a shared embedding space effectively eliminates representational differences between layers in the multi-layer network. This allows node features to interact and merge under 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, the parameters are shared by nodes in each layer to ensure consistency of cross-layer mapping, and Tanh is the activation function, which introduces non-linear characteristics.

[0090] 1.2.3 Inter-node feature fusion

[0091] Traditional aggregation algorithms typically use fixed cross-layer weights to quantify inter-layer influence. While this method is computationally convenient, it ignores the dynamic nature of inter-layer node relationships, especially the variability of cross-layer connection weights for the same node. Fixed weights fail to fully characterize high-order node interactions in complex networks, limiting the model's ability to express dynamic details of the network structure. To address this issue and enhance model encoding performance, a node-level attention mechanism is designed. This mechanism dynamically calculates custom attention weights for each cross-layer node relationship, thereby capturing richer high-order interaction patterns. By introducing an inter-layer attention mechanism, the model can flexibly adjust information propagation and aggregation strategies based on changes in node characteristics and network structure, adaptively adjusting the fusion weights of inter-layer features for each node. The specific formula is as follows:

[0092]

[0093] in This is the calculated inter-layer attention weight for node i from layer n to layer m, where W4 is the trainable weight matrix and b2 is the bias. This represents the Hadamard product, where `function` is the activation function, introducing non-linear characteristics. Note the significance of α. i It is asymmetric, that is... and The two are not equal, which means that a node in one layer may be important to another layer, but the reverse may not be true.

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

[0095]

[0096]

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

[0098] 1.3 Node Selection Module

[0099] In reinforcement learning, an agent learns the optimal policy by interacting with its environment. In multi-layer network problems, the environment state includes not only node information within a single layer but also the dynamic relationships between nodes across different layers. This module utilizes the... and nodes The feature representation serves as the state space. The agent uses a multi-head attention mechanism to calculate the selection probability of nodes, thereby determining the probability distribution of the next action. Each action corresponds to selecting a node in layer l of the network, thus completing the addition of an edge.

[0100] To integrate the current state with the node features of a specified layer l, this embodiment employs a multi-head attention (MHA) mechanism to compute a global view vector g. (l) It contains a global graph representation and local information about the currently selected node, representing the reinforcement learning agent's overall characterization of the environment state. This is achieved by using the state vector... As a query, all nodes in layer l are represented as keys and values. The correlation between each node and the state space is calculated, thereby extracting a vector g that reflects the global information within the current layer l. (l) This step dynamically focuses on the most useful information for decision-making in the state space and eliminates nodes that do not meet the conditions or are irrelevant (e.g., by setting the attention score of unselectable nodes to -∞). The specific formula is as follows:

[0101]

[0102]

[0103] Among them, g (l) It is a global perspective vector, and MHA represents multi-head attention mechanism. It is the set of nodes in the currently selected layer l. It represents the current state of the graph. It represents the currently selected node; when no node is selected... If there are already selected nodes

[0104] Then, obtain the global view vector g. (l) Then, it is transformed and mapped to the query vector q. (l) Simultaneously, a linear mapping is performed on each node of the l-th layer to generate a key vector. The specific formula is as follows:

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

[0106]

[0107] Among them, W5 and W6 are both trainable weight matrices. It is the representation of the final node in each layer of the network.

[0108] Next, this embodiment utilizes the obtained query vector q (l) and key vector By calculating q (l) and The dot product is normalized to measure the relationship between the current state and the node. The degree of matching between nodes is used to obtain the selection score for each node. The specific formula is as follows:

[0109]

[0110] Where C is a hyperparameter controlling the agent's exploration and utilization in reinforcement learning, d represents the query vector dimension, and Tanh is the activation function, introducing non-linear characteristics. Optional condition control: For nodes that do not meet the conditions (e.g., already selected or not selectable due to constraints), their scores are set to -∞ to ensure that the probability after softma is zero, thus preventing them from being sampled.

[0111] Finally, the score will be selected using the Softmax function. This is transformed into a probability distribution for node selection. The reinforcement learning agent samples or selects the node with the highest probability based on this distribution to make action decisions, thereby updating the network state and policy learning. The specific formula is as follows:

[0112]

[0113] 2. Strategy learning and robustness assessment

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

[0115] Figure 2 This is a schematic diagram of the multi-layer network reinforcement process provided in Embodiment 1 of the present invention. For a given edge budget `edges`, the agent's decision-making process 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 network G, and then performs an action to change the network state. The detailed process is as follows: For 0 ≤ t ≤ `edges-1`, state `s`... 2t It is a tuple in It is the decision result of the intelligent agent This was obtained by adding it to graph G. Let be the node selected by the agent at each step, 1≤l≤|L|, where l indicates that the selected node is at the l-th layer of the graph (l is the same for steps 2t and 2t+1). This indicates that initially no edges need to be added. When the agent observes state s... 2t Then, select a node in the l-th layer. Performing the action updates the state of the graph. At this point, the graph structure remains unchanged, i.e., G 2t+1 =G 2t The agent then continues to select the next node at layer l. At this point, the edge addition operation is performed, and the edge is... Add to image G 2t+1 The updated graph state is obtained. The agent repeatedly performs the above process within the edge budget, and the reward it can eventually obtain is graph G. 2*edges Improved robustness compared to the original graph G.

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

[0117]

[0118] The strategy parameter is π. θ (a|s) represents the probability of choosing action a in state s, and θ is the model parameter. τ is determined by policy π. θ The generated trajectory, R(τ) represents graph G 2*edges Improved robustness compared to the original graph G.

[0119] To better evaluate the robustness of multi-layer networks, this embodiment introduces a cascading failure mechanism. In a multi-layer network, there are coupling relationships between layers, meaning that a failure in a node in one layer may trigger failures in nodes in other layers that depend on it. Furthermore, this impact may recursively expand, leading to a chain reaction of failures. This embodiment also considers 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 those layers. This means that any two nodes must be reachable not only within a single layer but also across different layers. This analysis allows for a better assessment of which nodes have a profound and widespread impact on the entire network, thus enabling the development of more effective learning strategies. Therefore, robustness is defined as the minimum number of nodes that must be removed to make the MCCC of graph G equal to 1 under the cascading failure mechanism. The formula represents the strategy for deleting nodes, CF(·) represents the cascading failure mechanism, and q is the sequence of nodes to be deleted. 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 heterogeneity, dynamic coupling, and global optimization in multilayer networks. By introducing a cascading failure mechanism and maximum common connected component (MCCC), the model can effectively evaluate the robustness of multilayer networks. The reinforcement learning agent learns the optimal node or edge operation policy through interaction with the environment. The state is defined as the entire network, the action is to select nodes and add edges, and the reward is the improvement of network robustness.

[0122] The core modules of the model include initial feature construction, a multi-layer network representation module, and a node selection module. Initial feature construction comprehensively extracts local and global information of the network by capturing local structural features (such as degree centrality and clustering coefficients) and node pair relationship features (such as node pair distance and Jaccard coefficients), 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 ability to model complex coupling relationships. The node selection module uses a multi-head attention mechanism to calculate node selection probabilities, guiding the agent to select nodes and add edges, completing action decisions and updating the network state.

[0123] Overall, this embodiment addresses the limitations of traditional methods in enhancing multi-layer networks by synergistically combining reinforcement learning and graph representation learning, exhibiting 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 networking, and communication, further enhancing the robustness and adaptability of multi-layer networks and providing innovative solutions for the field of network optimization.

[0124] It is understood that the above embodiments are merely exemplary implementations used to illustrate the principles of the present invention, and the present invention is not limited thereto. For those skilled 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 to be within the scope of protection of the present invention.

Claims

1. A method for enhancing robustness of a multi-layer network based on representation learning, characterized in that, include: Initial features are constructed, and local and global information of the multi-layer network is comprehensively extracted by capturing local structural features and node pair relationship features. Nodes are represented within layers using multilayer perceptrons and graph neural networks. Node features from different layers are dynamically fused using interlayer attention mechanisms. A unified node representation is generated through intralayer feature learning, interlayer feature mapping, and interlayer attention mechanisms. The node selection probability is calculated using a multi-head attention mechanism. Based on the node selection probability, nodes are selected and edges are added to complete the action decision and update the network state. Through the inter-layer attention mechanism, the information propagation and aggregation strategy is flexibly adjusted according to changes in node characteristics and network structure. The fusion weights of inter-layer features are adaptively adjusted for each node, as shown in the following expression: ， wherein, is an inter-layer attention weight from layer n to layer m for node i, is a trainable weight matrix, h i (m) is an initial feature vector representation for node i in the mth layer, h i (n) is an initial feature vector representation for node i in the nth layer, h i (l) is an initial feature vector representation for node i in the lth layer, is a bias, denotes a Hadamard product, is an activation function used to introduce non-linear characteristics; By combining the intra-layer representations and inter-layer influence weights of nodes, the final node embeddings of the multi-layer network are generated. Simultaneously, the global representation of the graph in each layer of the multi-layer network is obtained based on the node embeddings, as shown in the following expression: ， ， wherein, represents the final feature representation of the ith node in the lth layer, is the feature vector representation of the ith node in the lth layer after uniform mapping, represents the feature weighted summation of the ith node except the lth layer, represents the feature representation of the lth layer network, Pooling is a function that aggregates the features of all nodes in the lth layer. 2.The method for enhancing robustness of a multi-layer network based on representation learning according to claim 1, wherein, Also includes: A reward function is designed based on the cascading failure mechanism and the maximum common connectivity component, and the robustness of the multilayer network is evaluated using the reward function. 3.The method for enhancing robustness of a multi-layer network based on representation learning according to claim 2, characterized in that, Also includes: The expression for obtaining the local structural features is as follows: ， wherein, represents the local structural feature of the ith node in the lth layer, and f is a feature extraction function, is the degree centrality of the ith node in the lth layer, is the average neighbor degree of the ith node in the lth layer, is the clustering coefficient of the ith node in the lth layer; The expression for obtaining the node-to-relationship feature is as follows: ， in, Let g represent the node pair relationship features between node i and node j in the l-th layer, and g be the feature extraction function. It is the physical distance between node i and node j in the l-th layer. It is the degree product of node i and node j at level l. It is the algebraic distance between node i and node j at level l. is the Jaccard coefficient of node i and node j in the l-th layer; A comprehensive node feature is formed based on the local structural features and the node pair relationship features. The node comprehensive feature X is normalized to obtain the feature matrix of each network layer: ， in, It is the feature matrix of the l-th layer, with dimension . n is the number of nodes. The feature dimension for each layer.

4. The method for enhancing the robustness of multilayer networks based on representation learning according to claim 3, characterized in that, Also includes: The initial feature vector representation of the multilayer network nodes is generated based on the multilayer perceptron, as expressed below: ， in, It is the initial feature vector representation of node i in the l-th layer. It is the feature matrix of node i in the l-th layer. It is a trainable weight matrix shared by multiple layers of the network. It is an activation function used to introduce nonlinear characteristics. It is a normalization operation used to adjust the scale of features; Using a graph neural network to capture the relationships between nodes and the global structure of the graph, the expression is as follows: ， in, This represents the characteristics of multiple network nodes during information propagation in the (k-1)th layer of a graph neural network, initially... = k represents the information of the k-hop neighbor that needs to be captured. This represents all neighboring nodes of node i in layer l of a multi-layer network. Describe the characteristics of node i's neighbor node u. This represents an aggregator, used to aggregate information from neighboring nodes. and It is the learnable weight matrix of the k-th layer of the graph neural network, used to adjust the contribution of the node's own features and the features of its neighboring nodes in feature updates. It is an activation function used to introduce nonlinear characteristics. It is a normalization operation used to adjust the scale of features; By embedding nodes from different layers into the same space, the representational differences between layers in a multi-layer network are eliminated, enabling node features to interact and merge under unified semantics and scale. The expression is as follows: ， Among them, h i (l) It is the initial feature vector representation of node i in the l-th layer. It is a trainable weight matrix. It's a bias. It is an activation function used to introduce nonlinear characteristics.

5. The method for enhancing the robustness of multilayer networks based on representation learning according to claim 4, characterized in that, Also includes: The state vector As a query, all nodes at level l are represented as keys and values. The correlation between nodes and the state space is calculated to obtain the global view vector. The expression is as follows: ， ， in, It is a global perspective vector. This indicates a multi-head attention mechanism. It is the set of nodes in the currently selected layer l. It represents the current state of the graph. It represents the currently selected node; when no node is selected... When a node has already been selected ; global view vector Transform mapping to query vector Generate key vectors by performing linear mapping on the nodes of the l-th layer. The expression is as follows: ， ， in, It is a query vector. It is a global perspective vector. It is a key vector. and It is a trainable weight matrix. It represents the final node in each layer of the network.

6. The method for enhancing the robustness of multilayer networks based on representation learning according to claim 5, characterized in that, Also includes: Based on the query vector and key vector Calculate the query vector With key vector The dot product is normalized to measure the relationship between the current state and the node. The degree of matching between nodes is used to obtain the selection score for each node. The expression is as follows: ， Where C is the control coefficient. This represents the query vector at level l. Let d represent the key vector of the l-th layer, and d represent the query vector dimension. It is an activation function used to introduce nonlinear characteristics; The score will be selected using the Softmax function. The process is transformed into a probability distribution for node selection. Based on this probability distribution, the node with the highest probability is selected to complete the action decision, update the network state and policy learning, and the expression is as follows: ， in, Let i represent the probability of selecting node i in the l-th layer. Let exp be the score for selecting node i in the l-th layer, and let exp be an exponential function. This represents the sum of the scores of all selected nodes.

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