Cascade failure-based double-layer international railway freight network toughness evaluation method

Abstract

The invention discloses a cascade failure-based double-layer international railway freight network toughness evaluation method, which belongs to the technical field of road traffic network evaluation, and comprises the following steps: constructing topology models of a basic line network and a transportation service network; constructing a network evaluation system and an importance evaluation system of each node in the network; constructing a network vulnerability evaluation model based on cascade failure, wherein the model adopts an improved CML node state model and a cargo redistribution strategy; and in combination with each network and the evaluation model, a cascade failure condition is simulated through a simulation experiment, and the network is evaluated in combination with an evaluation system. The characteristics of international railway freight are considered, a double-layer coupling network of a basic line network and a transportation service network is constructed, a multi-dimensional toughness evaluation index system and a cargo distribution mechanism are set up, and the cascade failure condition in the international railway freight network can be simulated.

Description

Technical Field

[0001] This invention relates to the field of road traffic network assessment technology, specifically to a method for assessing the resilience of a two-tier international railway freight network based on cascading failures. Background Technology

[0002] Currently, with the expansion of network scale and the increasing complexity of the operating environment, international railway freight networks are becoming increasingly vulnerable to unforeseen events. Analyzing the vulnerability and resilience of transportation networks has become a crucial academic issue in the transportation field. Existing research (CN202310305227.6) proposed a method for assessing the resilience of urban road traffic networks in response to unforeseen events. This method integrates absolute and relative resilience indices to establish a comprehensive road traffic network resilience assessment index. It utilizes the Space L method to construct the network topology and calculate the absolute resilience index; it also uses seepage theory to calculate the seepage threshold of the road network at different connectivity levels, serving as the relative resilience index. Furthermore, CN202510486979.6 provides a method for analyzing cascading failures of urban infrastructure based on a high-order network model. This method constructs a high-order network model of the urban infrastructure system and simulates cascading failures from both node and regional failure mechanisms to identify high-risk nodes in the high-order network model and improve the resilience of the urban infrastructure system.

[0003] While there is a certain research foundation for assessing the resilience of transportation networks, the following shortcomings still exist: Firstly, existing research is mostly focused on the same country or region, with a relative lack of research on cross-border transportation. As a complex system connecting different countries and regions, the international railway freight network is significantly different from that of the same country or region. For example, there are issues such as track switching (changing to different tracks) and border inspection, which leads to different considerations.

[0004] Secondly, cargo redistribution plays an important role in cascading failures, but existing research has not yet revealed the cargo redistribution rules of the international railway freight network, and has given little consideration to the interactive effects of the special and complex characteristics of the international railway freight network on the node states. Summary of the Invention

[0005] To address the aforementioned issues, this invention provides a resilience assessment method for a two-layer international railway freight network based on cascading failures. This invention constructs a two-layer coupled network model adapted to the characteristics of international railway freight, quantifies its cascading failure propagation mechanism, identifies key vulnerable nodes in the two-layer coupled network, simulates the vulnerability of the international railway freight network and key nodes when cascading failures occur using software, and proposes multi-level optimization strategies to improve network robustness, thereby promoting the high-quality development of international railway freight networks, exemplified by the China-Europe Railway Express freight network.

[0006] To achieve the above objectives, the present invention provides the following solution: A resilience assessment method for a two-tier international railway freight network based on cascading failures includes the following steps: S1. Based on the actual operating characteristics of the international railway freight network, topological models of the basic line network and the transportation service network are constructed respectively, forming a two-layer coupled network. The basic line network is a physical network composed of railway stations, transshipment stations, and border inspection stations as nodes, and the tracks connecting these stations as edges. The transportation service network is a service network composed of railway stations as nodes and the operating lines connecting these stations as edges. S2. Construct a network evaluation system and an evaluation system for the importance of each node in a two-layer coupled network; S3. Construct a network vulnerability assessment model based on cascading failures; the network vulnerability assessment model adopts an improved CML node state model and a cargo redistribution strategy; the cargo redistribution strategy is used to redistribute cargo when cascading failures occur and stations cannot provide sufficient transportation services; The improved node state model in CML is as follows: In the formula, , , These are the coupling coefficients for node degree, edge betweenness, and cargo flow, respectively. , They are nodes and nodes exist The status at any given moment reflects whether a node is capable of providing transportation services at that moment. and For nodes and nodes exist The local mapping function at time t reflects the evolution of the node's own state; The node state; For indicator functions, when node With nodes If there is an edge between them ,otherwise ; For nodes i The degree of the node; For nodes i betweenness; The edge betweenness; The number of nodes in a single network; For nodes i Freight volume; node and nodes The flow of goods between them; External interference quantity; For nodes At any moment The degree value; S4. Combining the two-layer coupled network and the evaluation model, the cascading failure situation is simulated through simulation experiments, and the network is evaluated in conjunction with the evaluation system.

[0007] As a specific embodiment of the present invention, in step S3, the cargo redistribution strategy includes cargo redistribution rules based on the load of adjacent nodes and cargo redistribution rules based on the distance between adjacent nodes. The cargo redistribution rules based on the load of adjacent nodes are as follows: The redistribution rules for goods based on the distance between adjacent nodes are as follows: In the formula, for Time Node The remaining load; For nodes The capacity; For nodes exist The load at any moment; for Time Node The load; for Time Node The remaining load; O is the failed node. i The set of neighboring nodes; for t Time Node j The set of failed nodes among the neighboring nodes; for t Failure Node at Any Time i Assigned to nodes j The load; for t At any given moment, all failed nodes send messages to the node. j The allocated load, i.e., the number of nodes. j Total load increment; For nodes i With nodes j The actual distance; For nodes i With nodesm The actual distance.

[0008] In a specific embodiment of the present invention, in step S2, the network evaluation system evaluates the basic line network using the maximum connectivity graph and network efficiency as indicators; and evaluates the transportation service network using transportation performance and the affected freight flow rate as indicators. The formula for calculating transportation performance is as follows: The formula for calculating the affected cargo flow rate is: In the formula, For network transport performance; For the flow of goods in the maximum connected graph; For transportation service network G The original flow of goods; For the initial network node i Freight volume; For the affected cargo flow rate; , They are nodes i、 j Freight volume; This is the set of failed nodes; This is the initial set of network nodes.

[0009] As a specific embodiment of the present invention, constructing the importance evaluation system for each node includes the following steps: (1) Construct a node importance evaluation index system to evaluate the basic line network and transportation service network. Its evaluation index includes node degree, betweenness, degree centrality, proximity degree centrality, betweenness centrality, eigenvector centrality and node efficiency. (2) In the basic line network and transportation service network, the weight of each indicator is determined by the Critic method, including constructing the original decision matrix, then performing positiveization and standardization on the elements in the original decision matrix in sequence, then calculating the variability and conflict of indicators, and finally calculating the final weight of each indicator. The formulas for calculating the weights of each indicator include: In the formula, As an indicator j The final weight; and They are nodes i Indicators j and m Data; , Indicators j , m The data mean, that is, the average value of this technical indicator across all nodes; , , , , , All are intermediate variables; M The number of indicators; and The elements in the decision matrix after positive transformation , The normalized value; As an indicator j The variability of the indicators; For Standard j ,index m Conflicting indicators; (3) In the basic line network and transportation service network, based on the weight matrix The importance evaluation value of each node was obtained using the TOPSIS method. S i ; (4) In the basic line network and transportation service network, the grey relational degree between each node and the reference object is determined based on the grey relational analysis method. G i ; (5) Determine the comprehensive importance evaluation value of each node; In the formula, M i For nodes i The overall importance evaluation value; S i For nodes i The importance evaluation value; Z is an intermediate variable.

[0010] Beneficial effects: Compared with the prior art, the technical solution of the present invention has the following beneficial technical effects: (1) This invention addresses the need for resilience evaluation of cross-regional and complex freight networks. It integrates network structure characteristics and cargo flow distribution to construct a two-layer coupled network of basic line network and transportation service network.

[0011] (2) Based on the characteristics of international freight network, this invention constructs a multidimensional resilience evaluation index system, introduces an improved Critic method and grey relational analysis to modify the traditional Topsis method, weakens the one-sidedness and bias of the evaluation results, fully considers the coordination of evaluation indicators, and accurately identifies key nodes.

[0012] (4) This invention characterizes the diffusion mechanism of cascade failure in the international railway freight network and introduces multiple coupling coefficients to design a dynamic node state model.

[0013] (5) This invention introduces a cargo allocation mechanism in the international railway freight network, which finds alternative routes for transportation when nodes are congested, and the simulation process is more realistic. Attached Figure Description

[0014] Figure 1 This is a trend chart of vulnerability index changes under random attacks with different node coupling coefficients; Figure 2 This is a trend chart of vulnerability index changes under deliberate attacks on different node coupling coefficients; Figure 3 This shows the changes in vulnerability indicators under different attack strategies when the node coupling coefficient is 0.4. Figure 4 This shows the changes in vulnerability indicators under random attack points of different cargo allocation rules; Figure 5 It shows the changes in vulnerability indicators of point attacks under different attack strategies; Figure 6 This shows the changes in vulnerability indicators under random edge attacks with different cargo allocation rules; Figure 7 This describes the changes in vulnerability indicators under deliberate attacks. Detailed Implementation

[0015] The present invention will be further described in detail below with reference to the embodiments and accompanying drawings, but the embodiments of the present invention are not limited thereto.

[0016] A resilience assessment method for a two-tier international railway freight network based on cascading failures includes the following steps: S1. Based on the actual operating characteristics of the international railway freight network, topological models of the basic line network and the transportation service network are constructed respectively to provide a structural basis for subsequent cascading failure research. In this invention, the basic railway network and the transportation service network together constitute a two-layer coupled network. The basic railway network is a physical network consisting of railway stations, transshipment stations, and border inspection stations as nodes, with the railway tracks connecting these stations as edges. The transportation service network is a service network consisting of railway stations as nodes, with the operating lines connecting these stations as edges. For example, when a train arrives at station A and goods pass through station B, there is an edge connecting stations A and B. Compared to the actual railway lines in the basic railway network, it reflects the flow of goods. A highly stable basic railway network has strong resilience and transformation capabilities, and can quickly absorb disturbances and maintain network performance, while a poorly functioning basic railway network may collapse rapidly under disturbances, affecting freight transportation. The transportation service network is the channel for the flow of goods between cities, and its operation affects railway lines and facilities. Therefore, the basic railway network and the transportation service network are interdependent and mutually influential.

[0017] S2. Construct a network evaluation system and an importance evaluation system for each node in a two-layer coupled network, including: S21. Construct a multi-dimensional vulnerability assessment system. For basic infrastructure networks, use maximum connectivity and network efficiency as indicators for evaluation; for transportation service networks, use transportation performance and affected freight flow rate as indicators for evaluation. The formula for calculating the maximum connected graph is: The formula for calculating network efficiency is: The formula for calculating transport performance is: The formula for calculating the affected cargo flow rate is: In the formula, It is a maximum connected graph; This represents the maximum number of nodes in the connected graph after a cascading failure. The number of nodes in a single network; For network efficiency; For nodes i With nodes j The shortest path length between; For network transport performance; For the flow of goods in the maximum connected graph; The original flow of goods for the transportation service network; For the initial network node i Freight volume; For the affected cargo flow rate; , They are nodes i, j Freight volume; This is the set of failed nodes; This is the initial set of network nodes; S22. Construct node importance evaluation indicators to evaluate the basic line network and transportation service network. The evaluation indicators include node degree, betweenness, degree centrality, proximity degree centrality, betweenness centrality, eigenvector centrality and node efficiency. Degree is an important indicator describing node attributes. In complex networks, node degree is the number of edges connected to the node, reflecting the local connectivity of the international railway freight network. Its calculation formula is: ; Betweenness includes node betweenness and edge betweenness. Node betweenness refers to the number of nodes that are traversed in all shortest paths in the network. The ratio of the number of shortest paths to the total number of shortest paths is calculated as follows: The edge betweenness refers to the number of edges. The ratio of the number of shortest paths between all pairs of nodes to the total number of shortest paths between all pairs of nodes is calculated as follows: ; Degree centrality refers to the degree of connection between nodes in a network. The number of directly connected edges and nodes The ratio of the number of edges connected to all other nodes is calculated as follows: ; Proximity centrality measures how closely nodes in a network are connected to other nodes. Defined as a node The average distance to the remaining nodes, and the proximity centrality is the reciprocal of the average distance, calculated as follows: , ; Betweenness centrality is used to evaluate the importance of a network node in information transmission. Betweenness centrality refers to the degree to which all other nodes are more important than a single node. The sum of the betweennesses is calculated as follows: ; Eigenvector centrality is used to evaluate the importance of network nodes in critical tasks. Eigenvector centrality is the sum of the values ​​of nodes. The importance of a network in all networks is calculated using the following formula: ; Node efficiency is the node When a node fails The communication efficiency between the first adjacent nodes is a measure of the node's performance. When a node fails, the information transmission capability between the remaining nodes is calculated as follows: ; In the formula, For nodes i The degree of the node; For indicator functions, when node With nodes If there is an edge between them ,otherwise ; node i betweenness; From node m To the node n The number of shortest paths; From node m To the node n Through nodes i The number of shortest paths; Edge betweenness; From node m via edge To the node n The number of shortest paths; Degree centrality of node i; node i Proximity centrality; For the node The average distance to the remaining nodes; node i Betweenness centrality; node s and nodes t The shortest path passes through the nodes i The number of paths; Connecting nodes s and nodes t The number of shortest paths; For nodes i eigenvector centrality; , They are nodes i, j The importance of; proportionality constant; A Represents the adjacency matrix of the network. x Representation matrix A The corresponding feature vector; node i Node efficiency; S23. Based on the evaluation indicators constructed in step S22, the importance of each node is comprehensively evaluated using the improved Critic-Grey Relationship Analysis-TOPSIS method. In this invention, the TOPSIS method is used to evaluate node importance, and an improved Critic method is used to determine the weight of each indicator, while also considering the variability and correlation of the indicators. Subsequently, grey correlation analysis is performed on each subsystem to clarify the numerical relationships between them. The maximum importance evaluation value of each single-layer complex network is combined as a reference object, and the grey correlation between other nodes and this reference object is calculated. Finally, the weights of TOPSIS and grey correlation analysis in the comprehensive importance evaluation are calculated to generate the final comprehensive evaluation value for each node. Specifically, the steps include the following: S231. Determine the weights of each indicator based on the Critic method. S2311. Construct the original decision matrix; In the formula, Original decision matrix; The number of nodes in the network. The number of indicators for evaluating node importance; For nodes i No. j The index values ​​of importance evaluation indicators, ; S2312. Perform forwarding and standardization processing on each element of the original decision matrix in sequence; The formula for forward processing is: in, a and b These are the lower and upper limits of the interval, respectively. c As an intermediate variable; The formula for standardization is: The standardized matrix is ; S2313, Based on Standardized Matrix Calculate the variability and conflict of each indicator; The formula for calculating the variability of the index is as follows: The formula for calculating the conflict of indicators is as follows: S2314. Calculate the probability matrix; S2315. Calculate the original information content and the original index weights.

[0018] S2316. Calculate the improved information content and the final weights of the indicators, where the entropy weight method is used to determine the information entropy value. .

[0019] S232. Determine the importance evaluation value of each node based on the TOPSIS method; S2321. Determine the ideal solution. and negative ideal solution .

[0020] weight matrix Multiply by the standardized decision matrix The weighted decision matrix is ​​obtained. Positive ideal solution and negative ideal solution It is determined by the maximum and minimum values ​​in the weighted decision matrix.

[0021] S2322. Calculate the evaluation index and the positive ideal solution. and negative ideal solution The distance.

[0022] S2323. Calculate the overall importance evaluation value of each node. Calculate the proximity of each solution to the positive ideal solution, for each node. The importance rating values ​​are as follows: S233. Determine the grey relational degree of each node based on grey relational analysis. S2331. Construct a sample matrix and standardize it.

[0023] For ease of calculation, a two-layer network is constructed in the China-Europe Railway Express freight network, consisting of nodes that exist simultaneously in both the basic route network and the transport service network. Therefore, it is assumed that the China-Europe Railway Express freight network includes... Each single-layer network contains [number] single-layer networks. Nodes, Nodes In the network layer Importance rating value The sample matrix is ​​obtained. .

[0024] For the sample matrix Normalization by row yields a new matrix. The normalization formula is as follows: S2332. Determine the reference objects and evaluation objects; For railway freight networks, choose a matrix. The largest element in each row is used as a reference to form a reference sequence. The importance evaluation values ​​of nodes in the basic railway network and transportation service network are used as the evaluation objects in grey relational analysis, and a matrix is ​​used... for.

[0025] in, S2333, Calculate the grey relational degree.

[0026] Based on the previous calculations, the grey relational degree formula is used to calculate the grey relational degree between each node in the network and the reference object.

[0027] S234. Calculate the comprehensive importance evaluation value of node i.

[0028] in, S3. Establish a network vulnerability assessment model based on cascading failures; Cascading failure refers to a chain reaction in complex networks where the failure of a node or edge causes load to shift to other nodes or edges, potentially leading to the collapse of the entire network. In international rail freight networks, cascading failure occurs when a node or edge is disrupted (due to natural disasters, terrorist attacks, or capacity restrictions), preventing goods from passing through that node or edge. Consequently, goods must be redistributed to other nodes, which may cause their capacity to exceed limits and lead to failure. This section introduces an improved Coupled Mapping Lattice (CML) model and discusses traffic redistribution rules, including two cargo redistribution strategies: one based on the load of neighboring nodes and the other based on the distance between neighboring nodes.

[0029] S31. Node state model based on improved CML In international railway freight networks, node states are influenced by structural characteristics and freight flow distribution. Therefore, both structural coupling coefficients and freight flow coupling coefficients are considered. The improved node state model is developed by considering factors such as node degree, betweenness coefficient, and freight volume, and effectively takes into account the node's own resilience. The corresponding model equations are as follows: Among them, parameters , , These are the coupling coefficients for node degree, edge betweenness, and cargo flow, respectively. and satisfy normalization constraints The combination of coupling coefficients can be optimized through grid search. .exist Under constraints, the parameter space is traversed with a step size of 0.1 to obtain all feasible combinations. Then, with the goal of minimizing the network efficiency degradation rate, the loss function is defined as follows: Screening through simulation experiments The minimum combination of coefficients. (Considering the nodes) When faced with failure events, it possesses a certain degree of resilience. Therefore, a node resilience coefficient based on node degree is added to the cargo-flow coupling part, and a standard Sigmoid activation function is introduced, with an output range of... Its general form is The node resilience function based on node degree is: At this time, the node's The larger the value, the stronger its ability to transfer new cargo flow and the weaker it is from the adverse effects of its neighboring nodes.

[0030] If the status of all nodes in the international rail freight network is... Within this range, it indicates that these nodes can provide stable and efficient transportation services, unaffected by abnormal loads, equipment failures, or congestion, meaning they are in normal operating condition. If within... Time to node Apply external interference Then the node It will fail. At that time, for nodes No external interference was applied. When Time Node If the equipment cannot operate normally due to equipment failure, overload, or other abnormal conditions, then The states of its neighboring nodes will also be affected accordingly. Therefore, at time... The network state is recalculated periodically. If the state of an adjacent node is greater than 1, the adjacent node also becomes invalid and is removed from the network.

[0031] This invention introduces parameters This increases the complexity of the model, but through constraints... The computational burden can be controlled. Simultaneously, the introduction of the standard Sigmoid activation function effectively reduces the scope and speed of failure propagation, making the flow changes at nodes more realistic. Traffic flow exhibits chaotic characteristics, which can be simulated using the Logistic chaotic mapping. Therefore, [the following option is chosen]. As a Logistic chaotic mapping, when , hour, This represents the evolutionary pattern of node capacity constraints.

[0032] S32, Rules for the Redistribution of Goods In a cascading failure scenario, when a station cannot provide sufficient transport services, the carrier will choose alternative transport routes, leading to a redistribution of goods. The node capacity is positively correlated with the initial load of the initial node, as shown below: S321. Cargo redistribution rules based on adjacent node loads The load on a failed node is distributed proportionally based on the remaining available capacity of its neighboring nodes (i.e., the difference between the current capacity and the available capacity). Specifically, nodes with larger remaining available capacity will take on more load from the failed node due to their higher carrying potential. Calculate all failed nodes at time t Assigned to its neighboring nodes Load increment: (2) Goods redistribution rules based on the distance between adjacent nodes Neighboring nodes closer to the failed node will receive more of the redistributed load. Therefore, at time t, all failed nodes Assigned to its neighboring nodes Load increment Use the following formula to determine: As the cascading failure process progresses, nodes exist The load at any given time is: S4. By combining various networks and evaluation models, cascading failure scenarios are simulated through simulation experiments, and the network is evaluated using an evaluation system. The impact of cascading failure is quantified, and the changing patterns of network efficiency with attack strategies (random attacks, intentional attacks) are revealed, providing a new tool for dynamic vulnerability analysis.

[0033] This embodiment takes the China-Europe Railway Express as an example for illustration. It is an important part of the international railway freight network, spanning 26 European countries. The transportation environment is complex and the transportation process is complex (requiring at least multiple track changes). This invention constructs a basic line network topology model with 280 nodes and 439 edges and a transportation service network topology model with 280 nodes and 1181 edges, and calculates the top 20 nodes of the China-Europe Railway Express freight network.

[0034] Table 1. Top 20 Cities / Ports by Importance of China-Europe Railway Express Freight Network S41. Vulnerability Simulation Based on Coupling Coefficient This paper analyzes the changes in network vulnerability under two attack strategies: random attack and intentional attack, as the node coupling coefficient changes ([0.1-0.7], with a step size of 0.1). The specific relationship between the node coupling coefficient and network vulnerability is as follows: Figure 1 , Figure 2 As shown in the figure. The results indicate that under the random attack strategy, the decline trend of each vulnerability index is relatively slow, and the change in the node coupling coefficient has little impact on the network vulnerability. However, when the proportion of network failure nodes exceeds 20%-30%, the network vulnerability becomes more obvious.

[0035] Under the deliberate attack strategy, different attack targets are set, targeting nodes with high degree, high betweenness, high freight volume, and high overall importance evaluation value. Figure 2 The changes in network performance under the attack node's cargo volume are presented.

[0036] The results show that under the deliberate attack strategy, the network exhibits minimal vulnerability when the node coupling coefficient is 0.4. Therefore, a coupling coefficient of 0.4 is chosen for further analysis, such as... Figure 3 As shown, the results indicate that under the deliberate attack strategy, all vulnerability indices show a significant downward trend. The network stability is strongest when nodes with high degree are attacked, while the network is most vulnerable when nodes with high overall importance are attacked. This demonstrates that nodes with high overall importance play a crucial role in the network. If these nodes are prioritized for attack during a deliberate attack, the overall network efficiency will be significantly impacted. Therefore, when optimizing network strategies, the protection and optimization of nodes with high overall importance should be considered first.

[0037] The network stability is optimal when the node coupling coefficient is 0.4. Therefore, we simulate the changes in network vulnerabilities under different cargo redistribution rules with a node coupling coefficient of 0.4.

[0038] S42. Vulnerability Simulation Based on Goods Redistribution Rules The network exhibits the best stability when the node coupling coefficient is 0.4. Therefore, with the node coupling coefficient remaining constant, this study investigates the relationship between cargo redistribution rules and network vulnerability under different attack strategies, including random and intentional attacks.

[0039] S421, Point Attack like Figure 4 As shown, under the goods redistribution rule based on node distance, the network vulnerability index shows a relatively stable trend, while under the goods redistribution rule based on node load, the vulnerability index shows a more significant downward trend. Therefore, to reduce network vulnerability, in the event of cascading failures, choosing the goods redistribution rule based on node distance is more conducive to maintaining network stability and delaying the increase in network vulnerability.

[0040] This study investigates the changes in network vulnerability under different cargo allocation rules, focusing on four attack targets: high node degree, high node betweenness, high node cargo volume, and high overall node importance evaluation value. Figure 5 As shown, the results indicate that under the deliberate attack strategy, the vulnerability indicators decrease relatively quickly. For example, when attacking nodes with high node degree, all vulnerability indicators decrease by about 20% when the proportion of failed nodes is 20%. Furthermore, the network is most vulnerable when attacking nodes with high node degree and high overall node importance evaluation value. This demonstrates that nodes with high node degree and high overall node importance evaluation value play a crucial role in the network.

[0041] S422, Side Attack Under random attacks, the changing trends of vulnerability index values ​​are simulated under different cargo allocation rules, such as... Figure 6 As shown, the results indicate that the vulnerability index values ​​change in a similar trend under the two cargo allocation rules. However, when the cargo redistribution scheme based on node load is selected, the network exhibits less vulnerability. Taking the maximum connectivity graph as an example, when the failure rate reaches 30%, the index value of the allocation scheme based on node load decreases by about 20%, while the index value of the other scheme decreases by nearly 40%.

[0042] This study investigates network vulnerability changes under different cargo allocation rules by launching edge attacks starting from targets with high edge betweenness. Figure 7As shown, the results indicate that the network is more stable under the load-based cargo redistribution rule when attacking edges. When the edge failure rate is 20%, the changes in various network vulnerability indicators are around 20%, while when the edge failure rate is 50%, the changes in vulnerability indicators reach around 60%. Continuing to attack the network further leads to a sharp decline in stability, highlighting the network's vulnerability.

[0043] In summary, under cascading failure scenarios, the China-Europe freight network is significantly more vulnerable to deliberate attacks compared to random attacks. Network congestion spreads rapidly; after only 3-4 runs, the failure rate reaches approximately 30%, network performance degrades by over 50%, and the network becomes virtually unusable. The network's vulnerability to cascading failures is extremely high. In cascading failure simulations, if the attack target is a node, the China-Europe freight network exhibits the lowest vulnerability when the node coupling coefficient is 0.4. Furthermore, the freight redistribution scheme based on node distance demonstrates the strongest stability for the China-Europe freight network. However, when attacking edges, the freight redistribution scheme based on node load is more effective in delaying overall network failure.

[0044] The above are merely preferred embodiments of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the embodiments of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for assessing the resilience of a two-tier international railway freight network based on cascading failures, characterized in that, Includes the following steps: S1. Based on the actual operating characteristics of the international railway freight network, topological models of the basic line network and the transportation service network are constructed respectively, forming a two-layer coupled network. The basic line network is a physical network composed of railway stations, transshipment stations, and border inspection stations as nodes, and the tracks connecting these stations as edges. The transportation service network is a service network composed of railway stations as nodes and the operating lines connecting these stations as edges. S2. Construct a network evaluation system and an evaluation system for the importance of each node in a two-layer coupled network; S3. Construct a network vulnerability assessment model based on cascading failures; the network vulnerability assessment model adopts an improved CML node state model and a cargo redistribution strategy; the cargo redistribution strategy is used to redistribute cargo when cascading failures occur and stations cannot provide sufficient transportation services; The improved node state model in CML is as follows: In the formula, , , These are the coupling coefficients for node degree, edge betweenness, and cargo flow, respectively. , They are nodes and nodes exist The status at any given moment reflects whether a node is capable of providing transportation services at that moment. and For nodes and nodes exist The local mapping function at time t reflects the evolution of the node's own state; The node state; For indicator functions, when node With nodes If there is an edge between them ,otherwise ; For nodes i The degree of the node; For nodes i betweenness; The edge betweenness; The number of nodes in a single network; For nodes i Freight volume; node and nodes The flow of goods between them; External interference quantity; For nodes At any moment The degree value; S4. Combining the two-layer coupled network and the evaluation model, the cascading failure situation is simulated through simulation experiments, and the network is evaluated in conjunction with the evaluation system.

2. The resilience assessment method for a two-tier international railway freight network based on cascading failure as described in claim 1, characterized in that, In step S3, the cargo redistribution strategy includes cargo redistribution rules based on the load of adjacent nodes and cargo redistribution rules based on the distance between adjacent nodes; The cargo redistribution rules based on the load of adjacent nodes are as follows: The redistribution rules for goods based on the distance between adjacent nodes are as follows: In the formula, for Time Node The remaining load; For nodes The capacity; For nodes exist The load at any moment; for Time Node The load; for Time Node The remaining load; O is the failed node. i The set of neighboring nodes; for t Time Node j The set of failed nodes among the neighboring nodes; for t Failure Node at Any Time i Assigned to nodes j The load; for t At any given moment, all failed nodes send messages to the node. j The allocated load, i.e., the number of nodes. j Total load increment; For nodes i With nodes j The actual distance; For nodes i With nodes m The actual distance.

3. The resilience assessment method for a two-tier international railway freight network based on cascading failure as described in claim 2, characterized in that, In step S2, the network evaluation system uses the maximum connectivity and network efficiency as indicators for the basic line network; and transportation service networks use transportation performance and the rate of affected freight flow as indicators. The formula for calculating transportation performance is as follows: The formula for calculating the affected cargo flow rate is: In the formula, For network transport performance; For the flow of goods in the maximum connected graph; For transportation service network G The original flow of goods; For the initial network node i Freight volume; For the affected cargo flow rate; , They are nodes i, j Freight volume; This is the set of failed nodes; This is the initial set of network nodes.

4. The resilience assessment method for a two-tier international railway freight network based on cascading failure as described in claim 2, characterized in that, As a specific embodiment of the present invention, constructing the importance evaluation system for each node includes the following steps: (1) Construct a node importance evaluation index system to evaluate the basic line network and transportation service network. Its evaluation index includes node degree, betweenness, degree centrality, proximity degree centrality, betweenness centrality, eigenvector centrality and node efficiency. (2) In the basic line network and transportation service network, the weight of each indicator is determined by the Critic method, including constructing the original decision matrix, then performing positiveization and standardization on the elements in the original decision matrix in sequence, then calculating the variability and conflict of indicators, and finally calculating the final weight of each indicator. The formulas for calculating the weights of each indicator include: In the formula, As an indicator j The final weight; and They are nodes i Indicators j and m Data; , Indicators j , m The data mean, that is, the average value of this technical indicator across all nodes; , , , , , All are intermediate variables; M The number of indicators; and The elements in the decision matrix after positive transformation , The normalized value; As an indicator j The variability of the indicators; For Standard j ,index m Conflicting indicators; (3) In the basic line network and transportation service network, based on the weight matrix The importance evaluation value of each node was obtained using the TOPSIS method. S i ; (4) In the basic line network and transportation service network, the grey relational degree between each node and the reference object is determined based on the grey relational analysis method. G i ; (5) Determine the comprehensive importance evaluation value of each node; In the formula, M i For nodes i The overall importance evaluation value; S i For nodes i The importance evaluation value; Z is an intermediate variable.

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