# Resilience Analysis of Traffic Network under Emergencies: A Case Study of Bus Transit Network
## 3 Methodology
### Transport Lines Resilience Analysis Model Based on Resilience Theory
Thus, the resilience of public transit routes can be further summarized as the ability to eliminate negative impacts before unexpected events occur, adapt to impacts after the events, and recover from the damage and changes brought about by the events. Based on the above research foundation, this study quantifies the abilities of the three stages of resilience (adaptation, absorption, and recovery) and forms calculation factors to calculate the resilience of the route. The calculation factors fully consider the changes in the resilience cycle. The public transit route resilience analysis model proposed in this study, based on the resilience cycle theory, is defined as follows:  
Firstly, a resilience indicator \(R_{L}\) is defined to quantify the overall resilience of a public transit route within a resilience cycle that has been impacted by an unexpected event. Based on the definition of the resilience hypothesis, important features are determined to quantify the overall resilience indicator in the resilience calculation process. These important features are quantified by a set of resilience factors, which are used to calculate the overall resilience of the public transit route, as shown in the equation. \[R_{L}= \begin{cases}\theta\rho\Big{[}\lambda+\sigma+1-\tau^{(\rho-\lambda)} \Big{]}&\quad\text{if }\rho-\lambda\geq 0\\ \theta\rho(\lambda+\sigma)&\quad\text{otherwise}\end{cases}  \tag{11}\]  
In the calculation of the overall resilience capability index of the bus route, \(\theta\) is a resilience quantification factor that can generally reflect the resilience changes in a complex system. However, in order to make the calculation of route resilience more sensitive to the resilience period under the impact of sudden events, we optimize this resilience calculation method by integrating a set of resilience factors. Among them, the recovery factor \(\rho\) reflects the system's final recovery state (i.e., its recovery performance) and reflects the system performance changes in the recovery stage of the resilience period. The absorption factor \(\lambda\) reflects the system's ability to absorb the impact of sudden events and reflects the system performance changes in the absorption stage of the resilience period. The volatility factor \(\sigma\) explains the volatility of indicator data, representing the system's ability to smoothly transition from one state to another. Due to the general data fluctuations in various indicators of the bus route, the volatility factor \(\sigma\) can reduce the interference caused by data fluctuations. The normalized recovery time factor \(\tau\) represents the system's response speed by the time required to reach a steady state after being affected by sudden  
Figure 7: Resilience analysis of bus routes (\(y\)-axis: indicator value). events. For the discrete indicator time series data in this study, \(\tau\) can be defined as the ratio of the observed number of the system in the steady state to the total number of observations. The influence of \(\tau\) decreases with the decrease in the indicator recovery amount (i.e., \(\rho-\lambda\)). The conditional statement in the formula ensures that the bus route can only reach a steady state quickly (i.e., with a low \(\tau\) value) when its recovery performance is better than its absorption performance. Only one \(R_{L}\) value is calculated within a resilience period (i.e., \(R_{L}\) is a function of the time interval from \(t_{S}\) to \(t_{E}\), not directly a function of time t). The following introduces the definition of each resilience quantification factor in the resilience calculation formula for bus routes. 1. Overall performance factor \(\theta\)  
The overall performance factor \(\theta\) reflects the system's resilience in a complete resilience cycle affected by sudden events, and its calculation formula is:  
\[\theta=\frac{\int_{t_{S}}^{t_{E}}y(t)dt}{(t_{E}-t_{S})y_{N}} \tag{12}\]  
Under normal circumstances, the value of \(\theta\) is [0, 1], and when the system performance does not decrease, \(\theta\) is 1.