## 4 Case Study
### Model Results
#### 4.4.2 Analyzing Sparse Road Network Resilience  
1. Sparse Road Network Structure  
Assuming that the nodes of the road network are randomly damaged without recovery, we simulate the network resilience under continuous natural disaster events or extreme environments (Figure 17). The results show that the network's performance gradually declines with the continuous destruction of nodes, and the rate of performance decline increases as more nodes are damaged. In the early stages of damage, Nodes 1 to 7 are destroyed. Although the damage to these nodes results in some performance decline, it does not lead to severe network failure, thus having a relatively minor impact on the overall performance of the sparse road network. However, the destruction of Nodes 9 to 16 significantly accelerates the decline in network performance. In the later stages, the destruction of Nodes 17 to 31 fragments the network into multiple isolated segments, severely affecting its usability and connectivity, causing a sharp performance decline, approaching zero. A detailed analysis of critical nodes (Figure 18) reveals that Nodes 26, 29, and 31, being interchanges, tunnels, and bridges, respectively, are the most vulnerable links in the network. In a sparse road network primarily composed of two-lane roads, these nodes are the only paths connecting two regions, having a decisive impact on network connectivity. Damage to these nodes rapidly weakens network connectivity and efficiency. Nodes 8, 9, 12, and 15, which are located in villages and county towns, are characterized by having only one main road with no alternative routes. Damage or disruption to these nodes leads to traffic bottlenecks or congestion, severely affecting the overall network flow and transport efficiency. 2. Duration of the Disaster Event  
To reflect the impact of disaster duration on the recovery phase of the road network, we assume an initial network performance of 1 and a rescue vehicle arrival time of 2 h. Based on the average debris clearance times for different disaster events shown in Figure 12, we adjusted disaster durations from 5 h to 70 h to simulate five different disaster scenarios. Figure 17: The resilience curve of the road network after a random node is broken. As shown in Figure 19, the fluctuation trend of network resilience over time \(t\) with varying disaster durations is displayed. From Table 5, it can be seen that the resilience of the sparse road network during collapse events is relatively high (0.6880), with good adaptability (0.80) but slow recovery speed (0.047). Frequent collapses in sparse road networks require large equipment to handle falling rocks, resulting in longer disaster durations. Debris flows have low survivability (0.45) but fast recovery speed (0.075) and adaptability (0.55). Subsidence disasters show moderate survivability (0.35) and evacuability (0.62), with a fast recovery speed (0.04) and good adaptability (0.65). Overall, the resilience is relatively balanced, exhibiting  
Figure 19: Road network resilience values under varying disaster durations. Figure 18: Resilience curve after random failures of critical nodes. some recovery capability within the sparse road network. Water damage shows moderate survivability, a slow recovery speed, and moderate adaptability. Due to the long disaster duration, the overall resilience is the lowest. Overall, when the disaster duration increases to 69 h, the resilience of the road network significantly decreases throughout the disaster period. For instance, the overall resilience of the network during a 69 h landslide event is nearly 10% lower than during a 6 h event. This is mainly because water damage often leads to extensive scour of bridge piers and roadbeds, requiring long-term repairs. Over time, unrepaired sections will lead to traffic congestion, increased delays, and higher accident probabilities, further weakening the network's resilience.