# 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
This indicator reflects the resilience capability during a complete period. This calculation method is also applied to the resilience of other complex systems. Although it can measure the resilience of a system in normal application scenarios, its shortcomings are also obvious. Its calculation result does not reflect the changes in the system during the absorption and recovery stages. For example, there are two systems A and B. When encountering sudden events, system A suffers a severe performance decline but recovers quickly, while system B has a slight decline in performance but recovers slowly. Using \(\theta\) to calculate the system's resilience may result in the same value. The fundamental reason is that \(\theta\) does not take into account the absorption rate, recovery rate, and data volatility of the bus network. Therefore, we need to incorporate other resilience quantification factors to identify the system's absorption rate, recovery rate, and data volatility. 2. Absorption factor \(\lambda\)  
Quantifying the ability of a bus route to absorb the impact of unexpected events is crucial. The absorption factor \(\lambda\) represents the ability of the system to absorb the impact of unexpected events on system performance, reflecting the absorption capacity of the route during the resilience cycle. Its calculation formula is:  
\[\lambda=\frac{y_{L}}{y_{N}} \tag{13}\]  
\(\lambda\) usually takes a value between 0 and 1 and is 1 under normal circumstances. The absorption factor can reflect the ability of the system to absorb negative impacts when facing unexpected events, and the smaller the value, the greater the degree of impact on the system. 3. Recovery factor \(\rho\)  
The calculation formula used to quantify the recovery capacity of bus routes during the resilience period is:  
\[\rho=\frac{y_{S}}{y_{N}} \tag{14}\]  
where \(0\leq\rho\leq\infty\), and \(y_{S}\) is the stable value of system performance after recovery. In this study, the commonly used augmented Dickey-Fuller (ADF) test in time series analysis was used to determine when the bus route reached stationarity, i.e., to determine \(t_{SS}\) and \(y_{S}\). For bus route indicators with obvious daily trends (morning and evening peak periods) and data drift, an autoregressive process with drift and trend terms was used for ADF testing, as shown in equation:  
\[Y_{t}=\omega+\beta t+\gamma Y_{t-1}+\sum_{i=t_{L}}^{k}C_{i}\Delta Y_{t-i}+ \varepsilon_{t},Y_{0}=0 \tag{15}\]  
where \(\omega\) is the constant term, \(\beta t\) is the time trend, \(\varepsilon_{t}\) is the random disturbance term, and \(t\) takes values from \([t_{L},t_{E}]\). The hypothesis is that there is a unit root in the time series, making it non-stationary for the null hypothesis, and the alternative hypothesis is that there is no unit root, indicating that the time series is stationary with an intercept and trend. If we strictly judge whether the sequence is weakly stationary, we can directly test whether it is stationary without intercept and trend. If the null hypothesis cannot be rejected (e.g., \(p>0.05\)), the sequence is non-stationary, and it is still necessary to test whether the sequence is trend stationary. The steady-state recovery time factor \(\tau\) is used to capture the speed at which the system recovers to a steady state during the recovery phase of the resilience cycle. This factor considers the recovery capacity of the bus route when preventive measures are taken. \[\tau=\frac{t_{SS}-t_{L}}{t_{E}-t_{S}} \tag{16}\]  
where \(0\leq\tau\leq 1\) and approaches 0 under normal circumstances. \[\text{ fluctuation coefficient }\sigma \tag{5}\]  
The fluctuation coefficient \(\sigma\) is used to address the highly unstable performance data of the bus routes. This study borrows the concept of signal-to-noise ratio from communication system research, which originally refers to the ratio of signal to noise in an electronic device or system.