A method and device for mitigating cascading failures in interdependent infrastructure networks

Abstract

The application provides a mutual dependence infrastructure network cascade failure mitigation method and device, wherein the method comprises the following steps: constructing a first node feature matrix of a whole network based on feature vectors of nodes of each sub-network; constructing a first adjacency matrix of the whole network based on connection relationships between nodes; using a state encoder to encode the first node feature matrix and the first adjacency matrix of the whole network to obtain a first encoded feature matrix; using a strategy network to process the first encoded feature matrix and a first global feature vector to obtain a first remedial action probability distribution; selecting an action with the maximum probability value from the first remedial action probability distribution as a remedial action; the remedial action comprises a remedial node and a remedial intensity level; determining a remedial capacity of the remedial node based on a system emergency capability of a sub-network to which the remedial node belongs and the remedial intensity level; and using an intelligent agent to schedule the remedial capacity to the remedial node. The application can reduce the cost of failure solution and improve the efficiency of failure mitigation.

Description

Technical Field

[0001] This application relates to the field of communication network technology, and in particular to a method and apparatus for mitigating cascading failures in interdependent infrastructure networks. Background Technology

[0002] Unlike the static coupling pattern of interdependent networks in conventional scenarios, extreme disturbances caused by disasters may trigger sudden resource sharing and functional substitution needs between networks, leading to the dynamic growth of cross-network dependency edges as failure propagates. This mechanism amplifies the high degree of uncertainty in failure paths, making cascading failures difficult to manage effectively. To address this challenge, existing emergency response strategies mainly fall into two categories: preventative hardening before failure and real-time mitigation during failure. While pre-identifying and hardening critical nodes can comprehensively improve the baseline resilience of the system, such passive defense strategies often face high economic costs under the reality of limited emergency resources. More importantly, the cascading failure process is difficult to predict in both time and space; the critical nodes dominating failure propagation may change significantly at different time points and along different propagation paths. For critical nodes identified statically using prior information, it is difficult to ensure they match the nodes that truly play a dominant role in the actual failure propagation process, resulting in a lack of targeted preventative strategies. In contrast, real-time proactive mitigation strategies can dynamically adjust emergency resource deployment plans based on the boundary conditions of failure propagation, thereby achieving more effective cascading failure suppression at a lower cost.

[0003] Developing optimal cascading failure mitigation strategies in real time is a complex task. Decision-makers need to accurately identify potentially high-risk nodes from a vast pool of candidate nodes during the highly urgent failure propagation process, and promptly deploy emergency resources to enhance their resilience before the failure impact reaches those nodes. Historically, this decision-making process has typically relied on heuristic models driven by topological characteristics such as the degree and centrality of nodes or edges, or functional attributes such as their carrying capacity, at a specific point in time during the failure propagation process. These methods are essentially based on a static ranking of node importance, which has significant limitations in addressing the complex evolution of cascading failures. Summary of the Invention

[0004] In view of this, this application provides a method and apparatus for mitigating cascading failures in interdependent infrastructure networks to solve the above-mentioned technical problems.

[0005] In a first aspect, embodiments of this application provide a method for mitigating cascading failures in interdependent infrastructure networks, including: Obtain the feature vectors of nodes in each sub-network of the infrastructure network at the current moment, as well as the connection relationships between nodes; The first node feature matrix of the entire network is constructed based on the feature vectors of the nodes in each sub-network; the first adjacency matrix of the entire network is constructed based on the connection relationships between nodes. The first encoded feature matrix is ​​obtained by encoding the first node feature matrix and the first adjacency matrix of the entire network using a pre-trained state encoder. The first encoded feature matrix and the first global feature vector are processed using a pre-trained policy network to obtain the probability distribution of the first remedial action; The action with the highest probability value is selected from the probability distribution of the first remedial action as the remedial action; the remedial action includes a remedial node and a remedial intensity level; The recovery capacity of a recovery node is determined based on the system emergency response capability and recovery intensity level of the sub-network to which the recovery node belongs. The agent is used to schedule the recovery capacity to the recovery node.

[0006] Secondly, an embodiment of this application provides a cascading failure mitigation device for interdependent infrastructure networks, comprising: The acquisition unit is used to acquire the feature vectors of nodes in each sub-network of the infrastructure network at the current time and the connection relationships between nodes; The construction unit is used to construct the first node feature matrix of the entire network based on the feature vectors of the nodes in each sub-network; and to construct the first adjacency matrix of the entire network based on the connection relationships between nodes. The encoding unit is used to encode the first node feature matrix and the first adjacency matrix of the entire network using a pre-trained state encoder to obtain the first encoded feature matrix; The processing unit is used to process the first encoded feature matrix and the first global feature vector using a pre-trained policy network to obtain the probability distribution of the first remedial action. A selection unit is used to select the action with the highest probability value from the first remedial action probability distribution as the remedial action; the remedial action includes a remedial node and a remedial intensity level; The determining unit is used to determine the recovery capacity of the recovery node based on the system emergency response capability and recovery intensity level of the sub-network to which the recovery node belongs; A failure mitigation unit is used to schedule recovery capacity to the recovery node using an agent.

[0007] Thirdly, embodiments of this application provide an electronic device, including: a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the method of embodiments of this application.

[0008] Fourthly, embodiments of this application provide a computer-readable storage medium storing computer instructions that, when executed by a processor, implement the methods of embodiments of this application.

[0009] Fifthly, embodiments of this application provide a computer program product, including a computer program / instructions, which, when executed by a processor, implement the method of embodiments of this application.

[0010] This application models the cascading failure mitigation problem as a discrete-time Markov decision process. By solving the Markov decision process, a failure mitigation scheme can be obtained, which can reduce the cost of failure resolution and improve efficiency. Attached Figure Description

[0011] To more clearly illustrate the technical solutions in the specific embodiments of this application or the prior art, the drawings used in the description of the specific embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of this application. For those skilled in the art, other drawings can be obtained from these drawings without creative effort.

[0012] Figure 1 A schematic diagram illustrating the propagation and mitigation of interdependent network cascading failures provided in an embodiment of this application; Figure 2 A schematic diagram of the MDP for cascading failure mitigation tasks provided in an embodiment of this application; Figure 3 A GA-PPO framework diagram provided for embodiments of this application; Figure 4 A flowchart of a method for mitigating cascading failures in interdependent infrastructure networks provided in this application embodiment; Figure 5 A schematic diagram illustrating the impact of different network structures on the algorithm convergence performance in embodiments of this application; Figure 6 A comparison chart of the convergence performance of GA-PPO and different DRL algorithms provided in the embodiments of this application; Figure 7(a) is a comparison of the overall network failure mitigation performance of GA-PPO and heuristic rule-driven methods provided in the embodiments of this application; Figure 7(b) is a comparison of the failure mitigation performance of GA-PPO and heuristic rule-driven subnetwork T provided in the embodiments of this application; Figure 7(c) is a comparison of the failure mitigation performance of GA-PPO and heuristic rule-driven subnetwork E provided in the embodiments of this application; Figure 8 A schematic diagram illustrating the impact of different K values ​​on the algorithm's convergence performance, provided in an embodiment of this application. Figure 9(a) shows the different embodiments provided in this application. M (k) The diagram illustrating the overall impact of network failure mitigation is shown below. Figure 9(b) shows the different embodiments provided in this application. M (k) A schematic diagram illustrating the impact of the failure mitigation effect on the corresponding subnetwork T; Figure 9(c) shows the different embodiments provided in this application. M (k) A schematic diagram illustrating the impact of the failure mitigation effect on the corresponding subnetwork E; Figure 10 Different embodiments provided for this application μ 2. Distribution of remedial strength of selected agents; Figure 11 Different embodiments provided for this application μ Schematic diagram of failure mitigation performance under condition 2; Figure 12 Different embodiments provided for this application μ A diagram showing the benefit comparison under option 2; Figure 13 Functional structure diagram of the interdependent infrastructure network cascade failure mitigation device provided in the embodiments of this application; Figure 14 This is a structural diagram of an electronic device provided in an embodiment of this application. Detailed Implementation

[0013] To make the objectives, technical solutions, and advantages of the embodiments of this application clearer, the technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, and not all embodiments. The components of the embodiments of this application described and shown in the accompanying drawings can generally be arranged and designed in various different configurations.

[0014] Therefore, the following detailed description of the embodiments of this application provided in the accompanying drawings is not intended to limit the scope of the claimed application, but merely to illustrate selected embodiments of the application. All other embodiments obtained by those skilled in the art based on the embodiments of this application without inventive effort are within the scope of protection of this application.

[0015] First, a brief introduction to the design concept of the embodiments of this application will be given.

[0016] Driven by the continuous process of large-scale urbanization, modern society is rapidly evolving into a complex mega-system composed of multiple infrastructure networks. These networks are no longer merely geographically adjacent, but rather interdependent through the dynamic exchange of pedestrian and physical flows, such as transportation-electricity, communication-electricity, water supply-electricity, and multimodal transportation networks. While this highly coupled model can significantly improve urban operational efficiency, it also weakens the fault isolation capabilities that should exist between different infrastructure networks. In recent years, global climate change has led to a continuous increase in the frequency and intensity of extreme weather disasters, and the intricate interdependence between infrastructures has become a channel for the cross-system propagation of risks. Taking the transportation-electricity interdependent network as an example, urban road flooding caused by extreme rainfall severely restricts the efficiency of power repair resource allocation, while power outages caused by flooding will in turn weaken the energy supply capacity of the transportation system, forming a typical spatiotemporal feedback loop, where the impact of local disturbances is continuously amplified. This phenomenon, known as cascading failure, has become a core challenge restricting the resilience of modern urban systems.

[0017] Deep Reinforcement Learning (DRL) offers a novel intelligent decision-making paradigm for overcoming the aforementioned challenges. While DRL has demonstrated potential in mitigating cascading failures in single physical flow networks, the mitigation logic fundamentally changes when a system contains multi-layered dependent networks composed of pedestrian and physical flows. For physical flow networks, creating islands through line cutoff is a common and effective risk mitigation strategy, aiming to limit the scope of load redistribution and thus suppress failure propagation. However, in pedestrian flow networks, simple line cutoff often means that people are stranded or trapped in localized areas, which is unacceptable from a public safety perspective. Meanwhile, the emergency response and resource allocation processes under disaster impacts may lead to the dynamic growth of dependencies between different networks, further increasing the difficulty of designing mitigation strategies. To address these challenges, this application designs a cascading failure mitigation framework for dependent networks based on DRL, enabling adaptive decision-making regarding the timing, location, and intensity of remedies.

[0018] In interdependent infrastructure networks, extreme disasters often first cause local node failures, and then, through the combined effects of load redistribution within the network and cross-network dependencies, trigger cascading effects and cause widespread degradation of network functionality. This application directly links the topological importance of nodes to their load-capacity relationship.

[0019] Let network m∈{T, E} represent different types of infrastructure subnetworks, where T represents a network primarily driven by pedestrian flow (such as a public transportation network), and E represents a network primarily driven by physical flow (such as a power grid). For any node in different subnetworks... Its load and capacity at time t are denoted as follows: and In the initial, intact state of the network, the node load is defined by normalized betweenness centrality as:

[0020] in, Let i be the betweenness centrality of node i. This represents the total number of network nodes. Node capacity is assumed to be proportional to its initial load to reflect the limited redundancy of the infrastructure system under cost constraints.

[0021] in, This is a tolerance factor, used to characterize the redundancy design commonly reserved in real-world infrastructure networks to ensure stable operation; its value is 0.5. During cascading evolution, when nodes... satisfy > If the failure occurs at time t, it is determined that the node has failed. Simultaneously, all its load will be redistributed to neighboring nodes, affecting nodes across the network that have established dependent edges with it.

[0022] Because different types of infrastructure networks differ significantly in node attributes, traffic transmission methods, and operating mechanisms, this application constructs differentiated intra-network load redistribution rules (e.g., ...) for sub-network T and sub-network E respectively. Figure 1 As shown in the upper module), and based on this, a growth mechanism for cross-network dependency edges was further designed (such as...). Figure 1 (As shown in the middle module).

[0023] For a network T dominated by pedestrian traffic, the failure load redistribution process follows the basic idea of ​​the gravity model. When nodes When a failure occurs at time t, the load will be distributed according to its relationship with neighboring nodes. The interaction intensity between them is redistributed:

[0024]

[0025] in, represent The set of neighboring nodes; Representative node and The interaction strength; α and β represent the scale parameter and the distance attenuation coefficient, respectively, both of which are set to 1; This represents the dynamic cross-impedance between node pairs, comprehensively reflecting the impact of factors such as spatial distance, network structure, and traffic congestion on load shifting. In this way, it demonstrates that in the event of node failure, pedestrian flow will preferentially shift to nearby nodes with stronger service attractiveness and lower travel resistance. This process is consistent with the rational behavioral choices of individuals in reality.

[0026] For a network E primarily driven by physical flows, load redistribution in case of failure follows a redundancy-driven mechanism. When a node When a failure occurs at time t, its load will be determined based on the neighboring nodes. The relative redundancy capacity available at the current moment is reallocated:

[0027] in, represent The set of neighboring nodes. This formula reflects the reality that the greater the redundancy, the more load is redistributed. This formula reflects the engineering operation rules set in reality to reduce the risk of physical flow network failure, that is, failed loads will be preferentially transferred to neighboring nodes with larger safety margins to improve the network's ability to absorb additional shocks.

[0028] Besides internal and external networks, cascading failures can also propagate between different subnetworks through cross-network dependency edges. For a failed node at time t... Its candidate dependent nodes in another network (in The nodes will be selected from the feasible space. This process switches between two selection rules with equal probability: (i) randomly selecting nodes within a certain spatial range to characterize the uncertainty of disaster propagation; and (ii) selecting the geographically closest nodes to reflect local spatial coupling effects. After the candidate nodes are determined, the growth of cross-network dependency edges is defined probabilistically as follows:

[0029] in, and To control the steepness coefficient and activation threshold of network dependency strength, they were set to 3.5 and 1.5, respectively. The dependency score incorporates three main factors: spatial proximity, topological importance, and proximity to the failure boundary. This design reflects the nonlinear growth mechanism of cross-network dependencies under disaster impact: when a candidate dependent node is located at the failure propagation boundary of another sub-network and can provide strong functional alternative support, the dependency will naturally emerge and rapidly strengthen.

[0030] Based on the magnitude of the dependency probability, the impact of dependency edges is divided into three modes: strong dependency, weak dependency, and no dependency. In the case of strong dependency ( , (At its maximum value), the coupling between nodes will directly trigger cross-network failure propagation. Dependent nodes will synchronously fail at this moment and further participate in the cascading failure evolution within their respective networks. In the case of weak dependencies ( < < , (At the minimum value), the coupling between nodes does not immediately trigger cross-network failure propagation, but rather works by weakening the carrying capacity of nodes. The capacity of dependent nodes will increase with the strength of the dependency according to (1-ρ). The proportional decay of ). ρ is the weak dependency influence strength coefficient, set to 0.5. In the case of no dependency ( Cross-network coupling effects are negligible, and node states are unaffected. Dependency mode transition thresholds are set to [values ​​to be filled in]. =0.8, =0.5.

[0031] System emergency capability (SEC) can be viewed as the redundancy a system possesses under abnormal operating conditions. In engineering practice, it corresponds to the combined effect of various emergency resources such as emergency evacuation vehicles and backup power supplies. These redundant resources are typically expected to be promptly deployed to critical nodes after a failure, enhancing their ability to withstand additional load shocks and thus suppressing the further propagation of cascading failures. It should be noted that, to focus on the upper limit of SEC's suppression of cascading failures, this study does not address the optimization of specific scheduling paths at the microscopic level. The latter can usually be addressed in engineering practice through reasonable pre-deployment of redundant resources and path planning. This application focuses on the macroscopic impact of finite SEC applied at different locations on the dynamic evolution of network cascading failures.

[0032] When the node When an SEC-based remedial action is implemented, its capacity will be temporarily increased to:

[0033] in, This represents the temporary capacity increment resulting from the SEC allocation operation. To ensure the comparability and adaptability of remedial actions across networks of different sizes and topologies, the percentage of a single operation is [not specified]. Defined as a function reflecting the overall capacity characteristics of the network. In real-world emergency response scenarios, the schedulable capacity limit (SEC) is constrained by real-world conditions in terms of both time and scale, making it difficult for remedial actions to impose arbitrary capacity increases on individual nodes within a short period. Based on this constraint, instead of directly setting a fixed capacity increment, the SEC is characterized from the perspective of the entire network. For any network... The baseline SEC can be given by the average total capacity of the nodes:

[0034] Meanwhile, sub-network The total number of schedulable SECs during the entire cascading failure process is finite, and its upper limit is determined by an integer multiple of the baseline SEC:

[0035] in, A positive integer parameter that reflects the network size, operational characteristics, and emergency response capabilities.

[0036] Remedial decision-making is a process of balancing economic feasibility with mitigating the risk of failure. Considering that remediation based on a fixed threshold may result in low SEC utilization, this application proposes a novel tiered remedial mechanism. Specifically, using a benchmark SEC as a reference, the intensity of a single remedial action is discretized into several levels to reflect the differentiated capacity improvement effects produced by SEC investments of different scales, such as... Figure 1 The module below is shown. When dealing with a node Implementation level is When remedial action is taken, the capacity increment is expressed as:

[0037] in, To compensate for the strength coefficient, corresponding to low strength ( , ), conventional strength ( ) and high strength ( This mechanism not only characterizes the SEC differences required for failure scenarios of different scales in reality, but also facilitates the implementation of priority-based remediation actions under cost-constrained conditions. Furthermore, discretization avoids the complexity of continuous adjustment of capacity increments, making it highly feasible in engineering practice.

[0038] The SEC (Security Conflict) is finite throughout the cascading failure evolution process, and its consumption is irreversible. Therefore, it is necessary to explicitly introduce a global SEC budget constraint into the model. (Subnetwork) The total budget available at the initial moment can be considered the SEC ceiling. Then at time Implement a level Following the remedial action, the remaining SECs are updated as follows:

[0039] in, This represents the maximum value operation, ensuring the budget is not negative. This budget constraint characterizes the gradual depletion characteristic of the SEC during its dynamic evolution. When A value of ≤0 indicates that the network's SEC has been completely consumed, and subsequent remedial actions will not be possible.

[0040] The task of mitigating cascading failures is essentially a sequential decision-making process unfolding gradually in a dynamically evolving environment. Each remedial action not only instantly changes the node's carrying capacity and SEC margin, but also continuously affects subsequent failure propagation paths, the set of feasible remedies, and the future decision space, thus having a long-term impact on the evolution of the failure state. To characterize this decision-environment interaction pattern with cumulative effects, this application models the cascading failure mitigation problem as a discrete-time Markov Decision Process (MDP), denoted as: .in, This represents the state space consisting of cascading failure environments; This represents the space for remedial action; This represents the state transition function jointly determined by the cascading failure propagation dynamics and the salvage update mechanism; This represents the reward function used to weigh the cost of remediation against the risk of failure.

[0041] Within the MDP framework, remedial actions require dynamic decision-making in a highly uncertain and constantly evolving environment, where traditional analytical modeling or fixed-rule optimization methods are ineffective. To address this, this application introduces DRL, treating the decision-maker as an intelligent agent capable of perceiving and interacting with the cascading failure environment. By observing the environmental state, executing remedial actions, and receiving immediate feedback, the agent can progressively approach the optimal mitigation strategy at different propagation stages. This learning process is achieved by maximizing the expected cumulative reward throughout the entire cascading failure evolution. What has been achieved. Among them, The strategy function has parameter θ; γ∈(0,1] is the discount factor used to weigh the immediate recovery benefits against the risk of future cascading failures. This indicates that the agent observes the state of the environment. Remedial action should be taken in time. And evolve to state according to the state transition probability p. The reward received afterward.

[0042] This section provides a detailed explanation of the state space, action space, state transition, and reward function in MDP, with an overall schematic diagram as shown below. Figure 2 As shown. In short, the environment state is first characterized by combining node attributes and global constraints, and then hierarchical remedial actions considering regional constraints are implemented accordingly. This leads to state transitions at different stages and generates immediate rewards that balance cost and efficiency.

[0043] Generally speaking, the state space should contain all the information necessary to fully characterize the state of a cascading failure environment. This means that the environment should be defined at time [time]. status It is divided into two dimensions: node-level features and global-level features.

[0044] First, for any subnetwork Nodes in At any moment The eigenvectors are defined as follows:

[0045] in, and Representative node The load and capacity attributes of a node reflect its functional importance; Representative node The betweenness centrality of a node is used to describe the structural importance of a node in the network topology. Representative node The degree of impact under cross-network dependencies, with no dependency, weak dependency, and strong dependency patterns corresponding to discrete codes {0, 0.5, 1} respectively; indicator function Representative node Whether it is a newly failed node after the previous round of failure propagation is indicated by a value of 1 if yes and 0 otherwise. Introducing this indicator variable effectively characterizes the phased changes of nodes during the cascading failure process. Based on the above node feature vectors, the entire network node feature matrix can be obtained by stacking them:

[0046] in, For sub-networks At any moment The node feature matrix.

[0047] To supplement the missing macro-environmental information at the node level, a global feature vector shared by all sub-networks is further defined:

[0048] in, Representative subnetwork m ∈{ T , E At that moment The remaining SEC; This indicates the number of remedial actions that have been performed within the current propagation round.

[0049] In summary, the environment is constantly changing. The complete state space is defined as: All of the above features have been normalized to ensure that their values ​​are within the range of [0,1], in order to eliminate the influence of different dimensions on the state representation.

[0050] The goal of failure mitigation is not to indiscriminately repair all nodes in the network, but rather to implement targeted remediation for potentially high-risk critical nodes that gradually emerge during the failure propagation process. In fact, cascading failures often spread along well-defined topological boundaries in the spatial dimension, and remedial actions performed near the failure propagation boundary are usually more efficient. Therefore, the feasibility of remedial actions is explicitly linked to the propagation boundary of cascading failures, thereby constructing a state-constrained action space.

[0051] remember For a given set of nodes in a subnetwork The first-order neighbor set on the topology search depth can then propagate failures across the topology search depth. All nodes reachable within are defined as:

[0052] in, The recursive relation satisfies ;parameter This is used to control the scope of remedial actions at the topology level. It's important to note that once a node fails, it loses its original functionality and cannot be recovered in the short term; therefore, remedial actions are only practically meaningful at the engineering level for nodes that have not yet failed. Thus, subnetworks... At any moment The set of candidate remedy nodes can be represented as:

[0053] in, Representative moment The set of newly failed nodes during a cascading failure process; Represents the deadline The set of nodes that have failed. Therefore, at time... The action space at time can be represented as:

[0054] in, This represents the selected remedial node in the current state; The level of remedial strength. When the SEC is exhausted, no remedial action is taken, thus ensuring the integrity of the MDP. It is a moment In a subnetwork T, the neighboring nodes of a failed node can only be used to select a remedy from among these neighboring nodes. It is a moment The neighboring nodes of the failed node in subnetwork E. Therefore, at time... The performed action satisfies ∈ And the dimensions of the action space satisfy .

[0055] The aforementioned definition of the action space allows remedial actions to be implemented closely along the propagation boundary of cascading failures, avoiding the dispersion of the SEC (Self-Controlled Decision Space) into low-risk areas where it is unlikely to have a substantial impact on the failure evolution in the short term. Furthermore, by screening decision options with practical remedial significance under the current environmental conditions, the scale of the action space can be effectively controlled, thereby improving engineering feasibility and efficiency.

[0056] The state transition of the environment is modeled as a phased process driven by both remedial actions and failure propagation. In real-world infrastructure networks, each round of cascading failure propagation is typically not instantaneous, but rather unfolds gradually along the network topology over time. Between two adjacent failure propagations, there is often a response time window, allowing for multiple remedial actions to be implemented consecutively to mitigate further failure propagation. This application utilizes propagation count parameters... To characterize this feature, that is, allowing the agent to execute actions before a round of cascading failure propagation is triggered. Q Secondary remedial action.

[0057] A single remedial action is considered a discrete time step. This occurs within the response time window before failure propagation is triggered. At this point, the evolution of the environmental state is driven solely by remedial actions. Specifically, given the current environmental state... Under the conditions, carry out remedial action The capacity level of the selected node will be updated immediately, consuming the corresponding SEC. Simultaneously, the propagation count parameter increments with each remedial action. When the cumulative number of remedial actions reaches the maximum, that is... At this point, the network will trigger a round of failure propagation during this state transition. The environment state will then be updated according to the defined cascading failure dynamics model, including load redistribution of failed nodes, identification of new failed nodes, and growth of cross-network dependency edges. After a round of failure propagation is completed, the propagation count parameter is reset. Then it enters the next state transition cycle.

[0058] In the above process, remedial actions are no longer limited to single remedies for different nodes, but rather allow the agent to continuously invest SEC (Security Decision) into potentially critical nodes based on the failure evolution. This is achieved through the rigid constraints of the environment. and Explicitly incorporate into the global state vector This can ensure that in a given Under the condition of state transition function The generated conditional probabilities The network no longer relies on other historical information, thus satisfying the MDP assumption. When a round of failure propagation does not trigger any new failed nodes, the network is considered to have reached a stable state. At this point, no further remedial actions are performed, and this state is considered the termination state of the MDP.

[0059] In emergency response to cascading failures, decision-makers typically face a classic engineering trade-off. On the one hand, implementing more aggressive remedial actions can effectively suppress the spread of cascading failures; on the other hand, SEC (Self-Controlled Remedial Action) is significantly constrained in terms of scale and scheduling, and overuse will lead to high costs and is difficult to implement in real-world engineering scenarios. This application explicitly embeds this trade-off into the design of the reward function, and divides the immediate reward into two parts: a remedial action cost component and a failure scale penalty component.

[0060] First, the concept of recovery cost is introduced to characterize the differences in SEC consumption under different recovery intensities. At time... t Perform remedial actions In this case, this operation can be considered to correspond to engineering inputs such as equipment, manpower, and scheduling, and therefore assigned a negative reward. The cost of a single remedial action is defined as:

[0061] in, Representative and remedial strength levels l The corresponding unit SEC consumption cost takes values ​​of {0.5, 0.75, 1.0}. To compensate for the cost weighting coefficient. Denominator term Q This is used to scale the remediation costs within a single failure propagation cycle proportionally, making them comparable to subsequent failure penalties on the same order of magnitude.

[0062] Secondly, to characterize the negative impact of cascading failures on the overall resilience of the network, a failure penalty is introduced when a round of failure propagation is triggered. The penalty value is defined by the degree of loss that cascading failures bring to the network at the current time step, i.e., the time-step cascading failure ratio (RTCF).

[0063] in, This represents the number of newly added failure nodes caused by this round of failure propagation; This is the failure penalty weighting coefficient.

[0064] In summary, the reward function In a given state, an action pair The corresponding instant reward value is as follows:

[0065] This reward design can guide agents to make dynamic trade-offs between minimizing remediation costs and maximizing cascading failure suppression under finite SEC constraints, thereby forming an engineering-rational mitigation strategy.

[0066] The GA-PPO solution framework achieves high-quality state representation by embedding a graph neural network within an Actor-Critic (AC) architecture, and combines it with the PPO algorithm for joint optimization of the policy and value networks. Its overall structure is as follows: Figure 3 As shown.

[0067] For an agent, the learning process of failure mitigation strategies relies on the high-quality representation of the environmental state by deep neural networks. In cascading failure scenarios, the key information upon which remedial actions depend not only comes from the local attributes of individual nodes but is also implicit in their topological location and the operational states of their neighboring nodes. If fully connected neural networks are directly used to process the environmental state, it will be difficult to characterize the structural correlations between nodes, thus limiting the agent's ability to identify potentially high-risk nodes.

[0068] This embodiment introduces a Residual Graph Convolutional Network (ResGCN) as a state encoder into the AC framework, and designs an end-to-end graph-enhanced AC network structure, such as... Figure 2 Modules 2 and 3 are shown in the diagram. The actor corresponds to the policy network, responsible for generating remedial actions based on the current environmental state; the critic corresponds to the value network, responsible for evaluating the potential value of a given environmental state in long-term rewards.

[0069] First, the adjacency matrix of the entire network is constructed as a block diagonal matrix to characterize the topological correlation between two sub-networks within a unified state coding framework:

[0070] in, Representative subnetwork k ∈{ T , E The adjacency matrix of}, when the node and When a connection exists =1, otherwise 0. and This represents an all-zero matrix. Cross-network dependency edges are not directly modeled using the adjacency matrix, but are reflected in the state transition process of cascading failure environments.

[0071] Graph Convolutional Neural Networks (GCNs), as a typical form of graph neural networks, are based on the idea of ​​constraining the information propagation paths between nodes using given network connections, and then aggregating and updating node features accordingly. The most basic GCN state encoding form can be represented as:

[0072] in, d Represents the index of the graph convolutional layer; Representative after the first d The node feature matrix updated after layer GCN; in particular, when d =0 This represents the initial node state feature matrix; W Represents the learnable parameter matrix; Represents a non-linear activation function; N hid This represents the dimension of the hidden layer. This form achieves information propagation along the graph structure through a simple weighted sum of the features of neighboring nodes, but it still has certain limitations. First, the node's own features are not explicitly preserved; second, it does not consider the impact of differences in degree between nodes on information propagation, which may lead to the amplification of features of high-degree nodes. Therefore, a self-loop mechanism is introduced into the construction of the adjacency matrix, and it is normalized using a symmetric normalization form:

[0073] in, I The identity matrix is ​​used to ensure that nodes retain their own information during the aggregation of neighborhood information. This represents the corresponding degree matrix. From the perspective of spectral graph theory, the graph convolution operation after symmetric normalization can be interpreted as a Laplace smoothing process. Specifically, In the spectral domain, this is equivalent to applying a low-pass filter to the node features, making the representations of adjacent nodes more consistent in the feature space. When graph convolutional layers are stacked multiple times, this smoothing effect may accumulate to some extent, leading to problems such as decreased node feature discriminability. To alleviate this phenomenon, this application improves the node feature aggregation form based on the idea of ​​residual connections, updating it as follows:

[0074] Among them, LayerNorm Representation layer normalization is used to normalize the distribution of node representations along the feature dimension, thereby improving training stability in dynamic evolution environments; Res Residual connections are used to preserve the original feature representations of the nodes themselves. This operation allows for the construction of state representations with region-aware information, providing a sound semantic foundation for subsequent remedial decision-making and optimization.

[0075] To effectively process global feature vectors, a late-stage fusion approach was adopted in the design of the corresponding decision heads. That is, macroscopic constraint information does not directly participate in the neighborhood aggregation process of ResGCN, but is introduced as supplementary information during the decision-making stage. The actor and critic decision heads each use fully connected layers with shared parameters as mapping functions.

[0076] The actor decision-maker is responsible for strategy generation. To inject macro-level constraints into the micro-level decision-making process, the global feature vector is... g This is obtained by copying and expanding at the node level. It is then concatenated with the node feature matrix to construct a fused feature matrix. :

[0077] in, This represents the feature concatenation operation. Through a linear transformation, the fused features of each node are independently mapped to... l The action score corresponding to each remedial level. Considering a noop Triggered solely by SEC exhaustion and weakly correlated with the state characteristics of specific nodes, a separate auxiliary branch was designed for it. This branch is based solely on global characteristics. g As input, calculate directly. a noop The action scores are then calculated. Finally, the Softmax function is used to normalize all action scores, resulting in the probability distribution of remedial actions in the action space.

[0078] The Critic decision head is responsible for value assessment. Unlike the Actor node-level output, the Critic requires a unified evaluation at the entire graph level. To do this, global average pooling is performed on the node feature matrix to obtain graph-level features, which are then combined with the global feature vector. g The features are concatenated to construct a graph-level fused feature vector:

[0079] The vector is input into a fully connected layer, and the regression output is a value estimate of the current environmental state.

[0080] PPO, a policy gradient-based same-policy optimization method, effectively limits the offset of policy distribution in a single update by introducing a probability ratio shearing mechanism during policy update, thereby significantly improving training stability in complex dynamic environments. Therefore, the constructed graph augmentation AC architecture is embedded into PPO for joint training, forming a GA-PPO framework for mitigating cascading failures.

[0081] First, the policy network aims to maximize the expected cumulative reward of remedial actions during the cascading failure mitigation process. Let... θ The network parameters of the policy to be updated are as follows. For the old policy network parameters used for sampling trajectories, its objective function can be expressed as:

[0082] Where, min This represents the minimum value operation; clip The shearing function represents the restriction that the sampling probability of the current policy for any action is at most amplified by (1+) compared to the old policy. ) times or reduced to (1- This multiplies the strategy by 10 times, preventing excessive shifts in the strategy during a single update. >0 represents the shearing threshold; Represented by parameters Defined strategy function. The importance sampling ratio is defined as:

[0083] This ratio measures the magnitude of the change in action selection probability between the current policy and the old policy.

[0084] Advantage function Used to measure the state of an agent in the environment. Perform remedial actions The degree of superiority relative to the baseline value is considered. A generalized advantage estimation (GAE) is used to approximate the advantage function to achieve a balance between bias and variance. Prior to this, the temporal difference error (TD) needs to be calculated.

[0085] in, Representative moment TD error under; Represents the state-value function approximated by the commentator network, with parameters as follows: This function represents the agent's response when observing the state of the environment. Then, the expected cumulative return that can be obtained by implementing subsequent remedial actions according to the current strategy. It can be approximated by the following formula:

[0086] in, T Represents the total number of decision time steps in a round; nRepresents relative to the current moment t Forward offset; λ ∈[0, 1] represents the GAE attenuation parameter. When λ When = 0, the dominance estimation degenerates into a single-step TD error; when λ When the ratio is 1, the advantage estimate approaches the Monte Carlo return. GAE (General Advantage Estimation) allows for a weighted integration of the immediate benefits of current remedial actions with their cumulative impact in subsequent failure propagation rounds, thus avoiding overly optimistic assessments of remedial actions based solely on short-term effects.

[0087] Secondly, the value network is trained by minimizing the mean squared error between the predicted value and the return target, and its loss function is defined as:

[0088] in To align with GAE's value objectives, it is defined as:

[0089] In summary, the overall optimization objective of GA-PPO can be expressed as maximizing the following objective function:

[0090] in, This represents policy entropy, used to encourage agents to explore the unknown. , This is the loss weight hyperparameter.

[0091] During GA-PPO training, the policy network parameters With value network parameters Joint updates are performed under the constraints of their respective optimization objectives. (In ResGCN) W The gradient of the state encoding parameters, shared by actors and critics, is obtained through backpropagation of both policy loss and value loss. This parameter optimization method ensures that the feature encoding layer can simultaneously capture policy-sensitive features that aid in remedial decisions and value-sensitive features that contribute to network-wide evaluation. The entire process is summarized below: The agent processes the current environment state through a graph-reinforced AC architecture. s t The actors output a policy function, and the critics output a state value function. The agent then takes remedial actions based on the probability distribution. a t After receiving the action, the environment evolves to the next state. s t+1 Simultaneously provide instant rewards r t Place local samples of these trajectories into the trajectory buffer until the sample size meets the requirements.

[0092] Backtracking is performed from the end of the trajectory backwards. The time difference error is calculated using the predictive value of future states by the commentator network. t And obtain the advantage function through GAE. Subsequently, the value objectives used to supervise the critics' learning were calculated. .

[0093] The data in the trajectory buffer is shuffled and divided into several mini-batches. For each mini-batch, the policy optimization objective is calculated separately. Value mean square error loss And the entropy regularization term that enhances exploration capabilities The weighted combination of the three factors constructs the overall objective function. The network parameters are optimized collaboratively using the gradient descent algorithm. W , θ , .

[0094] After introducing the application scenarios and design concepts of the embodiments of this application, the technical solutions provided by the embodiments of this application will be described below.

[0095] like Figure 4 As shown in the figure, this application provides a method for mitigating cascading failures in interdependent infrastructure networks, including: Step 101: Obtain the feature vectors of nodes in each sub-network of the infrastructure network at the current moment and the connection relationships between nodes; Step 102: Construct the first node feature matrix of the entire network based on the feature vectors of the nodes in each sub-network; construct the first adjacency matrix of the entire network based on the connection relationships between nodes; Step 103: Encode the first node feature matrix and the first adjacency matrix of the entire network using the pre-trained state encoder to obtain the first encoded feature matrix; Step 104: Use the pre-trained policy network to process the first encoded feature matrix and the first global feature vector to obtain the probability distribution of the first remedial action; Step 105: Select the action with the highest probability value from the probability distribution of the first remedial action as the remedial action; the remedial action includes the remedial node and the remedial intensity level; Specifically, the level of remediation intensity Including: low strength, normal strength and high strength.

[0096] Step 106: Determine the recovery capacity of the recovery node based on the system emergency response capability and recovery intensity level of the sub-network to which the recovery node belongs; For example, For the remedial strength level The remedial strength coefficient.

[0097] Step 107: Use the agent to schedule the recovery capacity to the recovery node.

[0098] In some embodiments, the feature vector of a node includes: the node's load, the node's capacity, the node's betweenness centrality, the encoded value of the degree of impact the node suffers under cross-network dependencies, and the identifier value of whether the node is a newly failed node after the previous round of failure propagation; the first global feature vector includes: the remaining system emergency response capability of each sub-network at the current time and the number of remedial actions performed before the current time.

[0099] In some embodiments, constructing the first adjacency matrix of the entire network based on the connection relationships between nodes includes: Obtain the connection relationships between nodes in subnetwork T and subnetwork E; The connection value is determined based on the connection relationship. If two nodes are connected, the connection value is 1; otherwise, the connection value is 0. Generate the adjacency matrix of subnetwork T based on the connection values ​​between nodes in subnetwork T. ; Generate the adjacency matrix of subnetwork E based on the connection values ​​between nodes in subnetwork E. ; Construct the first adjacency matrix of the entire network :

[0100] in, and All are zero matrices.

[0101] In some embodiments, the state encoder employs a residual graph convolutional neural network, which includes D-layer graph convolutional layers. The first encoded feature matrix is ​​obtained by encoding the first node feature matrix and the first adjacency matrix of the entire network using a pre-trained state encoder; including: For the first adjacency matrix Processing is performed to obtain a matrix. :

[0102] in, I It is the identity matrix; calculate the matrix according to the following formula. :

[0103] in, This is the corresponding degree matrix; The feature matrix output by the (d+1)th graph convolutional layer for:

[0104] in, This represents the feature matrix output by the graph convolutional layer at layer d. ;when d =0 Represents the feature matrix of the first node; This represents the learnable parameter matrix of the (d+1)th graph convolutional layer; LayerNorm represents a non-linear activation function. Representation layer normalization operation; Res Indicates a residual join operation; The feature matrix output by the (D-1)th graph convolutional layer This is determined to be the first encoding feature matrix.

[0105] In some embodiments, the method further includes: Step S1: Obtain historical moments The feature vectors of nodes in each subnetwork of the infrastructure network and the remaining emergency response capacity of each subnetwork; where the initial value of k is 1; Step S2: Construct the second node feature matrix of the entire network based on the feature vectors of the nodes in each sub-network; construct the second global feature vector based on the remaining emergency response capability of each sub-network and the number of remedial actions taken; construct the second adjacency matrix of each sub-network based on the connection relationships between nodes; Step S3: Determine the state of the environment by using the second node feature matrix and the second global feature vector of the entire network. ; Step S4: Encode the second node feature matrix and the second adjacency matrix of the entire network using the state encoder to obtain the second encoded feature matrix; Step S5: Use the policy network to process the second encoded feature matrix and the second global feature vector to obtain the probability distribution of the second remedial action; Step S6: The agent randomly selects a remedial action from the probability distribution of the second remedial action. And perform remedial actions. The entire infrastructure network undergoes state transition evolution to time [time]. status Simultaneously provide instant rewards Thus, the time is obtained. samples ; Step S7: Determine whether the number of samples is less than the preset threshold. If yes, update k to k+1 and proceed to step S1; otherwise, proceed to step S8. Step S8: For each batch of samples, calculate the policy optimization objective value, the mean squared error of value loss, and the entropy regularization term to construct the overall objective value. ; Step S9: Update the parameters of the state encoder, policy network, and value network based on the total target value.

[0106] In some embodiments, instant reward value for:

[0107] in, The cost of a single remedial action; For a historic moment The cumulative number of remedial actions; The maximum number of remedies;

[0108] in, In order to match the remedial strength level l The corresponding cost of the unit system's emergency response capabilities; The cost weighting coefficient is used to remedy the situation. The cascade failure rate of time steps is calculated using the following formula. :

[0109] in, This is the failure penalty weighting coefficient. This represents the number of newly added failure nodes caused by this round of failure propagation; Let T be the number of nodes in the subnetwork. Let E be the number of nodes in subnetwork E.

[0110] In some embodiments, the strategy optimizes the target value. for:

[0111] in, θ For policy network parameters; min This represents the minimum value operation; clip The shearing function represents the restriction that the sampling probability of the current policy for any action is at most amplified by (1+) compared to the old policy. ) times or reduced to (1- ) times; >0 represents the shearing threshold; Importance sampling ratio; Calculate according to the following formula The estimated value :

[0112] Where N represents the total number of decision time steps in a round; λ∈[0, 1] is the GAE decay parameter; Discount factor; The time difference error is calculated using the following formula. :

[0113] in, Represents the value network function. These are the parameters of the value network.

[0114] In some embodiments, the value mean squared error loss value for:

[0115] in, To align with the GAE's target value, the calculation formula is as follows: .

[0116] In some embodiments, the total target value The calculation formula is:

[0117] in, Represents policy entropy; and All of these are loss weight hyperparameters. This represents the policy function with policy network parameters θ.

[0118] This embodiment considers two different scales of dependent networks to evaluate the effectiveness of GA-PPO. (1) Large-scale network, including a subnetwork T with 600 nodes and a subnetwork E with 300 nodes; (2) Small-scale network, including a subnetwork T with 400 nodes and a subnetwork E with 200 nodes. During network generation, the connection parameters of newly added nodes are all set to 1, that is, each newly added node generates an edge with an existing node. The number of remedies allowed between each round of failure propagation is positively correlated with the network size and is set to 8 and 5 times respectively, accounting for about 10% of the total number of nodes. It should be noted that the experiment mainly focuses on the relative performance differences between algorithms and the impact of parameter changes on policy learning. Such conclusions are not sensitive to the specific network size. To avoid experimental redundancy, the small-scale network is only used to test the scalability of GA-PPO in dependent networks of different sizes.

[0119] Traditional force-directed layout (BA) models can only characterize the network's connectivity topology and cannot directly generate the spatial coordinates of nodes. To achieve geographic mapping of dependent networks and characterize the differences in spatial layout mechanisms among different types of infrastructure, BA models are enhanced by embedding them into a specific force-directed layout algorithm. For networks primarily driven by pedestrian flow, their layout is heavily influenced by travel demand and often exhibits non-uniform clustering characteristics. Therefore, the Fruchterman-Reingold algorithm is selected to generate subnetworks. T The algorithm, by introducing a balance mechanism of attraction and repulsion between nodes, can spontaneously form spatial distributions with local cluster structures (such as urban CBDs and transportation hubs). In contrast, for networks primarily driven by physical flows (such as power systems), their spatial layout is typically constrained by engineering factors such as line length and transmission loss, leading them to avoid excessive stretching and high concentration of local spaces in their geometric structure. To characterize this feature, the Kamada-Kawai algorithm is used to generate subnetworks. E The algorithm optimizes the spatial layout of the network by minimizing the total potential energy of the springs across the entire network, ensuring that the Euclidean distance between nodes is as consistent as possible with the shortest path length in the topology.

[0120] The training process for GA-PPO begins with the most severe single-node damage scenario, simulating a disaster impact that causes the initial failure of the nodes with the highest betweenness centrality in the two sub-networks. The state encoder of GA-PPO consists of a multi-layer ResGCN with ReLU activation. The decision heads for the Actor and Critic are constructed using a Multilayer Perceptron (MLP) with Tanh activation. After extensive preliminary experiments, the specific model training hyperparameters are shown in Table 1. Table 1

[0121] The proposed GA-PPO framework is compared with corresponding baseline methods from multiple dimensions to verify its advantages in cascade failure mitigation tasks.

[0122] First, we explore the impact of graph convolutional encoding depth on agent learning results and verify the effectiveness of the proposed graph-augmented AC architecture in cascade failure mitigation tasks. Specifically, the number of ResGCN layers is set to 2, 3, and 4, while the number of MLP layers in the decision head is fixed at 3 to ensure comparability. Under the same experimental settings, each configuration is trained independently using 3 different random seeds, and the mean and standard deviation of the results are calculated, such as... Figure 5As shown (all subsequent experiments used this configuration). The results show that GA-PPO exhibits good robustness across different numbers of ResGCN layers. GA-PPO with 3 ResGCN layers performs more stably throughout training, with a faster rate of increase in cumulative reward per round. In contrast, when the number of ResGCN layers is set to 2 or 4, the convergence speed decreases slightly, but the difference is not significant. In the later stages of training, the cumulative reward per round obtained by GA-PPO under all three configurations converges to approximately [value missing]. Around 11.5, the final difference was only about 0.3.

[0123] On the other hand, a significant performance gap exists between the GA-PPO architecture with graph-enhanced AC and the original PPO architecture using MLP AC. The original PPO learns almost no effective mitigation strategies, and its cumulative reward for each round remains at an extremely low level for a long time. This result reveals the essential difficulty in learning cascading failure mitigation strategies: the dynamic evolution of the failure boundary is highly dependent on the node correlations brought about by the network topology, and it is difficult to explicitly characterize such structural information using only MLP. Based on a comprehensive consideration of convergence speed, number of parameters, and overall performance, this embodiment selects a graph-enhanced AC architecture with 3 ResGCN layers and 3 MLP layers as the default configuration for GA-PPO in subsequent experiments.

[0124] Further exploring the applicability of different DRL algorithms in cascading failure mitigation tasks from the perspective of modeling paradigms, this study investigates their suitability from a modeling paradigm perspective. Four representative value-based DRL algorithms were selected as baselines for comparative experiments: Deep Q-Learning (DQN), Double DQN (DDQN), Dueling DQN, and a combination of both (Dueling DDQN). To ensure fairness in the evaluation process, all comparative algorithms employed the same graph-enhanced state coding structure as GA-PPO, and their training results are shown below. Figure 6 As shown.

[0125] Clearly, the value-based DQN family of algorithms failed to learn effective mitigation strategies during training, with their cumulative reward remaining consistently low (around -16 to -18) and exhibiting significant oscillations. This phenomenon is not due to differences in parameter settings, but rather stems from insufficient adaptability of the algorithm mechanism to the cascading failure mitigation problem. Specifically, because the degree distribution of the BA network exhibits a significant power-law characteristic, when a hub node fails, the action space faced by the agent in each decision stage expands dramatically, and the feasible action domain dynamically evolves with failure propagation. In this scenario, a single remedial action often has a profound impact on subsequent failure propagation paths and potential risk areas, resulting in a highly non-stationary and complex state-action value function. When faced with such situations, the Bellman objective of the DQN family, which relies on Q-value approximation, is prone to severe drift, causing the agent to fall into continuous policy oscillations and making it difficult to form stable value estimates. In contrast, GA-PPO directly optimizes the probability distribution of the policy and suppresses excessive policy updates caused by the uncertainty of failure propagation through a shearing mechanism, thus demonstrating a more stable learning process in this task.

[0126] Finally, to verify the advantages of the proposed GA-PPO over traditional heuristic methods, rule-based ranking mitigation strategies in existing research were selected as the baseline for comparison. These mainly include two categories: (1) Network topology-driven strategy: Based on degree centrality, betweenness centrality, or a weighted fusion of both (each accounting for 50%), the normal neighbor nodes of the failed node are ranked, and priority is given to remediating the nodes with higher rankings. (2) Network capacity-driven strategy: Decisions are made based on node capacity information, prioritizing the high-capacity normal neighbors of high-capacity failed nodes or the low-capacity normal neighbors of low-capacity failed nodes for remediation. To ensure the comparability between different mitigation strategies, all strategies are implemented at each decision stage. l =1 as a remedial measure. In addition, all experimental results were obtained by averaging 1000 rounds of Monte Carlo simulation to reduce the impact of randomness on the results (this setting was adopted for all subsequent related experiments).

[0127] The effectiveness of mitigation strategies is evaluated using the ratio of cascading failures (RCF), which is defined as:

[0128] Where the numerator represents the total number of nodes; the denominator represents the cutoff time. t The total number of nodes that failed during the experiment. t Take the moment when the failure propagation ends, i.e. RCF This reflects the magnitude of the final failure. Correspondingly, RCF (T) and RCF (E) Each corresponds to one of the two sub-networks.

[0129] Figures 7(a), 7(b), and 7(c) show the overall network and sub-networks, respectively. T Hezi Network E The failure scale distribution under different failure mitigation strategies. It can be seen that GA-PPO achieved the best performance across all three metrics. Through repeated interaction with the environment and continuous learning, the agent can autonomously and dynamically adjust its mitigation strategy based on the potential characteristics of failure propagation throughout the entire process, thereby effectively identifying key risk nodes controlling the cascading failure evolution path. Compared to the no-mitigation scenario, RCF , RCF (T) , RCF (E) These reduced performance by 0.427, 0.361, and 0.562 respectively. In contrast, all rule-based heuristic mitigation methods exhibited varying degrees of performance degradation. On one hand, the high-capacity-first strategy generally outperformed the low-capacity-first strategy, indicating that high-capacity nodes play a more important role in absorbing failure loads. On the other hand, the differences in mitigation effectiveness among topology-based ranking strategies were relatively limited. RCF The decrease was around 0.2.

[0130] From the perspective of sub-networks, different mitigation strategies RCF (E) All significantly higher than RCF (T) This means that the sub-networks related to physical flows contributed the vast majority of failed nodes in the cascading failures. This indicates that the redundancy-driven load redistribution mechanism played a dominant role in the failure propagation process, and the overall network failure scale was rapidly amplified as dependent edges grew.

[0131] Sensitivity analysis was conducted on the three key parameters of the proposed GA-PPO, with a focus on exploring the topology search depth. K Reward function weights μ And parameters for controlling the total amount of SEC M Impact on failure mitigation effectiveness. These parameters control the spatial scope of remedial actions, the trade-off between remedial costs and failure suppression, and the total number of SECs available for scheduling in the network. In practical applications, these parameters can be dynamically adjusted based on network topology characteristics, emergency response capabilities, and the available SEC levels to adapt to different network and cascading failure scenarios.

[0132] Figure 8 Showing different topology search depths K ∈{1, 2, 3, 4, 5, 6} and unconstrained (i.e.K The convergence performance of GA-PPO differs under the condition that the agent can select a remedy node from across the entire network (where the action space is infinitely large). It can be observed that in the unconstrained case, the cumulative reward for each round remains at a low level for a long time. This is mainly because the dimension of the action space expands rapidly under unconstrained conditions, requiring the agent to select a remedy target from a large number of nodes across the entire network, making it difficult to form an effective state-action mapping. In contrast, when the remedy action space is topologically constrained, the agent can learn effective failure mitigation strategies under different conditions and achieve stable convergence. Furthermore, with… K As the value increases, the convergence performance of the algorithm shows a slight downward trend.

[0133] The above results indicate that imposing reasonable constraints on the action space is a key prerequisite for ensuring efficient agent learning in cascading failure mitigation tasks. Furthermore, under conditions of limited emergency resources, emergency response strategies should focus more on nodes near the current failure propagation boundary (i.e.,... K The timely remedy for (=1) is to avoid prematurely allocating limited resources to remote nodes that have not yet been impacted by the failure (i.e., ...). K >1).

[0134] Figures 9(a), 9(b), and 9(c) illustrate the parameters controlling the total amount of SEC. For the overall network and sub-networks T Hezi Network E of RCF Impact. It should be noted that when When the value is 50, the SEC is relatively abundant, meaning that there will still be remaining SEC after completing the remedial action for the entire cascading failure. Only configuration is required. =10 of SEC, RCF , RCF (T) , RCF € This resulted in reductions of 0.142 (0.799-0.657), 0.123 (0.7-0.577), and 0.182 (1-0.818), respectively. These results fully demonstrate the feasibility of the proposed method even under conditions of severe SEC restrictions.

[0135] along with As the magnitude of failure increases, the failure scale of both the overall network and the two subnetworks exhibits a monotonically decreasing trend. Crucially, the magnitude of this decrease differs significantly. A clear critical value appears... The stage of expanding from 20 to 30. At this point, RCF , RCF (T) , RCF €They decreased by 0.107, 0.087, and 0.145 respectively, significantly higher than others. The case where the increase is equal. In contrast, when... As the level increases, the failure mitigation benefits diminish significantly. Specifically, when... When increasing from 40 to 50, RCF , RCF (T) , RCF € The decreases were only 0.027, 0.023, and 0.036, respectively. This decrease accounts for only about 25% of the aforementioned critical range.

[0136] also, The degree to which the increase in [amount] mitigates the failure of the two sub-networks also differs. When When increasing from 10 to 50, RCF (T) , RCF € They decreased by 0.198 and 0.33 respectively. This result is consistent with the previous analysis, namely, the subnetwork E It plays a leading role in the spread of cascading failures across the entire network, thus making it a major beneficiary of SEC investments.

[0137] The above findings indicate that the total SEC allocation exhibits diminishing marginal returns in cascading failure mitigation. In real-world network disaster scenarios, due to constraints such as cost, emergency response capabilities, and resource availability, networks often struggle to maintain a high SEC ceiling in the long term. In such cases, priority should be given to increasing the SEC allocation to a critical benefit level to maximize failure mitigation effectiveness under limited SEC conditions.

[0138] Finally, the impact of the weight parameters in the reward function on the agent's mitigation policy learning was discussed. Fixed, but adjusted To analyze its moderating effect on the choice of remedial intensity, the results are as follows: Figure 10 As shown. In the default configuration At 20, 67.7% of remedial actions were achieved through low-intensity treatment. l =1 to implement, and high intensity l Remedial actions with a value of 3 account for only 2.5%. At this point, the agent tends to strictly control the cost of using SEC while mitigating failures. With... Increasing to 30, the agent begins to focus more on suppressing the overall failure scale, medium intensity. l =2 and high strength l =3 The proportion of remedial actions used will increase accordingly. When This trend becomes even more pronounced when the value increases further to 40. Compared to =20, medium strength l =2 and high strength l The proportion of remedial actions with a value of 3 increased by approximately 0.092 and 0.139, respectively.

[0139] To quantitatively evaluate the failure mitigation benefits of different mitigation strategies per unit of SEC consumption, a cost-effectiveness (CE) metric was designed and defined as follows:

[0140] in, RCF Represents before and after the implementation of mitigation strategies RCF The differences; ratio1, ratio2, and ratio3 represent the values ​​at a given time interval. Below, different remedial intensities l =1, 2, 3 represent the proportion of SEC used throughout the failure mitigation process. The denominator reflects the unit SEC used by the mitigation strategy by scaling the SEC consumption corresponding to different remedy intensities to a uniform range of [0.8, 1.2]. From Figure 11 and Figure 12 It can be seen that, although the intelligent agent is The mitigation strategies learned at =30 compared to =20 can be further reduced RCF However, its CE However, it is at a relatively low level. In contrast, when When the value is 40, the agent maximizes the reduction. RCF At the same time CE It is also at the highest level, achieving the highest SEC utilization efficiency.

[0141] The above findings indicate that larger This encourages agents to consume more SEC (Security Conflict) during decision-making in exchange for strong suppression of the scale of cascading failures. Smaller SEC... This guides the agent to adopt a more conservative, SEC-saving mitigation strategy. Through reasonable adjustment... The proposed GA-PPO framework can not only adaptively adjust the selection distribution of remedy intensity, but also effectively balance the reduction of failure scale with the consumption of remedy costs, so as to maximize the utilization benefits of SEC.

[0142] The above results demonstrate that the GA-PPO framework proposed in this embodiment is not limited by a specific number of network nodes or topological relationships, and can learn effective cascading failure mitigation strategies in dependent networks of different sizes, exhibiting good scalability. Furthermore, these results provide important insights for cascading failure mitigation under SEC-constrained conditions. On the one hand, limited SEC should be preferentially concentrated on a very few critical nodes that have a decisive impact on the propagation of cascading failures, rather than being evenly distributed among multiple nodes. On the other hand, in dependent network scenarios, it is crucial to identify and remedy subnetworks that play a dominant role in the overall network failure scale to avoid systemic risks caused by network coupling effects.

[0143] Based on the same inventive concept, embodiments of this application provide a cascading failure mitigation device for interdependent infrastructure networks, see reference. Figure 13 As shown, the cascading failure mitigation device 200 for interdependent infrastructure networks provided in this application embodiment includes at least: The acquisition unit 201 is used to acquire the feature vectors of the nodes of each sub-network of the infrastructure network at the current time and the connection relationships between the nodes; Construction unit 202 is used to construct the first node feature matrix of the entire network based on the feature vectors of the nodes of each sub-network; and to construct the first adjacency matrix of the entire network based on the connection relationship between nodes. The encoding unit 203 is used to encode the first node feature matrix and the first adjacency matrix of the entire network using a pre-trained state encoder to obtain the first encoded feature matrix. Processing unit 204 is used to process the first encoded feature matrix and the first global feature vector using a pre-trained policy network to obtain the probability distribution of the first remedial action; The selection unit 205 is used to select the action with the highest probability value from the first remedial action probability distribution as the remedial action; the remedial action includes a remedial node and a remedial intensity level. The determining unit 206 is used to determine the recovery capacity of the recovery node based on the system emergency response capability and recovery intensity level of the sub-network to which the recovery node belongs; Failure mitigation unit 207 is used to schedule recovery capacity to the recovery node using an agent.

[0144] It should be noted that the principle of the interdependent infrastructure network cascading failure mitigation device 200 provided in this application embodiment to solve the technical problem is similar to the method provided in this application embodiment. Therefore, the implementation of the interdependent infrastructure network cascading failure mitigation device 200 provided in this application embodiment can refer to the implementation of the method provided in this application embodiment, and the repeated parts will not be described again.

[0145] Based on the same inventive concept, embodiments of this application also provide an electronic device, such as... Figure 14 As shown, it includes a memory and a processor, wherein the memory stores an executable program and the processor executes the executable program to implement the steps of the cascading failure mitigation method for interdependent infrastructure networks provided in the above embodiments.

[0146] The aforementioned processor can be a general-purpose processor, a digital signal processor, an application-specific integrated circuit (ASIC), a programmable logic device (PLD), or a combination thereof. The aforementioned PLD can be a complex programmable logic device (CPLD), a field-programmable gate array (FPGA), a generic array logic (GAL), or any combination thereof. The general-purpose processor can be a microprocessor or any conventional processor, etc.

[0147] Since the electronic device described in this application embodiment is an electronic device equipped with a memory that implements the interdependent infrastructure network cascading failure mitigation method disclosed in this application embodiment, those skilled in the art can understand the structure and variations of the electronic device described in this application embodiment based on the interdependent infrastructure network cascading failure mitigation method described in this application embodiment, and therefore will not be described again here.

[0148] This application also provides a computer-readable storage medium storing a computer program thereon, which, when executed by a processor, implements the steps of the interdependent infrastructure network cascading failure mitigation method provided in the above embodiments.

[0149] The storage medium in this embodiment may be included in an electronic device; or it may exist independently and not be assembled into an electronic device. The storage medium carries one or more computer programs, which, when executed, implement the steps of the cascading failure mitigation method for interdependent infrastructure networks provided in the above embodiments.

[0150] It should be understood that the various solutions in this embodiment have the same technical effects as those in the above method embodiments, and will not be repeated here.

[0151] According to embodiments of this application, the computer-readable storage medium can be a non-volatile computer-readable storage medium, such as including but not limited to: portable computer disks, hard disks, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or flash memory), portable compact disk read-only memory (CD-ROM), optical storage devices, magnetic storage devices, or any suitable combination thereof. Optionally, specific examples in this embodiment can refer to the examples described in any embodiment of this application, which will not be repeated here. Obviously, those skilled in the art should understand that the various modules or steps of this application described above can be implemented using general-purpose computing devices. They can be centralized on a single computing device or distributed across a network of multiple computing devices. Optionally, they can be implemented using computer-executable program code, thereby storing them in a storage device for execution by a computing device. In some cases, the steps shown or described can be performed in a different order than those presented here, or they can be fabricated as separate integrated circuit modules, or multiple modules or steps can be fabricated as a single integrated circuit module. Thus, this application is not limited to any particular hardware and software combination.

[0152] This application also provides a computer program product, including a computer program / instruction, which, when executed by a processor, implements the steps of the interdependent infrastructure network cascading failure mitigation method provided in the above embodiments.

[0153] The flowcharts and block diagrams in the accompanying drawings illustrate the architecture, functionality, and operation of possible implementations of systems, methods, and computer program products according to various embodiments of this application. In this regard, each block in a flowchart or block diagram may represent a module, segment, or portion of code containing one or more executable instructions for implementing a specified logical function. It should also be noted that in some alternative implementations, the functions targeted in the blocks may occur in a different order than those targeted in the drawings. For example, two consecutively represented blocks may actually be executed substantially in parallel, and they may sometimes be executed in reverse order, depending on the functions involved. It should also be noted that each block in the block diagrams and / or flowcharts, and combinations of blocks in the block diagrams and / or flowcharts, can be implemented using a dedicated hardware-based system that performs the specified function or operation, or using a combination of dedicated hardware and computer instructions.

[0154] Furthermore, while the operations are described in a specific order, this should not be construed as requiring these operations to be performed in the specific order shown or in a sequential order. Multitasking and parallel processing may be advantageous in certain environments. Similarly, while several specific implementation details are included in the above discussion, these should not be construed as limiting the scope of this application. Certain features described in the context of individual embodiments may also be implemented in combination in a single embodiment. Conversely, various features described in the context of a single embodiment may also be implemented individually or in any suitable sub-combination in multiple embodiments.

Claims

1. A method for mitigating cascading failures in interdependent infrastructure networks, characterized in that, include: Obtain the feature vectors of nodes in each sub-network of the infrastructure network at the current moment, as well as the connection relationships between nodes; Construct the first node feature matrix of the entire network based on the feature vectors of the nodes in each sub-network; Construct the first adjacency matrix of the entire network based on the connection relationships between nodes; The first encoded feature matrix is ​​obtained by encoding the first node feature matrix and the first adjacency matrix of the entire network using a pre-trained state encoder. The first encoded feature matrix and the first global feature vector are processed using a pre-trained policy network to obtain the probability distribution of the first remedial action; The action with the highest probability value is selected from the probability distribution of the first remedial action as the remedial action; the remedial action includes a remedial node and a remedial intensity level; The recovery capacity of a recovery node is determined based on the system emergency response capability and recovery intensity level of the sub-network to which the recovery node belongs. The agent is used to schedule the recovery capacity to the recovery node.

2. The method according to claim 1, characterized in that, The feature vector of a node includes: the node's load, the node's capacity, the node's betweenness centrality, the encoded value of the degree of impact the node suffers under cross-network dependencies, and the identifier value of whether the node is a newly failed node after the previous round of failure propagation; the first global feature vector includes: the remaining system emergency response capability of each sub-network at the current time and the number of remedial actions performed before the current time.

3. The method according to claim 1, characterized in that, Construct the first adjacency matrix of the entire network based on the connection relationships between nodes, including: Obtain the connection relationships between nodes in subnetwork T and subnetwork E; The connection value is determined based on the connection relationship. If two nodes are connected, the connection value is 1; otherwise, the connection value is 0. Generate the adjacency matrix of subnetwork T based on the connection values ​​between nodes in subnetwork T. ; Generate the adjacency matrix of subnetwork E based on the connection values ​​between nodes in subnetwork E. ; Construct the first adjacency matrix of the entire network : in, and All are zero matrices.

4. The method according to claim 1, characterized in that, The state encoder employs a residual graph convolutional neural network, which includes D layers of graph convolutional layers. The first encoded feature matrix is ​​obtained by encoding the first node feature matrix and the first adjacency matrix of the entire network using a pre-trained state encoder; including: For the first adjacency matrix Processing yields a matrix. : in, I It is the identity matrix; calculate the matrix according to the following formula. : in, This is the corresponding degree matrix; The feature matrix output by the (d+1)th graph convolutional layer for: in, This represents the feature matrix output by the graph convolutional layer at layer d. ;when d =0 Represents the feature matrix of the first node; This represents the learnable parameter matrix of the (d+1)th graph convolutional layer; LayerNorm represents a non-linear activation function. Representation layer normalization operation; Res Indicates a residual join operation; The feature matrix output by the (D-1)th graph convolutional layer This is determined to be the first encoding feature matrix.

5. The method according to claim 1, characterized in that, The method further includes: Step S1: Obtain historical moments The feature vectors of nodes in each subnetwork of the infrastructure network and the remaining emergency response capacity of each subnetwork; where the initial value of k is 1; Step S2: Construct the second node feature matrix of the entire network based on the feature vectors of the nodes in each sub-network; construct the second global feature vector based on the remaining emergency response capability of each sub-network and the number of remedial actions taken; construct the second adjacency matrix of each sub-network based on the connection relationships between nodes; Step S3: Determine the state of the environment by using the second node feature matrix and the second global feature vector of the entire network. ; Step S4: Encode the second node feature matrix and the second adjacency matrix of the entire network using the state encoder to obtain the second encoded feature matrix; Step S5: Use the policy network to process the second encoded feature matrix and the second global feature vector to obtain the probability distribution of the second remedial action; Step S6: The agent randomly selects a remedial action from the probability distribution of the second remedial action. And perform remedial actions. The entire infrastructure network undergoes state transition evolution to time [time]. status Simultaneously provide instant rewards Thus, the time is obtained. samples ; Step S7: Determine whether the number of samples is less than the preset threshold. If yes, update k to k+1 and proceed to step S1; otherwise, proceed to step S8. Step S8: For each batch of samples, calculate the policy optimization objective value, the mean squared error of value loss, and the entropy regularization term to construct the overall objective value. ; Step S9: Update the parameters of the state encoder, policy network, and value network based on the total target value.

6. The method according to claim 5, characterized in that, Instant reward value for: in, The cost of a single remedial action; For a historic moment The cumulative number of remedial actions; The maximum number of remedies; in, To match the remedial strength level l The corresponding cost of the unit system's emergency response capabilities; The cost weighting coefficient is used to remedy the situation. The time-step cascade failure ratio is calculated using the following formula. : in, This is the failure penalty weighting coefficient. This represents the number of newly added failure nodes caused by this round of failure propagation; Let T be the number of nodes in the subnetwork. Let E be the number of nodes in subnetwork E.

7. The method according to claim 5, characterized in that, Strategy optimization target value for: in, θ For policy network parameters; min This represents the minimum value operation; clip The shearing function represents the restriction that the sampling probability of the current policy for any action is at most amplified by (1+) compared to the old policy. ) times or reduced to (1- ) times; >0 represents the shearing threshold; Importance sampling ratio; Calculate according to the following formula The estimated value : Where N represents the total number of decision time steps in a round; λ∈[0, 1] is the GAE decay parameter; Discount factor; The time difference error is calculated using the following formula. : in, Represents the value network function. These are the parameters of the value network.

8. The method according to claim 7, characterized in that, The value of mean square error loss for: in, To align with the GAE's target value, the calculation formula is as follows: 。 9. The method according to claim 8, characterized in that, Total target value The calculation formula is: in, Represents policy entropy; and All of these are loss weight hyperparameters. This represents the policy function with policy network parameters θ.

10. A cascading failure mitigation device for interdependent infrastructure networks, characterized in that, include: The acquisition unit is used to acquire the feature vectors of nodes in each sub-network of the infrastructure network at the current time and the connection relationships between nodes; The building unit is used to construct the first node feature matrix of the entire network based on the feature vectors of the nodes in each sub-network; Construct the first adjacency matrix of the entire network based on the connection relationships between nodes; The encoding unit is used to encode the first node feature matrix and the first adjacency matrix of the entire network using a pre-trained state encoder to obtain the first encoded feature matrix; The processing unit is used to process the first encoded feature matrix and the first global feature vector using a pre-trained policy network to obtain the probability distribution of the first remedial action. A selection unit is used to select the action with the highest probability value from the first remedial action probability distribution as the remedial action; the remedial action includes a remedial node and a remedial intensity level; The determining unit is used to determine the recovery capacity of the recovery node based on the system emergency response capability and recovery intensity level of the sub-network to which the recovery node belongs; A failure mitigation unit is used to schedule recovery capacity to the recovery node using an agent.

Images