## 2 Related Work
### Design of the Algorithm for Measuring Railway Network Resilience  
In this study, we distinguish between two complementary network representations of the railway system. Figure 2: Resilience measurement system diagram. Physical Network: This represents the actual infrastructure of the railway system, where nodes are physical stations and edges are the physical rail connections between adjacent stations. This network captures the spatial topology and geographical constraints of the railway infrastructure, focusing on the physical connectivity between locations. The physical network is modeled using the Space L method, where stations are represented as nodes and direct rail connections between adjacent stations are represented as edges. Service Network: This represents the operational and functional aspects of the railway system, where nodes are still stations but edges represent direct service connections (trains running between stations without intermediate stops). Unlike the physical network, the service network captures how trains actually operate and serve passengers, reflecting scheduled services rather than mere physical connections. This network is modeled using the Space P method, accounting for service frequency, capacity, and operational patterns. The difference between the two networks in resilience analysis is shown in Figure 3.  
The Algorithm for Measuring the Accessibility of Physical Network Resilience  
The accessibility measure of the resilience of the physical network reflects the ability of the railway network to maintain interconnectivity among its components and ensure the continued operation of transportation functions under various disturbance scenarios. This measure encompasses factors such as the number of alternative paths that allow effective connections between stations despite disruptions. The accessibility measure is directly related to the ability of the railway network to consistently deliver essential transportation services during disturbances and serves as a critical indicator for assessing the resilience of the physical railway infrastructure. The corresponding characteristic operator is defined as follows:  
(1) Reachable Node Pairs  
Taking the reachable node pairs in the initial railway network model as a reference baseline, the status of the remaining node pairs after simulating disturbance events is observed and analyzed to evaluate changes in network accessibility. \[\text{K}=\frac{\text{k}^{\prime}}{\text{k}} \tag{2}\]  
where \(K\) denotes the accessibility metric based on reachable node pairs, k' represents the number of reachable node pairs after disturbance, and k represents the total number of reachable node pairs in the initial undisturbed network. (2) Clustering Coefficient  
When the nodes (stations) and edges (routes) in the railway network model are subjected to disturbances, previously connected nodes may become isolated, leading to  
Figure 3: Railway service vs. physical network in resilience analysis. a reduction in local node aggregation and consequently affecting the overall clustering coefficient of the network. \[G=\frac{1}{N}\sum_{i=1}^{N}G_{i}=\frac{1}{N}\sum_{i=1}^{N}\frac{2E_{i}}{k_{i}(k_ {i}-1)} \tag{3}\]  
where G represents the average clustering coefficient of the network, N is the total number of nodes, G_i is the clustering coefficient of node i, k_i denotes the degree of node i, and E_i indicates the actual number of edges among the nodes adjacent to node i. (3) Network Efficacy  
Network efficiency is a crucial metric for evaluating network connectivity and overall operational efficacy. It is defined by comparing the alterations in the sum of the reciprocals of the actual shortest path lengths between all node pairs before and after a disturbance. By analyzing the variation in network efficiency after the removal of a specific node, one can evaluate the impact of that node on network connectivity and operational performance.