# Vulnerability Analysis of the China Railway Express Network Under Emergency Scenarios
## 5 Vulnerability Analysis of CRN Under Emergency
### Disruption of CRN Edges
#### 5.2.3 Analysis of Coupling Coefficients During Edge Disruption  
This section primarily investigates the impact of changes in coupling coefficients under deliberate attacks on network vulnerability. We selected edges with higher weight ranking (Xi'an-Chongqing) and those with higher betweenness centrality ranking (Wuhan-Changsha) as the main edges for analysis of the effects of coupling coefficient changes on network vulnerability.  
(1) Topological coupling coefficient analysis  
As shown in Figure 15, it can be observed that, regardless of whether under edge strength attack or edge betweenness attack, the cascading failure propagation process of the  
Figure 14: The cascading failure propagation process of several edges with common points in the CRN under deliberate attack: (**a**) the edge of the shared node Xi’an; and (**b**) other edges that share a node.  
Figure 13: The cascading failure propagation process in the edges of CRN under deliberate attack: (**a**) deliberate attacks based on edge weights; and (**b**) deliberate attack based on edge betweenness.  
network, the range of the failure peak time, and the time required for all nodes to fail are almost unaffected by the change in \(\varepsilon_{1}\) (the time of failure peak is \(t\ \in\ [5,\ 6]\), the proportion of failure peaks within a small range is \(p\ \in\ [0.176,\ 0.203]\) and \(p\ \in\ [0.162,\ 0.176]\), and the time for all nodes to fail is \(t=10\) and \(t=11\)). Topological coupling is a network characteristic between nodes in the network, representing the topological relationship between two nodes in the network. When deliberately attacking the edges in the network, changing the topological coupling coefficient cannot produce significant changes in the vulnerability of the network. In summary, changing the value of \(\varepsilon_{1}\) does not have a significant impact on intentional attacks based on network edges.  
(2) Analysis of cargo flow coupling coefficient  
As shown in Figure 16, when \(\varepsilon_{2}<\varepsilon_{1}\), the range of cascading failures caused by edge strength attacks or edge betweenness attacks is reduced. When \(\varepsilon_{2}\geq\varepsilon_{1}\), the process of cascading failure propagation in the network is almost synchronized (the time of failure peaks is \(t=5\) and \(t=7\), the proportion of failure peak values within a small range is \(p\ \in\ [0.189,\ 0.202]\) and \(p\ \in\ [0.162,\ 0.202]\), and the time for all nodes to fail is \(t\ \in\ [9,\ 10]\) and \(t=11\)). Notably, compared to edge betweenness attacks, edge strength attacks result in larger cascading failure peaks, and the time required for all nodes to fail is also advanced. In summary, changing \(\varepsilon_{2}\) has an impact on deliberate edge attacks; the network demonstrates a higher stability against edge betweenness attacks.  
Figure 15: The cascading failure propagation process on edges in CRN under different \(\varepsilon_{1}\): (**a**) deliberate attack based on edge weight; and (**b**) deliberate attack based on edge betweenness.