## 2 Related Work
### Design of the Algorithm for Measuring Railway Network Resilience
Drawing on the research methodologies and conceptual frameworks from the relevant literature, this paper employs the load-capacity model as the basis for developing the cascading failure model of the railway transportation network in China. In this model, the initial time period is denoted as t, and the initial load \(Vi(t)\) of each node i in the network at time t is defined based on the node betweenness \(B(t)\):  
\[Vi(t)=B(i) \tag{8}\]  
This approach uses the betweenness centrality of a node--defined as the frequency with which the node appears in all shortest paths--as a measure of its load, thereby assessing the importance and carrying capacity of each node in the network. The initial load represents the initial flow of the nodes in the network, and its capacity \(Ci\) has a direct proportional relationship with the initial load \(Vi(t)\) as follows:  
\[Ci=(1+\sigma)*Vi(t) \tag{9}\]  
In the load-capacity model, \(\sigma\) acts as the enhancement coefficient of node capacity, manifesting the capacity variances of different nodes due to their own facilities, equipment statuses, and renovation circumstances. This coefficient can be adjusted according to the specific requirements and conditions of the network. When studying the cascading failure phenomena in railway networks, the overload conditions of nodes can be simulated to mirror congestion events in real life. If, after a disturbance event, the number of overloaded nodes increases, it indicates a deterioration in the network's congestion level, highlighting the network's limited ability to resist and recover from sudden events, and reflecting its poor stability. To quantify this process, an overload function \(Fi(T)\) is introduced to depict the load conditions endured by each node. The larger the function value \(Fi(T)\), the more severe the congestion at the node, which further reflects the vulnerability of the network when confronted with disturbances. The overload function \(Fi(T)\) can be expressed by a piecewise function. As the load borne by the node increases, the value of the overload function \(Fi(T)\) also increases correspondingly, denoted as  
\[F_{i}(T)= \begin{cases}1,V_{i}(T)\leq V_{i}(t)\\ 1+\frac{V_{i}(T)-V_{i}(t)}{C_{i}-V_{i}(t)},V_{i}(t)\leq V_{i}(T)\leq C_{i}\\ N,V_{i}(T)>C_{i}\end{cases}  \tag{10}\]  
The overload function \(Fi(T)\) is used to quantify the network load of node i during the current period T, where Vi(T) represents the genuine network load of node i in period T. If the network load of the node does not surpass its initial load level, the value of the overload function is set to 1, indicating that the node is operating normally. As the load increments, the value of the overload function will increase in a linear relationship with the node load; yet, once the load of the node exceeds its maximum capacity threshold, the node is regarded as entering an overload state. In such a circumstance, \(Fi(T)\) will be designated as a higher value N, signifying the overload condition. During the process of path selection in the network, by using the overload function as the weight, it is possible to effectively evade those overloaded nodes, thereby optimizing the fluidity of the network and mitigating congestion. By applying the overload function \(Fi(T)\), load distribution strategies can be established to effectively simulate the cascading failure phenomenon within the network. To reduce the complexity of the model, the logic of load allocation can be represented by the following mathematical expressions: Suppose there is an overloaded node n and its neighbor node set \(N_{n}\). The current load of node n is \(V_{n}\) and its capacity is \(C_{n}\). For each neighbor node \(i\in N_{n}\), the current load is \(V_{i}\) and the capacity is \(C_{n}\). When node n experiences \(V_{n}>C_{n}\), the load amount \(T_{i}\) allocated to each neighbor node \(i\in N_{n}\) is calculated as follows:  
Step 1. Compute the residual capacity of each neighboring node of the node: \(R_{i}=C_{i}-V_{i}\);  
Step 2.