## 1 Introduction  
### Motivation  
With the large-scale deployment of low earth orbit (LEO) satellite constellations, heterogeneous satellite networks are emerging as a key architectural paradigm for next-generation space-air communications. Reference [1] provides a detailed overview of their advantages, such as global coverage, low latency, and high capacity. It also highlights key technical challenges, including cross-orbit coordination, interference management, and efficient resource scheduling across LEO and geostationary earth orbit (GEO) satellite layers. These heterogeneous satellite networks combine LEO and GEO satellite systems to exploit their complementary strengths. LEO satellites enable high data rates and low communication latency, while GEO satellites deliver reliable coverage in a wide area [2]. However, this heterogeneous integration also introduces new technical challenges. In particular, the coexistence of LEO and GEO systems operating within overlapping frequency bands can lead to severe inter-system interference, making spectrum sharing a critical issue to be addressed [3]. In addition, to fully exploit the capacity of the system, satellite networks are required to allocate communication resources, such as frequency bands, transmission power, time slots, and beam directions in a highly adaptive and dynamic way. This requirement is particularly critical for LEO satellites, which have greater flexibility in resource scheduling [4]. However, interdependence among these resources across the temporal, spatial, spectral, and power domains results in strong multidimensional coupling, substantially increasing the complexity of joint resource optimization [5].  
In conventional multi-beam satellite communication systems, each beam typically provides fixed coverage over a specific geographic region, lacking the flexibility to adapt resource allocation in response to dynamic user distribution. This often leads to resource shortages in high-traffic areas and resource underutilization in low-demand regions, resulting in inefficiencies and waste in the overall allocation of resources of the system [6]. Beam-hopping (BH) technology offers a promising solution to the resource allocation problem characterized by time-space-frequency coupling in heterogeneous satellite networks. By dividing operational time into discrete slots, the BH algorithm enables satellites to dynamically activate beams across different regions based on user demand and channel conditions, thus achieving multiplexed time-divided coverage and adapting to spatio-temporal variations in traffic load [7].  
BH technology facilitates multi-dimensional resource allocation across the temporal, spatial, spectral, and power domains. Through this flexible and adaptive resource management paradigm, the system can dynamically schedule and optimize resources, ensuring rapid responsiveness to non-uniform traffic distributions and maintaining service quality guarantees [8]. Furthermore, by leveraging dynamic beam steering capabilities, BH technology enables precise control over spatial coverage, which not only increases the number of users that can be served simultaneously but also improves the efficiency of spatial resource utilization [9]. In addition, BH technology supports dynamic frequency and power allocation across beams, allowing the system to flexibly adapt to diverse user-traffic demands and service requirements [10].