# Vulnerability Analysis of the China Railway Express Network Under Emergency Scenarios
## 3 Methodology
### Cascading Failure Model of CRN Based on CML  
CML is a dynamical system with discrete variables for time and space, and continuous variables for state. Traditional CML models only consider one topological coupling factor ([PERSON], 1986). However, in CRN, the cause of cascading failures is related not only to the network topology but also to the volume of cargo flow within the network. Therefore, this paper constructs a CML model that takes into account cargo flow to analyze the cascadingfailures of the CRN. This model provides a theoretical basis for the analysis of the CRN. The following Equation (7) represents this model:  
\[w\big{(}\epsilon_{ij}\big{)}=x_{i}(t+1)=\left| \begin{matrix}(1-\epsilon_{1}- \epsilon_{2})f(x_{i}(t))+\frac{\epsilon_{1}\sum_{j=1,j\
eq i}^{N}a_{ij}f\big{(} x_{j}(t)\big{)}}{\epsilon_{i}}\\ +\frac{\epsilon_{2}\sum_{j=1,j\
eq i}^{N}w_{ij}f\big{(}x_{j}(t)\big{)}}{s(i)} \end{matrix} \right|,i=1,2,\ldots,N \tag{7}\]  
In this context, \(t\) represents the concept of days. \(x_{i}(t+1)\) represents the state value of node \(i\) at time \(t~{}+~{}1\). When \(x_{i}(t+1)\) is within the range of (0, 1), it indicates that the node is operating normally; if \(x_{i}(t+1)\geq 1\), node \(i\) will fail. \(\epsilon_{1}\) represents the topological coupling strength coefficient between nodes, indicating the correlation between adjacent cities; \(\epsilon_{2}\) represents the cargo flow coupling strength coefficient between nodes, which measures the coupling strength of network cargo flow through actual cargo flow between cities. For \(\epsilon_{1}\) and \(\epsilon_{2}\), they are in the range of (0, 1), and \(\epsilon_{1}+\epsilon_{2}\)\(<\) 1; for example, \(\epsilon_{1}\)= 0.3 means that 30% of the state change of this node comes from the influence of its neighboring nodes' topological structure changes, and \(\epsilon_{2}\) = 0.3 means that 30% of the state change of this node comes from the influence of its neighboring nodes' cargo flow changes. \(a_{ij}\) represents the connection between node \(i\) and node \(j\); if there is a connection, then \(a_{ij}\) = 1, otherwise \(a_{ij}\) = 0; \(k_{i}\) represents the degree of node \(i\). \(w_{ij}\) represents the cargo flow from node \(i\) to node \(j\), and \(s(i)\) represents the total cargo flow directly connected to node \(i\). \(f(x)\) represents the chaos mapping function, establishing the connection between the state values of nodes at time \(t\) and \(t~{}+~{}1\), with \(f(x)~{}=~{}4x(1-x)\), when \(0~{}<~{}x~{}<~{}1\), \(0~{}<~{}f(x)~{}<~{}1\). The initial state of the nodes: \(x_{i}=f_{i}/C_{i}\) where \(f_{i}\) represents the cargo flow of node \(i\) when not disturbed, and \(C_{i}\) represents the capacity of node \(i\). If there is no external disturbance to the network at this time, all nodes will always maintain a normal state; otherwise, if an emergency occurs at node \(i\) at moment \(l\) (\(x_{i}(l)\geq 1,l>1\)), then node \(i\) fails at this time, and the state of the failed node is assumed to be \(x_{i}(t)\) = 0 in subsequent moments. At the \(t\)-th moment, due to sudden incidents, the node \(i\) of the Central European train operation network is unable to function. An external disturbance \(R\) (\(R\geq 1\)) is added to node \(i\) to represent the effect of the sudden incident, and the model after adding the disturbance is shown in Equation (8). When \(x_{i}(t)\geq 1\) at time \(t\), it means that node \(i\) fails at time \(t\). At time \(t+1\), the adjacent nodes of \(i\) will be affected by the failed node \(i\) at time \(t\), and the status values of the adjacent nodes will be updated according to Equation (7), and the cargo flow will be redistributed according to Equations (9) and (10). The updated node status value may also exceed 1, which could lead to new node failures. This process may continue, causing the scale of node failures to further spread, ultimately leading to network collapse. \[x_{i}(t)=\left|