## 1 Introduction  
In the context of escalating climate change and frequent public health incidents, the transportation system faces increasing uncertainties and unknown risks [1, 2, 3]. This issue is particularly pronounced in remote and rural areas [4], where geological and climatic conditions often limit road networks, resulting in sparse road infrastructure. Characteristics of these sparse networks include a scarcity of critical nodes, widespread node distribution, long distances between nodes, low traffic volumes, low network density, and limited alternative routes. Despite these limitations, these sparse networks are the only means of regional connectivity, emergency service provision, and economic support [5]. Therefore, ensuring the reliability and resilience of these sparse networks is crucial for the safety of individuals and the sustainable development of regions. Currently, research on the resilience of sparse road networks under the impact of emergencies faces several challenges. On the one hand, natural disasters (such as mudslides, landslides, earthquakes, etc. ), severe weather conditions (dense fog, heavy snow, icy roads, etc. ), traffic accidents, and planned interruptions can cause random damage to segments and critical nodes within the sparse road network. These disruptions can lead to complete closures or one-way traffic, significantly altering the road network's topology compared to regular conditions [6; 7]. Additionally, temporary traffic control measures or road closures are often necessary in response to sudden disaster events, with normal traffic resuming once the temporary state ends. This results in frequent, localized, and time-limited changes in the network topology [8]. Therefore, it is crucial to develop a road network model that accurately reflects the structural characteristics of sparse road networks and their dynamic changes during disaster events. Such a model would provide a foundation for assessing traffic conditions, planning emergency rescue routes, and optimizing resilience measures, thereby improving the timeliness and accuracy of emergency responses in sparse road networks affected by disasters. On the other hand, geological disaster events in extreme environments and severe weather pose unique challenges to sparse road networks [9; 10; 11]. Unlike rare large-scale events such as earthquakes [12], tsunamis [13], and floods [14; 15], frequent geological disasters often occur before the previous incident is fully repaired, causing continuous service interruptions and long-term recovery efforts. This results in the cumulative degradation of road network infrastructure and service performance, severely impacting the stability and safety of the road network. Compared to dense road networks, sparse road networks lack alternative bypass routes, making them more vulnerable to disasters. When a disaster occurs, it severely hampers emergency response efforts and significantly delays the arrival of emergency relief and recovery resources. Moreover, the critical components of these networks, including bridges and tunnels, may require months or even years for repair or reconstruction [16; 17]. During this period, utilizing the remaining functionality of damaged infrastructure for intermittent passage is crucial for maintaining the operability and accessibility of the road network. Previous research on the resilience of transportation networks has primarily focused on the development of frameworks for quantifying and optimizing the resilience of urban roads and dense networks [9; 18; 19]. These studies often assume that road networks operate in environments with infrequent natural disasters, such that disaster events occur simultaneously, and that the service functionality of segments or nodes is completely disrupted [20]. However, these assumptions do not always reflect reality, neglecting the unique challenges faced by sparse road networks, including the frequent occurrence of geological disasters, limited recovery resources, and the lack of alternative routes.