## 1 Introduction  
### Background  
The economic cluster has become a crucial model for global economic development, due to the rapid pace of globalization, urbanization, and spatial reconstruction [1]. In contrast to the historically isolated growth of individual cities, the formation of city clusters is intended to enhance resource efficiency through the consolidation of production factors and the establishment of well-connected spatial structures among cities [2]. This developmental approach places increased requirements on the intra-regional transportation infrastructure. However, the construction and operation of the road network are accompanied by a large amount of energy consumption [3] and cannot be built indefinitely. Consequently, assessingthe effectiveness of transportation networks at the city cluster level emerges as a crucial issue in fostering efficient interconnectivity among urban clusters [4]. For an extended period, road transportation has been the primary mode of intercity transportation in China, primarily due to the absence of efficient long-distance transportation options [5]. The rapid expansion of transit systems in China is being driven by a large population, rapid economic growth, and policy reforms [6]. High-speed railways (HSRs) are experiencing rapid development and construction globally, as a novel mode of transportation [7]. To date, HSRs have become a more competitive mode of intercity transportation, offering economic advantages, environmental friendliness, and superior ride quality, compared to road transportation. The investment in HSRs forms an integral component of China's national development strategy, designed to meet the rapidly increasing transportation needs, particularly for intercity travel [8]. The construction of HSRs has a significant impact on both medium- and long-distance passenger and cargo transportation, as well as on the spatial organization of cities and regions [9]. Concurrently, significant advancements have been achieved in highway construction. By the conclusion of 2021, the total length of highways in China had extended to 177,300 km. Rail transit and road transportation have emerged as the predominant modes of intercity transportation in China. The significant impact of substantial investments in transportation infrastructure on the effectiveness of intercity transportation within regions has garnered considerable attention from scholars. The significant cost associated with them raises a question that is of considerable interest to proponents and detractors of transport infrastructure alike. There exists a plethora of studies examining mobility and the extensive effects of developing transport infrastructure. Europe [8], Japan [10], and the United States [11] have conducted comprehensive and valuable studies on their potential, offering exemplary cases in various fields to the global community. Previous research conducted on developed economies can provide valuable insights regarding theories and analytical approaches. These studies have explored various fields, such as intercity travel time, cost, and distance [12; 13]; network connectivity [14; 15]; spatial accessibility [5; 16]; and agglomeration effects [17]. However, the majority of these studies concentrated on specific cities, provinces, or countries. There is a limited amount of research available on city clusters, despite their significance as a primary driver of economic development in the current stage. Therefore, there is an urgent need to investigate the present state of intercity traffic efficiency in light of the ongoing development of traffic infrastructure, which serves as the impetus for this study. Trade-offs exist among coverage, accessibility, connectivity, and directness within transportation networks [14]. It is important to note that high-density transport infrastructure does not guarantee efficient operations. Elevated levels of indirectness intensify trip detours, consequently diminishing traffic efficiency.