## 3 Methodology
### Quantifying the Performance of Sparse Road Networks  
The performance of a road network is represented by its accessibility and travel time. Equation (7) provides the performance representation of the road network at time \(t\):  
\[F(t)=\alpha_{s}G(t)+\alpha_{f}ATD(t) \tag{7}\]  
In this equation, \(F(t)\) denotes the performance of the road network at time \(t\), \(\alpha_{s}\) represents the weight of the network's structural performance, and \(\alpha_{f}\) denotes the weight of the network's functional performance. In this study, it is assumed that the structural and functional aspects of the road network are equally important; therefore, both \(\alpha_{s}\) and \(\alpha_{f}\) are set to 0.5. Existing research widely utilizes the resilience indicator proposed [46; 47]. This indicator dynamically represents resilience as a function of time (Equation (8)), quantifying resilience as the ratio of recovery to loss:  
\[\mathcal{H}_{F}\Big{(}t\mid z^{j}\Big{)}=\frac{F\big{(}t\mid z^{j}\big{)}-F \big{(}t_{d}\mid z^{j}\big{)}}{F(t_{0})-F\big{(}t_{d}\mid z^{j}\big{)}} \tag{8}\]  
Here, \(\mathcal{H}_{F}\big{(}t\mid z^{j}\big{)}\) represents the resilience of the system at time \(t\) after the occurrence of disaster \(z^{j}\). The numerator represents recovery up to time \(t\), while the denominator represents the total loss due to the disruption. This resilience indicator reflects the ratio of the system's recovery during the recovery period to the performance loss due to the disturbance. As described in Section 3.2, the resilience of sparse road networks encompasses three aspects: the ability to continuously resist disasters, the ability to maintain basic structure and functionality before emergency rescue vehicles arrive, and the ability to recover from disasters. Therefore, the resilience indicator for sparse road networks should consider the overall performance of the system throughout the disaster event lifecycle. Thus, as illustrated in Figure 6, this study quantifies the resilience of sparse road networks as the average cumulative performance from the occurrence of the first disaster event until the system performance recovers to match the current traffic volume. A comprehensive resilience indicator for evaluating the entire disaster event lifecycle is proposed, with the resilience of the sparse road network at time \(t\) given by Equation (8):  
\[\psi(t)=\frac{\int_{t_{\text{e}}}^{t}F(\tau)\text{d}\tau}{F_{0}(t)(t-t_{\text{e} })}\cdot t_{\text{e}}\leqslant t\leqslant t_{\text{r}}, \tag{9}\]  
In this equation, \(F_{0}(t)\) is the expected performance of the system at time \(t\), and \(F_{0}(t)=F(t)\). The numerator represents the cumulative performance from the time of the first disaster event \(t_{\text{e}}\) to time \(t\), while the denominator represents the ideal performance the system is expected to achieve from \(t_{\text{e}}\) to \(t\). Considering multiple successive disaster events, the modified resilience indicator is given by Equation (10):  
\[\psi\Big{(}t\mid z^{j}\Big{)}=\frac{\sum_{j=1}^{j}\int_{t(e\mid z^{j})}^{t(z^{ j-1})}F(t\mid z^{j})dt}{F\Big{(}t\mid z^{1}\Big{)}(t(z^{j})-t(e\mid z^{1}))} \tag{10}\]  
Here, \(F\big{(}t\mid z^{j}\big{)}\) denotes the system performance function after each disaster, where \(j=1,2,\ldots,J\), \(J\) represents the sequence of disaster events, and \(t\) denotes time. \(F\Big{(}t\mid z^{1}\Big{)}\) is the initial expected performance of the system at time \(t\), \(F\Big{(}t\mid z^{1}\Big{)}=F_{0}\) is the time of the first disaster event, and \(t\big{(}z^{j}\big{)}\) is the time when the last disaster is fully repaired, during which the system's performance gradually recovers. The numerator represents the cumulative performance from the time of the first disaster event \(t_{\text{e}}\) to time \(t\), while the denominator represents the ideal performance expected from \(t\big{(}e\mid z^{1}\big{)}\) to \(t\big{(}z^{j}\big{)}\).