Subway network toughness evaluation method and device

Abstract

The invention discloses a subway network toughness evaluation method and device, and relates to the technical field of traffic toughness evaluation. The subway network toughness evaluation method comprises the following steps: constructing a directed weighted network model of a subway network according to subway stations and running routes; determining a metro network variable index according to the directed weighted network model, and performing node initial load distribution of the directed weighted network model according to the metro network variable index; iteratively attacking the directed weighted network model to which the initial load is distributed, redistributing the node load after each attack, and calculating the subway network efficiency after each attack according to a redistribution result; and evaluating the subway network toughness according to the subway network efficiency after each attack. According to the method, the key nodes in the network can be identified more accurately, the network toughness evaluation result is more comprehensive and reliable, and the operation efficiency and the management level of the subway network are improved.

Description

Technical Field

[0001] This invention relates to a method and apparatus for assessing the resilience of subway networks, belonging to the field of transportation resilience assessment technology. Background Technology

[0002] Subways are a backbone of urban integrated transportation systems, playing an irreplaceable role in enhancing urban resilience. With rapid urban development, subway construction has received increasing attention, and the safety of subway network operation has also garnered widespread concern. How to address risks and external interventions has become an urgent issue to be resolved.

[0003] In extreme situations such as war or emergencies, the resilience of subway systems is directly related to a city's emergency response capabilities and the safety of its residents. Therefore, in-depth research on the resilience of subway networks is not only of theoretical significance but also has important practical application value.

[0004] Currently, assessment techniques for the resilience of subway networks are mostly limited to the analysis of static topology, such as graph-based metrics like degree and betweenness. While these methods can reflect the inherent structural properties of the network, their main drawbacks are: they cannot effectively simulate the dynamic distribution and transfer of passenger flow in real-world scenarios, they struggle to quantify the nonlinear failure mechanisms following node overload, and they fail to capture the cascading effects of failure events propagating within the network. Therefore, traditional assessment results differ significantly from the actual response behavior of subway systems under extreme conditions, making it difficult to provide accurate quantitative basis for developing resilience improvement strategies. Summary of the Invention

[0005] The purpose of this invention is to provide a method and apparatus for assessing the resilience of a subway network. By establishing a reasonable subway network model and applying the theory of complex network cascading failure, the operation of the subway network is simulated under two different attack modes: random attack and deliberate attack. The resilience of the subway network is assessed, and the inherent vulnerabilities in the network are identified and prioritized for reinforcement, thereby achieving "resilience-oriented" planning and design optimization.

[0006] To achieve the above objectives, the present invention is implemented using the following technical solution.

[0007] On one hand, the present invention provides a method for assessing the resilience of a subway network, comprising:

[0008] Construct a directed weighted network model of the subway network based on subway stations and operating routes;

[0009] Based on the directed weighted network model, the variable indicators of the subway network are determined, and the initial load allocation of the nodes in the directed weighted network model is carried out based on the variable indicators of the subway network.

[0010] An iterative attack is performed on a directed weighted network model that initially distributes the load. The node load is redistributed after each attack, and the subway network efficiency after each attack is calculated based on the redistribution results.

[0011] Assess the resilience of the metro network based on its efficiency after each attack.

[0012] Optionally, in the directed weighted network model, subway stations are used as nodes, running routes are used as edges, and edges are added between nodes corresponding to transfer stations.

[0013] The weight of an edge is a composite function of passenger flow and traffic capacity between nodes. The composite function value increases with increasing passenger flow and decreases with increasing traffic capacity.

[0014] Optional variables for the metro network include degree centrality, betweenness centrality, proximity centrality, eigenvector centrality, transfer station rating, geographic centrality, neighbor importance, and line intersection degree.

[0015] Degree centrality The calculation formula is:

[0016] ;

[0017] In the formula, Let i be the degree of node i; The total number of nodes;

[0018] Betweenness centrality The calculation formula is:

[0019] ;

[0020] In the formula, From node s to node The total number of shortest paths; The number of shortest paths passing through node i;

[0021] The proximity centrality The calculation formula is:

[0022] ;

[0023] In the formula, For nodes To the node The shortest path length;

[0024] The formula for calculating the centrality of the eigenvector is:

[0025] ;

[0026] In the formula, It is a directed weighted network adjacency matrix; for The largest eigenvalue; The eigenvector corresponding to the largest eigenvalue, its components That is, a node eigenvector centrality;

[0027] The approximate values ​​of the eigenvectors are obtained using the power iteration method:

[0028] ;

[0029] In the formula, For the first Approximate value of the eigenvector at the nth iteration For the first The approximate eigenvector value at the nth iteration, and the eigenvector corresponding to the largest eigenvalue obtained after the iteration converges. It is the Euclidean norm;

[0030] The geographic centrality Calculated based on the exponential decay function, the calculation formula is as follows:

[0031] ;

[0032] In the formula, Center point The weights; For nodes To the center point Geographical distance; The attenuation coefficient; This refers to the number of center points, where the center points are the geographical coordinates of the main passenger distribution centers determined according to the city's overall plan.

[0033] Among them, geographical distance The calculation is based on a model of the Earth's spherical surface, and the formula is as follows:

[0034] ;

[0035] ;

[0036] ;

[0037] In the formula, and They are nodes and center point latitude, For nodes and center point Latitude difference; For nodes and center point Difference in longitude; The radius of the Earth; As an intermediate variable; For nodes and center point The central angle of the sphere between them;

[0038] The importance of the neighbors is obtained by calculating the degree centrality of all the neighboring nodes of the node;

[0039] The line intersection degree is the number of subway lines passing through the node.

[0040] The transfer station rating Scoring based on degree Known transfer station ratings Line intersection score Clustering score Neighbor importance rating get:

[0041] ;

[0042] The degree score is calculated based on the degree of the transfer station:

[0043] ;

[0044] The known transfer station score is obtained based on predefined transfer station information in the subway network;

[0045] The line intersection score is the number of subway lines passing through the transfer station.

[0046] The neighbor importance score is obtained by calculating the degree centrality of all neighbor nodes of the transfer station;

[0047] The formula for calculating the cluster score is as follows:

[0048] ;

[0049] In the formula, For nodes The number of edges between neighboring nodes.

[0050] Optionally, the initial load allocation of nodes in the directed weighted network model is performed based on the metro network variable indicators, including:

[0051] Calculate node importance based on subway network variable indicators;

[0052] Calculate the initial load of the node based on its importance and initial load factor;

[0053] Calculate the load difference coefficient based on the node's maximum initial load and minimum initial load;

[0054] If the load difference coefficient is greater than the threshold, the initial load corresponding to the first X nodes is amplified, and the amplified initial load is used as the final initial load of the first X nodes; where the first X nodes are the X nodes with the highest node importance ranking.

[0055] Optionally, the formula for calculating the importance of a node is:

[0056] ;

[0057] In the formula, For nodes The importance of; For the first Weights of individual variable indicators; For the normalized first Individual variable indicators; The total number of variable indicators;

[0058] The initial load of a node is obtained by multiplying the node's importance by an initial load coefficient;

[0059] The load difference coefficient The calculation formula is:

[0060] ;

[0061] In the formula, The maximum initial load of the node; This represents the minimum initial load for the node.

[0062] Optionally, an iterative attack is performed on the directed weighted network model that initially allocates the load. After each attack, the node load is redistributed, and the subway network efficiency after each attack is calculated based on the redistribution results, including:

[0063] Attack the directed weighted network model that initially distributes the load. In each iteration, attack the nodes in the directed weighted network to make them fail, redistribute the load of the failed nodes, and calculate the node load and node capacity after redistribution.

[0064] Check if the node load is overloaded after redistribution, redistribute the load of overloaded nodes until all nodes are under normal load, and obtain the redistribution result.

[0065] The efficiency of the metro network after each attack is calculated based on the redistribution results.

[0066] Optionally, the process of redistributing the load of the failed node is as follows: the load of the failed node is proportionally distributed to neighboring nodes, and the load redistribution calculation formula is as follows:

[0067] ;

[0068] In the formula, Neighboring nodes Increased load; The load redistribution ratio; Failed node The load; Failed node The number of active neighbors;

[0069] The formula for determining whether the node load is overloaded after redistribution is as follows:

[0070] ;

[0071] In the formula, For nodes The load; For nodes The capacity;

[0072] ;

[0073] In the formula, For nodes The initial load; This represents the node capacity coefficient.

[0074] When a node fails due to overload, the load redistribution process continues for the overloaded node until all nodes are under normal load, and the redistribution result is obtained.

[0075] Optionally, the formula for calculating the efficiency of the subway network is as follows;

[0076] ;

[0077] In the formula, To improve network efficiency at step t; and For time-varying weights; The number of edges in the directed weighted network at step t is used to attack. The number of active network nodes at step t of the attack; The average path length of the directed weighted network at step t is used to attack. This is the path efficiency adjustment coefficient;

[0078] The time-varying weight adjustment formula is as follows:

[0079] ;

[0080] ;

[0081] ;

[0082] In the formula, Threshold for critical attack steps; This represents the total number of attack steps.

[0083] Optionally, the process for assessing the resilience of the metro network is as follows:

[0084] The robustness index measures the dynamic efficiency resilience of the metro network. This robustness index is obtained through parameter sensitivity analysis, the process of which is as follows:

[0085] ;

[0086] In the formula, This represents the node capacity coefficient. for discrete value points, for The number of discrete value points; For the load redistribution ratio, for discrete value points, for The number of discrete value points;

[0087] The robustness response matrix is ​​represented as:

[0088] ;

[0089] In the formula, Node capacity coefficient The first in Each value point, For load redistribution ratio The first in Each value point; ; ; For robustness evaluation function; elements of robustness response matrix For parameter combination The robustness index is calculated below;

[0090] By robustness index Structural vulnerability penalty factor and parameter optimization gain factor Multiplication yields a subway network resilience score. :

[0091] ;

[0092] ;

[0093] ;

[0094] ;

[0095] ;

[0096] In the formula, and These are the variances of degree centrality and betweenness centrality, respectively. The Gini coefficient represents the importance of a node; This represents the maximum number of failed cascade nodes in the network. The total number of nodes; This represents the critical attack ratio. This represents the number of attack steps required when network efficiency drops to 50%. and These are the robustness indices for the optimal and worst parameter combinations, respectively.

[0097] In a second aspect, the present invention provides a subway network resilience assessment device, comprising:

[0098] The network construction module is used to: construct a directed weighted network model of the subway network based on subway stations and operating routes;

[0099] The resilience assessment module is used to: determine the metro network variable indicators based on the directed weighted network model, and perform initial load allocation of nodes in the directed weighted network model based on the metro network variable indicators.

[0100] An iterative attack is performed on a directed weighted network model that initially distributes the load. The node load is redistributed after each attack, and the subway network efficiency after each attack is calculated based on the redistribution results.

[0101] Assess the resilience of the metro network based on its efficiency after each attack.

[0102] Compared with the prior art, the beneficial effects achieved by the present invention are as follows:

[0103] 1. This invention establishes a reasonable subway network model and applies the complex network cascading failure theory to simulate the operation of the subway network under two different attack modes: random attack and deliberate attack. It conducts a resilience assessment of the subway network, identifies and prioritizes the reinforcement of inherent vulnerabilities in the network, and realizes "resilience-oriented" planning and design optimization. The constructed resilience scoring formula makes the assessment results more comprehensive and reliable.

[0104] 2. The multi-dimensional node importance evaluation algorithm of this invention integrates degree centrality, betweenness centrality, proximity centrality, eigenvector centrality, transfer station score, geographic centrality, neighbor importance, and line intersection degree to calculate node importance. This evaluation method, which integrates multiple centralities and introduces geographic factors and functional enhancements, can more accurately identify key nodes in the network.

[0105] 3. This invention defines specific attack strategies (deliberate attacks and random attacks) and cascading failure triggering and propagation rules. The network efficiency assessment comprehensively considers connectivity efficiency and path efficiency, and can more comprehensively reflect the degradation of network performance. Attached Figure Description

[0106] Figure 1 This is a flowchart of the subway network resilience assessment method of the present invention;

[0107] Figure 2 This is a schematic diagram showing the geographical distribution of Beijing subway stations according to the present invention;

[0108] Figure 3 This is a flowchart of the evaluation process for key nodes in this invention;

[0109] Figure 4 This is a diagram showing the ranking of the importance of key nodes in this invention;

[0110] Figure 5 This is a schematic diagram comparing the efficiency of the dynamically adjusted weight network during the attack phase of this invention.

[0111] Figure 6 This is a schematic diagram comparing network efficiency before and after different weight adjustment schemes during the attack phase of this invention.

[0112] Figure 7 This diagram illustrates the comparison of network efficiency under different attack methods according to the present invention. Detailed Implementation

[0113] The technical solution of the present invention will be described in detail below with reference to the accompanying drawings and specific embodiments. It should be understood that the embodiments of the present invention and the specific features in the embodiments are detailed descriptions of the technical solution of the present invention, rather than limitations thereof. In the absence of conflict, the embodiments of the present invention and the technical features in the embodiments can be combined with each other.

[0114] Example 1:

[0115] This embodiment introduces a method for assessing the resilience of a subway network, such as... Figure 1 As shown, it includes:

[0116] Construct a directed weighted network model of the subway network based on subway stations and operating routes;

[0117] Based on the directed weighted network model, the variable indicators of the subway network are determined, the importance of nodes is calculated based on the variable indicators, and the initial load allocation of nodes in the directed weighted network model is carried out; and the load difference of each node's initial load is evaluated.

[0118] An iterative attack is performed on the directed weighted network model after the initial load difference assessment and adjustment. The node load is redistributed after each attack, and the subway network efficiency after each attack is calculated based on the redistribution results.

[0119] Robustness indices are calculated based on node capacity coefficients and node redistribution ratios. The resilience of the metro network is then assessed based on the metro network efficiency and robustness indices after each attack.

[0120] By setting robustness metrics for different node capacity coefficients and node redistribution ratios, the optimal parameters are found, and a comprehensive report and optimization suggestions are output.

[0121] I. Network Modeling

[0122] like Figure 2 As shown, a directed weighted network model is constructed based on the actual operation data of Beijing Metro. In the directed weighted network model, metro stations are used as nodes, the running routes are used as edges, and edges are added between the nodes corresponding to transfer stations.

[0123] The weight of an edge is a composite function of passenger flow and traffic capacity between nodes. The composite function value increases with increasing passenger flow and decreases with increasing traffic capacity.

[0124] Network Modeling: Nodes: Beijing subway stations (partial subway lines), including Lines 10, 11, 12, 13A, 13B, 13, 14, 15, 16, 17, 19, 1, 22, 28, 2, 3, 4, 5, 6, 7, 8, 9, Yizhuang Line, Daxing International Airport Line, Fangshan Line, Changping Line, Yanfang Line, Xijiao Line, and Capital Airport Line. Edges: Adjacent stations are connected according to line order, with additional connections between transfer stations (forming a triangular structure to improve the clustering coefficient).

[0125] II. Initial Load Allocation of Nodes in a Directed Weighted Network Model

[0126] The initial load allocation of nodes in the directed weighted network model is performed based on the subway network variable indicators, including:

[0127] Calculate node importance based on subway network variable indicators;

[0128] Calculate the initial load of the node based on its importance and initial load factor;

[0129] Calculate the load difference coefficient based on the node's maximum initial load and minimum initial load;

[0130] If the load difference coefficient is greater than the threshold, the initial load corresponding to the top 10% of nodes in the node importance ranking is amplified, and the amplified initial load is used as the final initial load of the top 10% of nodes in the node importance ranking.

[0131] The variables and indicators of the metro network include degree centrality, betweenness centrality, proximity centrality, eigenvector centrality, transfer station rating, geographic centrality, neighbor importance, and line intersection degree.

[0132] Degree centrality reflects the scale of direct connections, betweenness centrality reflects its mediating role in network flow allocation, eigenvector centrality measures the influence weight of its neighboring nodes, geographic centrality assesses its spatial location advantage, and transfer station score quantifies its functional premium as a multi-line convergence hub. Each dimension is standardized and weighted to form a comprehensive scoring system that accurately characterizes the global contribution and vulnerability level of a node.

[0133] Degree centrality The calculation formula is:

[0134] ;

[0135] In the formula, Let i be the degree of node i; The total number of nodes;

[0136] Betweenness centrality The calculation formula is:

[0137] ;

[0138] In the formula, From node s to node The total number of shortest paths; The number of shortest paths passing through node i;

[0139] The proximity centrality The calculation formula is:

[0140] ;

[0141] In the formula, For nodes To the node The shortest path length;

[0142] The formula for calculating the centrality of the eigenvector is:

[0143] ;

[0144] In the formula, It is a directed weighted network adjacency matrix; for The largest eigenvalue; The eigenvector corresponding to the largest eigenvalue, its components That is, a node eigenvector centrality;

[0145] The approximate values ​​of the eigenvectors are obtained using the power iteration method:

[0146] ;

[0147] In the formula, For the first Approximate value of the eigenvector at the nth iteration For the first The approximate eigenvector value at the nth iteration, and the eigenvector corresponding to the largest eigenvalue obtained after the iteration converges. It is the Euclidean norm;

[0148] The geographic centrality Calculated based on the exponential decay function, the calculation formula is as follows:

[0149] ;

[0150] In the formula, Center point The weights; For nodes To the center point Geographical distance; , is the attenuation coefficient; This refers to the number of center points, where the center points are the geographical coordinates of the main passenger distribution centers determined according to the city's overall plan.

[0151] Among them, geographical distance The calculation is based on a model of the Earth's spherical surface, and the formula is as follows:

[0152] ;

[0153] ;

[0154] ;

[0155] In the formula, and They are nodes and center point latitude, For nodes and center point Latitude difference; For nodes and center point Difference in longitude; The radius of the Earth; As an intermediate variable; For nodes and center point The central angle of the sphere between them;

[0156] The importance of the neighbors is obtained by calculating the degree centrality of all the neighboring nodes of the node;

[0157] The line intersection degree is the number of subway lines passing through the node.

[0158] The transfer station rating Scoring based on degree Known transfer station ratings Line intersection score Clustering score Neighbor importance rating get:

[0159] ;

[0160] The degree score is calculated based on the degree of the transfer station:

[0161] ;

[0162] The known transfer station score is obtained based on predefined transfer station information in the subway network;

[0163] The line intersection score is the number of subway lines passing through the transfer station.

[0164] The neighbor importance score is obtained by calculating the degree centrality of all neighbor nodes of the transfer station;

[0165] The formula for calculating the cluster score is as follows:

[0166] ;

[0167] In the formula, For nodes The number of edges between neighboring nodes.

[0168] Identifying and protecting critical nodes plays a decisive role in improving the resilience of the entire network: 20% of critical nodes carry 60% of the network functions. Protecting the top 5 critical nodes can improve the resilience of the entire network by 30-40%. Collaborative protection among critical nodes is more effective than strengthening a single node.

[0169] Based on the variables of the subway network, the importance of nodes is calculated. The node importance index comprehensively considers the coreness of a node in the topology and passenger flow distribution.

[0170] like Figure 3 As shown, firstly, weights are assigned to each variable indicator, and the values ​​of each indicator are calculated and normalized. Then, the normalized indicator values ​​are summed according to their weights to obtain the initial importance of each node. Based on this, the importance of nodes identified as traditional hubs is increased. Finally, the importance of all nodes is re-normalized and sorted, and the key nodes are output.

[0171] The calculation formula is:

[0172] ;

[0173] In the formula, For nodes The importance of; For the first The weights of each variable indicator, The total number of variable indicators; the weight allocation ratio of each variable indicator. ; For the normalized first Individual variable indicators;

[0174] like Figure 4 As shown, this calculation process identified 20 super-critical nodes, including Xidan, Dongzhimen, Jianguomen, Fuxingmen, and Dongdan. Dongzhimen Station ranked first in comprehensive score, with its betweenness centrality and transfer bonus significantly higher than other stations, becoming a key bottleneck in network operation. Xizhimen and Jianguomen followed closely behind. The Beijing Metro network exhibits a significant core-periphery hierarchical characteristic in both its topology and passenger flow distribution: the central urban area forms the core layer of the network with high-density, highly connected transfer hubs, while the outer areas form peripheral corridors with radial lines, exhibiting typical characteristics of a scale-free network. A few key transfer nodes and line intersections with high betweenness centrality and high centrality play a decisive role in overall connectivity efficiency, passenger flow distribution capacity, and disturbance resistance; their failure can easily lead to a sharp drop in network efficiency, connectivity disruptions, or even cascading failures. Therefore, key nodes should be prioritized for cybersecurity protection, risk management, and resilience enhancement. By strengthening structural redundancy, improving emergency preparedness, and optimizing passenger flow control, the overall network's resilience and rapid recovery capabilities under extreme events and daily disturbances can be significantly improved.

[0175] The initial load of a node is obtained by multiplying the node's importance by an initial load coefficient;

[0176] The node importance is multiplied by 500, and then normalized and multiplied by 1000 to obtain the base load. The load redistribution ratio is 0.6, meaning that 60% of the load of a failed node will be redistributed to neighboring nodes. In addition, in the cascading failure simulation, the maximum number of attack steps is 60% of the total number of nodes (but not exceeding 15 steps), and an early termination condition is set (network efficiency is lower than 0.05 or the proportion of failed nodes exceeds 95%).

[0177] After the initial load distribution at each node, the uniformity of the load distribution is checked by calculating the load difference coefficient. The calculation formula is:

[0178] ;

[0179] In the formula, The maximum initial load of the node; This represents the minimum initial load for the node.

[0180] When the load difference coefficient exceeds the threshold (set to 8), adjust the load of each node.

[0181] III. Network Attack and Resilience Assessment

[0182] An iterative attack is performed on the directed weighted network model that initially allocates the load. After each attack, the node load is redistributed, and the subway network efficiency after each attack is calculated based on the redistribution results, including:

[0183] Random and deliberate attacks are performed on the directed weighted network model that initially distributes the load. In each iteration, the nodes in the directed weighted network are attacked and rendered ineffective. The load of the ineffective nodes is redistributed, and the node load and node capacity after redistribution are calculated.

[0184] Check if the node load is overloaded after redistribution, redistribute the load of overloaded nodes until all nodes are under normal load, and obtain the redistribution result.

[0185] Calculate the metro network efficiency after each attack based on the redistribution results;

[0186] In both random and deliberate attack scenarios, random attacks randomly remove nodes from the directed weighted network; deliberate attacks prioritize removing the top 10% of nodes in the directed weighted network based on their intermediateness centrality, in order to reveal the dominant role of critical hubs in system crashes.

[0187] The process of redistributing the load of a failed node is as follows: the load of the failed node is proportionally distributed to its neighboring nodes. The load redistribution calculation formula is as follows:

[0188] ;

[0189] In the formula, Neighboring nodes Increased load; The load redistribution ratio; Failed node The load; Failed node The number of active neighbors;

[0190] The formula for determining whether the node load is overloaded after redistribution is as follows:

[0191] ;

[0192] In the formula, For nodes The load; For nodes The capacity;

[0193] ;

[0194] In the formula, For nodes The initial load; This represents the node capacity coefficient.

[0195] When a node fails due to overload, the load redistribution process continues for the overloaded node until all nodes are under normal load, and the redistribution result is obtained.

[0196] The formula for evaluating the efficiency of a subway network is as follows:

[0197] ;

[0198] In the formula, To improve network efficiency at step t; and For time-varying weights; The number of edges in the directed weighted network at step t is used to attack. The number of active network nodes at step t of the attack; The average path length of the directed weighted network at step t is used to attack. , which is the path efficiency adjustment coefficient.

[0199] The time-varying weight adjustment formula is:

[0200] ;

[0201] ;

[0202] ;

[0203] In the formula, and For time-varying weights; This represents the number of attack steps. Threshold for critical attack steps; This represents the total number of attack steps.

[0204] like Figure 5 As shown, with the increase in the number of attack steps, the weights of network efficiency and path efficiency gradually decrease, while the weight of connectivity increases accordingly. In the later stages of the attack, the network connectivity plays a crucial role in the overall operation.

[0205] like Figure 6 As shown, when the number of attack steps t reaches 50% of the total number of attack steps T, the weights of network efficiency and path efficiency are... The weight of connectivity decreases from 0.6 to 0.35. The weighting then increases from 0.4 to 0.7. At this point, some critical nodes in the network may have failed, and broken paths have weakened the connectivity between local areas and the overall network. If network efficiency and path efficiency are still the primary evaluation indicators, the actual operational risks of the network will not be accurately reflected. By increasing the weighting of connectivity, the focus can be placed on whether the network still maintains overall or regional connectivity, avoiding the core issue of connectivity loss being masked by a decline in local efficiency. This time-varying weighting mechanism allows resilience assessment results to dynamically match the network characteristics at different stages of an attack, improving the accuracy and practicality of the assessment.

[0206] like Figure 7 As shown, the network efficiency of random attacks, deliberate attacks, degree-centrality attacks, and betweenness-centrality attacks is compared under different attack steps. Degree-centrality attacks refer to the strategy of attacking nodes sequentially according to their degree, while betweenness-centrality attacks refer to the strategy of attacking nodes sequentially according to their betweenness-centrality. Simulation results show that when the number of attack steps is approximately 10, the network efficiency decays non-linearly with the number of attack steps. In the case of deliberate attacks, the rate of network efficiency decline decreases rapidly after a certain number of steps, significantly faster than in random attacks. This means that the failure of high betweenness-centrality nodes can quickly trigger cascading failures, causing a sharp deterioration in the overall network connectivity. In contrast, under random attacks, the network resilience is relatively stable, and the rate of efficiency decay is relatively gradual, demonstrating fault tolerance for the failure of non-critical nodes. In attacks targeting betweenness-centrality, the rate of network efficiency decline is the fastest among the four attack methods. By comparing the inflection points of the resilience curves under the two attack modes, the system's vulnerability threshold can be identified, thus providing a quantitative basis for developing emergency plans and infrastructure hardening strategies.

[0207] Furthermore, considering the spatiotemporal distribution characteristics of passenger flow on the Beijing Metro, some transfer stations are approaching their capacity thresholds during peak hours, making them highly susceptible to widespread outages should a localized failure occur. Therefore, dynamic monitoring and flexible scheduling should be prioritized for hub stations with both high betweenness coefficients and high passenger flow to improve the overall network robustness.

[0208] The process of assessing the resilience of a subway network is as follows:

[0209] The robustness index measures the dynamic efficiency resilience of the metro network. This robustness index is obtained through parameter sensitivity analysis, the process of which is as follows:

[0210] ;

[0211] In the formula, This represents the node capacity coefficient. for discrete value points, for The number of discrete value points; For the load redistribution ratio, for discrete value points, for The number of discrete value points;

[0212] The robustness response matrix is ​​represented as:

[0213] ;

[0214] ;

[0215] ;

[0216] In the formula, Node capacity coefficient The first in Each value point, For load redistribution ratio The first in Each value point; ; ; For robustness evaluation function; elements of robustness response matrix For parameter combination The robustness index is calculated below;

[0217] By robustness index Structural vulnerability penalty factor and parameter optimization gain factor Multiplication yields a subway network resilience score. :

[0218] ;

[0219] ;

[0220] ;

[0221] ;

[0222] ;

[0223] In the formula, and These are the variances of degree centrality and betweenness centrality, respectively. The Gini coefficient represents the importance of a node; This represents the maximum number of failed cascade nodes in the network. The total number of nodes; This represents the critical attack ratio. This represents the number of attack steps required when network efficiency drops to 50%. and These are the robustness indices for the optimal and worst parameter combinations, respectively.

[0224] Based on the robustness matrix, the optimal parameter recommendations and parameter optimization suggestions are obtained:

[0225] The recommended optimal parameters are: capacity factor of 1.00; redistribution ratio of 0.10; and expected robustness index of 0.1838.

[0226] Parameter optimization suggestions are as follows:

[0227] 1. Implement key protection measures for critical nodes to improve capacity coefficient;

[0228] 2. Establish a rapid load redistribution mechanism;

[0229] 3. Conduct network stress tests and sensitivity analyses regularly;

[0230] 4. Establish a multi-layered redundant backup system;

[0231] Finally, the optimal parameter recommendations and parameter optimization suggestions are generated into a comprehensive visual report.

[0232] Example 2:

[0233] Based on the same inventive concept as Embodiment 1, this embodiment introduces a subway network resilience assessment device, comprising:

[0234] The network construction module is used to: construct a directed weighted network model of the subway network based on subway stations and operating routes;

[0235] The resilience assessment module is used to: determine the metro network variable indicators based on the directed weighted network model, and perform initial load allocation of nodes in the directed weighted network model based on the metro network variable indicators.

[0236] An iterative attack is performed on a directed weighted network model that initially distributes the load. The node load is redistributed after each attack, and the subway network efficiency after each attack is calculated based on the redistribution results.

[0237] Assess the resilience of the metro network based on its efficiency after each attack.

[0238] The specific functions of each module described above are explained in the relevant content of the method in Embodiment 1, and will not be repeated here.

[0239] Those skilled in the art will understand that embodiments of the present invention can be provided as methods, systems, or computer program products. Therefore, the present invention can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present invention can take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0240] This invention is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart illustrations and / or block diagrams. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0241] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0242] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0243] The embodiments of the present invention have been described above with reference to the accompanying drawings. However, the present invention is not limited to the specific embodiments described above. The specific embodiments described above are merely illustrative and not restrictive. Those skilled in the art can make many other forms under the guidance of the present invention without departing from the spirit and scope of the claims. All of these forms are within the protection scope of the present invention.

Claims

1. A method for assessing the resilience of a subway network, characterized in that, include: Construct a directed weighted network model of the subway network based on subway stations and operating routes; Based on the directed weighted network model, the variable indicators of the subway network are determined, and the initial load allocation of the nodes in the directed weighted network model is carried out based on the variable indicators of the subway network. An iterative attack is performed on a directed weighted network model that initially distributes the load. The node load is redistributed after each attack, and the subway network efficiency after each attack is calculated based on the redistribution results. Assess the resilience of the metro network based on its efficiency after each attack.

2. The subway network resilience assessment method according to claim 1, characterized in that, In the directed weighted network model, subway stations are used as nodes, running routes are used as edges, and edges are added between nodes corresponding to transfer stations. The weight of an edge is a composite function of passenger flow and traffic capacity between nodes. The composite function value increases with increasing passenger flow and decreases with increasing traffic capacity.

3. The subway network resilience assessment method according to claim 1, characterized in that, The variables and indicators of the metro network include degree centrality, betweenness centrality, proximity centrality, eigenvector centrality, transfer station rating, geographic centrality, neighbor importance, and line intersection degree. Degree centrality The calculation formula is: ； In the formula, Let i be the degree of node i; The total number of nodes; Betweenness centrality The calculation formula is: ； In the formula, From node s to node The total number of shortest paths; The number of shortest paths passing through node i; The proximity centrality The calculation formula is: ； In the formula, For nodes To the node The shortest path length; The formula for calculating the centrality of the eigenvector is: ； In the formula, It is a directed weighted network adjacency matrix; for The largest eigenvalue; The eigenvector corresponding to the largest eigenvalue, its components That is, a node eigenvector centrality; The approximate values ​​of the eigenvectors are obtained using the power iteration method: ； In the formula, For the first Approximate value of the eigenvector at the nth iteration For the first The approximate eigenvector value at the nth iteration, and the eigenvector corresponding to the largest eigenvalue obtained after the iteration converges. It is the Euclidean norm; The geographic centrality Calculated based on the exponential decay function, the calculation formula is as follows: ； In the formula, Center point The weights; For nodes To the center point Geographical distance; The attenuation coefficient; This refers to the number of center points, where the center points are the geographical coordinates of the main passenger distribution centers determined according to the city's overall plan. Among them, geographical distance The calculation is based on a model of the Earth's spherical surface, and the formula is as follows: ； ； ； In the formula, and They are nodes and center point latitude, For nodes and center point Latitude difference; For nodes and center point Difference in longitude; The radius of the Earth; As an intermediate variable; For nodes and center point The central angle of the sphere between them; The importance of the neighbors is obtained by calculating the degree centrality of all the neighboring nodes of the node; The line intersection degree is the number of subway lines passing through the node. The transfer station rating Scoring based on degree Known transfer station ratings Line intersection score Clustering score Neighbor importance rating get: ； The degree score is calculated based on the degree of the transfer station: ； The known transfer station score is obtained based on predefined transfer station information in the subway network; The line intersection score is the number of subway lines passing through the transfer station. The neighbor importance score is obtained by calculating the degree centrality of all neighbor nodes of the transfer station; The formula for calculating the cluster score is as follows: ； In the formula, For nodes The number of edges between neighboring nodes.

4. The subway network resilience assessment method according to claim 3, characterized in that, The initial load allocation of nodes in the directed weighted network model is performed based on the subway network variable indicators, including: Calculate node importance based on subway network variable indicators; Calculate the initial load of the node based on its importance and initial load factor; Calculate the load difference coefficient based on the node's maximum initial load and minimum initial load; If the load difference coefficient is greater than the threshold, the initial load corresponding to the first X nodes is amplified, and the amplified initial load is used as the final initial load of the first X nodes; where the first X nodes are the X nodes with the highest node importance ranking.

5. The subway network resilience assessment method according to claim 4, characterized in that, The formula for calculating the importance of the node is: ； In the formula, For nodes The importance of; For the first Weights of individual variable indicators; For the normalized first Individual variable indicators; The total number of variable indicators; The initial load of a node is obtained by multiplying the node's importance by an initial load coefficient; The load difference coefficient The calculation formula is: ； In the formula, The maximum initial load of the node; This represents the minimum initial load for the node.

6. The subway network resilience assessment method according to claim 1, characterized in that, An iterative attack is performed on the directed weighted network model that initially allocates the load. After each attack, the node load is redistributed, and the subway network efficiency after each attack is calculated based on the redistribution results, including: Attack the directed weighted network model that initially distributes the load. In each iteration, attack the nodes in the directed weighted network to make them fail, redistribute the load of the failed nodes, and calculate the node load and node capacity after redistribution. Check if the node load is overloaded after redistribution, redistribute the load of overloaded nodes until all nodes are under normal load, and obtain the redistribution result. The efficiency of the metro network after each attack is calculated based on the redistribution results.

7. The subway network resilience assessment method according to claim 6, characterized in that, The process of redistributing the load of a failed node is as follows: the load of the failed node is proportionally distributed to neighboring nodes, and the load redistribution calculation formula is as follows: ； In the formula, Neighboring nodes Increased load; The load redistribution ratio; Failed node The load; Failed node The number of active neighbors; The formula for determining whether the node load is overloaded after redistribution is as follows: ； In the formula, For nodes The load; For nodes The capacity; ； In the formula, For nodes The initial load; This represents the node capacity coefficient. When a node fails due to overload, the load redistribution process continues for the overloaded node until all nodes are under normal load, and the redistribution result is obtained.

8. The subway network resilience assessment method according to claim 6, characterized in that, The formula for calculating the efficiency of the subway network is as follows: ； In the formula, To improve network efficiency at step t; and For time-varying weights; The number of edges in the directed weighted network at step t is used to attack. The number of active network nodes at step t of the attack; The average path length of the directed weighted network at step t is used to attack. This is the path efficiency adjustment coefficient; The time-varying weight adjustment formula is as follows: ； ； ； In the formula, Threshold for critical attack steps; This represents the total number of attack steps.

9. The subway network resilience assessment method according to claim 8, characterized in that, The process for assessing the resilience of the subway network is as follows: The robustness index measures the dynamic efficiency resilience of the metro network. This robustness index is obtained through parameter sensitivity analysis, the process of which is as follows: ； In the formula, This represents the node capacity coefficient. for discrete value points, for The number of discrete value points; For the load redistribution ratio, for discrete value points, for The number of discrete value points; The robustness response matrix is ​​represented as: ； In the formula, Node capacity coefficient The first in Each value point, For load redistribution ratio The first in Each value point; ; ; For robustness evaluation function; elements of robustness response matrix For parameter combination The robustness index is calculated below; By robustness index Structural vulnerability penalty factor and parameter optimization gain factor Multiplication yields a subway network resilience score. : ； ； ； ； ； In the formula, and These are the variances of degree centrality and betweenness centrality, respectively. The Gini coefficient represents the importance of a node; This represents the maximum number of failed cascade nodes in the network. The total number of nodes; This represents the critical attack ratio. This represents the number of attack steps required when network efficiency drops to 50%. and These are the robustness indices for the optimal and worst parameter combinations, respectively.

10. A subway network resilience assessment device, characterized in that, include: The network construction module is used to: construct a directed weighted network model of the subway network based on subway stations and operating routes; The resilience assessment module is used to: determine the metro network variable indicators based on the directed weighted network model, and perform initial load allocation of nodes in the directed weighted network model based on the metro network variable indicators. An iterative attack is performed on a directed weighted network model that initially distributes the load. The node load is redistributed after each attack, and the subway network efficiency after each attack is calculated based on the redistribution results. Assess the resilience of the metro network based on its efficiency after each attack.

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