# Vulnerability Analysis of the China Railway Express Network Under Emergency Scenarios
## 5 Vulnerability Analysis of CRN Under Emergency
### CRN Node Failures
#### 5.1.3 Analysis of Coupling Coefficient Under Node Failure  
The previous section's analysis indicates that the CRN exhibits greater vulnerability to deliberate attacks, which can significantly impact the network. Therefore, this section primarily investigates the influence of changes in the coupling coefficient under deliberate attacks on network vulnerability. To better reflect the differences among the three indicators, we selected Xi'an as the node with the highest strength, Taiyuan as the node with the highest betweenness, and Wuhan as the main node with a high degree for deliberate attacks, setting the external disturbance R = 2. (1) Topological coupling coefficient analysis  
The study examined the vulnerability of the CRN under different topological coupling coefficients (\(\varepsilon_{1}\) is 0.1, 0.2, 0.3, 0.4, 0.5, and 0.6) when \(\varepsilon_{2}=0.3\). As shown in Figure 9, when \(\varepsilon_{2}>\varepsilon_{1}\), only 9% of cities with faulty nodes in the CRN are affected when the city with the highest node betweenness is deliberately attacked. However, when the city with the highest intensity of deliberate attack and the highest node degree is targeted, the network almost completely paralyzes. When \(\varepsilon_{2}\leq\varepsilon_{1}\), regardless of whether it is the city with the highest node strength or the highest node betweenness that is attacked, its cascading failure propagation process, the range of peak failure times, and the time required for all nodes to fail are almost unaffected by \(\varepsilon_{1}\) (the peak moment is t = 6, the peak ratio \(P\in[0.162,\ 0.243]\), and all nodes fail at \(t=[11,12]\) as shown in the figure). When the city with the highest node degree is deliberately attacked, the peak value of node failure is advanced by three time steps compared to when \(\varepsilon_{2}>\varepsilon_{1}\). It can be concluded that increasing \(\varepsilon_{1}\) will accelerate the  
Figure 8: Cascading failure propagation process of major cities under different indicators under intentional attacks with different external disturbances. Figure 7: Comparison of different indicators in the same city with the same node degree. propagation of cascading failures in the network when the city with the highest node degree is deliberately attacked. When \(\varepsilon_{1}\) = 0.6, the time for all nodes to fail is also advanced by two time steps. In summary, when \(\varepsilon_{2}>\varepsilon_{1}\), the CRN has strong stability against betweenness deliberate attacks; and, when \(\varepsilon_{2}\leq\varepsilon_{1}\), the CRN has strong stability against node strength and betweenness attacks. (2) Analysis of cargo flow coupling coefficient  
The study examined the vulnerability of the CRN under different cargo flow coupling coefficients (\(\varepsilon_{2}\) are 0.1, 0.2, 0.3, 0.4, 0.5, and 0.6, respectively) when \(\varepsilon_{1}\) = 0.3. As shown in Figure 10, when \(\varepsilon_{2}<\varepsilon_{1}\), only 6% of cities with the highest node betweenness suffer from intentional attacks in the CRN; however, when the city with the highest node betweenness or node strength is deliberately attacked, the proportion of network failure nodes reaches  
Figure 9: Cascading failure propagation process in CRN under deliberate attacks on nodes with different \(\varepsilon_{1}\): (**a**) deliberate attacks based on node degrees; (**b**) deliberate attacks based on node strength; and (**c**) deliberate attacks based on node betweenness. 100%. When \(\epsilon_{2}\geq\epsilon_{1}\), the process of cascading failure propagation is almost synchronized when the city with the highest node degree or node strength is attacked (the time of failure peak is \(t=5\) and \(t=7\), the proportion of failure peak within a small range is \(p\in[0.230,\ 0.243]\) and \(p\in[0.176,\ 0.189]\), and all nodes fail at \(t\in[10,\ 11]\) and \(t=10\)).