## 2 Related Work
### Design of the Algorithm for Measuring Railway Network Resilience
This metric enables the identification of critical nodes whose removal would significantly impair network functionality and efficiency. The calculation formula for network efficiency is given as follows:  
\[E=\frac{1}{N(N+1)}\sum i,j\in N\eta_{ij} \tag{4}\]  
where E represents the network efficiency, N is the total number of nodes in the network, and \(\eta_{ij}=1/d_{ij}\) is the reciprocal of the shortest path length \(d_{ij}\) between nodes \(i\) and \(j\). The shorter the path between two nodes, the higher their contribution to overall network efficiency. Upon completion of the aforementioned indicator calculations, the resulting data are normalized to the range [0, 1] using the maximum value normalization method. Let \(x\) represent a value in the original dataset, and \(x_{max}\) denote the maximum value within that dataset. The normalized value \(x_{norm}\) is then computed using the following formula:  
\[x_{norm}=\frac{x}{x_{max}} \tag{5}\]  
Table 3 elaborates the quantitative implications of these three feature operators and the corresponding real-world network performance. After normalizing the aforementioned indicators, the measure of resilience and accessibility can be represented as  
\[T=aK+bG+cL \tag{6}\]  
where a, b, and c are conversion coefficients, all being taken as \(1/3\). The specific design procedures of the metric algorithm for the resilience and accessibility of the physical network are as follows: \begin{table}
\begin{tabular}{c c c} \hline
**Measuring of Accessibility** & **Quantitative Implication** & **Characterizing Network Performance** \\ \hline Reachable node pairs & Referring to the proportion of reachable & Node accessibility \\  & node pairs of the initial network & Degree of network clustering \\ Clustering coefficient & The coefficient characterizing the extent of & Degree of network clustering \\ Clustering coefficient & Clustering among the vertices in the graph & The velocity and stability of data \\ Network efficiency & Transmission and communication within & The overall efficiency of the network network systems \\ \hline \end{tabular}
\end{table} 
Table 3: Reachability measurement feature operators. Step 1. Export the adjacency matrix A of the geographical model of the railway transportation network in China using Gephi (v0.9.2) software and import it into MATLAB (R2022b). Step 2. Calculate and sort the betweenness centrality and degree of each node in the network. Step 3. Calculate the indicators, such as the number of reachable node pairs, network efficiency, and clustering coefficient, for the initial network using the specified formulas. Step 4. Simulate the railway network perturbation events on the network. Step 5. Calculate indicators such as the number of reachable node pairs, network efficiency, and clustering coefficient in the newly generated network according to the formulas. Traverse whether all nodes and edges in the network have undergone the simulation of railway network perturbation events. If not, return to Step 4. If yes, proceed to Step 6. Step 6. Normalize the simulation results and calculate the accessibility metric of the network according to the formulas and output the results. (2) The algorithm for measuring the invulnerability (or resilience) of the physical network  
The invulnerability of a network describes the ability of a system to remain functional with little or no impact when confronted with perturbation events. Drawing upon relevant studies, this paper adopts a cascading failure model to analyze the changes in invulnerability of railway transportation networks under disturbance scenarios. As a measurement index, the ratio of the number of nodes in the maximum connected subgraph to the number of nodes in the initial state of the network is selected in this paper. This indicator effectively reflects the extent to which the connectivity and functionality of the network are preserved after experiencing perturbations, providing a practical tool for evaluating and enhancing the resilience of railway networks.