## 2 Related Work
### Design of the Algorithm for Measuring Railway Network Resilience
\[L=\frac{B^{\prime}}{B} \tag{7}\]  
The cascading failure of a network is a common phenomenon in complex network systems, characterized by the propagation of faults following an initial disturbance. When one or more nodes (such as power stations in an electrical grid or stations in a railway network) or connections (i.e., links between nodes, such as power lines or railway tracks) experience perturbation events, the resulting failure can trigger a sequence of subsequent faults. This adverse impact can trigger a series of further faults, thereby influencing other parts of the network and ultimately possibly leading to the large-scale paralysis of the entire network. The occurrence of cascading failure begins with the failure of a node or connection in the network, which can be induced by external attacks, natural disasters, technical issues, or inadequate maintenance. Once a node or connection fails, the load it was bearing (for example, electricity, traffic, or information) is redistributed to other parts of the network. This redistribution might cause certain nodes or connections to bear a load exceeding their designed capacity. Due to the redistribution of the load, some nodes or connections that were originally operating normally might fail because of overload. These new failures further exacerbate the burden on the remaining components of the network, triggering a cycle of failure, load redistribution, and subsequent failure. This process can continue iteratively, resulting in a chain reaction that escalates network failures. In the worst-case scenario, this chain reaction may lead to the failure of the majority or all of the network, seriously affecting the normal functions and services of the network. The cascading failure process of the railway transportation network can be subdivided into two scenarios: node failure and edge failure. At a superficial level, this process represents a series of chain reactions initiated by the failure of a single or a few components. However, it reflects a deeper aspect of the complexity and interdependence inherent in the entire railway network system. This phenomenon is not confined to physical-level faults, such as damage to tracks and bridges, but also extends to technical-level issues, such as signal system malfunctions and power supply disruptions, and even includes operational management decisions and response strategies. Any component in the railway network does not exist in isolation. When an operational failure occurs at a station, or a certain section of the railway is temporarily closed due to maintenance, natural disasters, and other reasons, the direct impact might be the cancellation or delay of train services at that station or section. Although this primary impact is localized, due to the close connection of the railway network, its indirect impact quickly spreads. As the primary fault spreads, the railway management department is required to adjust the operational plan. These adjustments not only have an impact on the directly affected sections but might also affect the operational efficiency and carrying capacity of other parts of the network. For instance, the closure of a major line might force trains to take longer detours, which not only increases operational costs but might also cause overcrowding on the detour routes, thereby triggering new operational problems. The risk of cascading failure is particularly pronounced during peak operational periods when the railway network operates under high resource utilization and capacity constraints. During such times, the network's ability to buffer disruptions is limited, amplifying the likelihood and severity of cascading failures. Failures occurring during peak hours are not only more difficult to resolve promptly but also have more significant socio-economic consequences, given the high volume of passengers and goods in transit.