## 2 Related Work
### The Construction of Railway Complex Network Model and the Fundamental Theory of Resilience  
(1) Complex Network Models of Railways  
A topological graph of a railway transportation network can consist of a collection of stations \(G\) and a collection of line segments between stations, which can be expressed as \(G=(V,E)\), where \(V=(v_{1},v_{2},v_{3},\ldots,v_{n})\) represents the set of all nodes (stations) in the railway transportation network and \(E=(e_{1},e_{2},e_{3},\ldots,e_{n})\) represents the set of all edges (lines) in the railway transportation network. Therefore, it can be represented by an N-order square matrix as follows:  
\[\text{A}= \begin{bmatrix}0&\text{a}_{1,2}&\ldots&\text{a}_{1,n-1}&\text{a}_{1, n}\\ \text{a}_{2,1}&0&\ldots&\text{a}_{2,n-1}&\text{a}_{2,n}\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ \text{a}_{n-1,1}&\text{a}_{n-1,2}&\ldots&0&\text{a}_{n-1,n}\\ \text{a}_{n,1}&\text{a}_{n,2}&\ldots&\text{a}_{n,n-1}&0\end{bmatrix}  \tag{1}\]  
where A represents the adjacency matrix of the railway network, with elements \(a_{ij}\) (i,j = 1, 2, 3, \(\ldots\), n - 1, n) indicating the presence (non-zero value) or absence (zero value) of a direct connection between stations i and j. The diagonal elements are zero as self-loops do not exist in railway networks. If the elements in the square matrix \(a_{ij}\) (\(i,j=1,2,3,\ldots,n-1,n\)) are non-zero constants, it indicates that there is a direct line connection between stations; otherwise, there is no connection. Moreover, since there is no case in the real world where a certain station is self-connected to form a loop, all the diagonal elements are zero. Simultaneously, when there are different research purposes, different modeling methods are often required to construct the railway topological network. The commonly used model construction methods for building railway topological networks include Space B, Space C, Space L, and Space P.  
The characteristics of each modeling approach are presented in Table 2.  
Among these, the Space L method and the Space P method are the most commonly used in the modeling of the topological model of rail transit networks. (2) Railway Network Resilience Measurement \begin{table}
\begin{tabular}{c c} \hline \hline
**Construction Methods** & **Model Characteristics** \\ \hline Space B & Both the stations and routes in the transportation network are regarded as nodes of the topological model. The adjacent station nodes to a route node represent all the stations covered by that route; the adjacent route nodes to a station node represent all the routes served by that station. \\  & The routes in the transportation network are regarded as the nodes of the topological model. Moreover, if a direct transfer is possible between two routes, then in the network graph, there will be a connection between the nodes corresponding to these two routes. \\ Space L & The stations in the transportation network are taken as nodes in the topological model, and the sections between adjacent stations on the same route are abstracted as edges (or arcs). \\ Space P & The stations in the transportation network are regarded as the nodes of the topological model, and the sections between adjacent stations on the same route, that is, all the pairwise connections among stations on the same route. \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Common methods of model building for transportation topology networks. The resilience of railway networks refers to the adaptability and recovery capability of the railway system when encountering unexpected events. Research on railway network resilience aims to provide a theoretical basis for the management and mitigation of disturbance events. Railway operations generally adhere to a predefined train schedule, requiring seamless coordination across various subsystems. However, when disruptions occur--impacting certain functional components--the railway management authorities must implement repair and rescheduling strategies to restore normal operations.