## 2 Related Work
### Design of the Algorithm for Measuring Railway Network Resilience
Calculate the sum of the residual capacities of all the neighboring nodes of the node: \(R_{all}=\sum_{i\in N_{n}}R_{i}\);  
Step 3. For each neighboring node \(i\in N_{n}\), the load amount \(T_{i}\) is computed based on the proportion of its residual capacity. If \(R_{all}>0\), that is, at least one neighboring node has residual capacity, then  
\[T_{i}=(V_{n}-C_{n})*\frac{R_{i}}{R_{all}} \tag{11}\]  
Step 4. Update the load of each neighbor: \(V^{\prime}_{i}=V_{i}+T_{i}\);  
Step 5. Update the load of the overloaded node to its capacity: \(V_{n}=C_{n}\). This allocation approach is more equitable and efficient compared to other commonly employed load allocation methods such as average allocation, path weight allocation, and residual capacity allocation. This allocation mechanism ensures that neighboring nodes with greater residual capacity assume a larger portion of the additional load. (3) The Resilience-Based Measurement Algorithm of Service Network Flexibility  
Restorability refers to a network's ability to return to its normal operational state after experiencing a perturbation. This study posits that the restorative capacity of the railway service network is closely tied to the integrity of the underlying physical infrastructure. The resilience of the physical network is manifested in its ability to preserve structural and functional integrity when subjected to external shocks, thereby facilitating the prompt recovery of the service network. The restorative capacity of the service network reflects the physical network's ability to maintain and restore its functions after withstanding external impacts. When examining the restorative capacity of the service network, the network's restorative resilience is primarily influenced by the recovery strategies implemented, for instance, the priority arrangement for restoring damaged nodes or the sequence of restoring the edges connecting nodes. Additionally, factors such as the repair time and maintenance costs are significant considerations in evaluating restorative capacity. This section will embark from the perspective of network function restoration to analyze the recovery of network performance under different recovery strategies. Using quantitative analysis methods, this study will employ the global network efficiency of the service network as the primary metric for assessing restorative capacity. This approach facilitates the evaluation of network recovery performance based on topological characteristics and adopted strategies. Moreover, it provides valuable insights into the effects of different recovery strategies and offers theoretical guidance for developing effective network recovery plans. Meanwhile, the recovery strategies can be classified as random recovery strategies, degree-based recovery strategies, betweenness-based recovery strategies, etc. (1) Random Recovery Strategy. Once a failed node is removed from the network, a novel connectivity structure emerges. In such cases, determining the optimal nodes for recovery to restore the network can be challenging. A viable solution is to adopt a random recovery strategy, namely assuming that in the network, the probability of any newly added node rejoining the network is identical. This supposition guarantees that each node has an equal opportunity of being reconnected. The random recovery strategy does not take into account the specific attributes or locations of the nodes within the network, instead selecting nodes for recovery randomly with equal probability. The random recovery strategy simplifies the decision-making process in the recovery procedure, yet it might not offer the optimal solution for network operational efficiency or resilience restoration. (2) Degree/Betweenness Restoration Strategy. In a network, nodes with higher degrees or betweenness values typically play more crucial roles in the network structure. Therefore, prioritizing the restoration of such nodes is one of the most widely adopted strategies in real-world network recovery scenarios.