## 1 Introduction  
In today's information age, the advancement of satellite communication technology has become a pivotal driving force in the global communication landscape. The widespread adoption of the Internet and rapid digitalization necessitate increasingly urgent demands for high-speed, efficient, and reliable global communication services [1]. Satellite communication, with its expansive coverage, rapid transmission speeds, and cost-effective communication capabilities, emerges as an ideal choice to meet these demands. Inter-satellite communication, as the core of satellite networks, plays a decisive role in the overall performance and efficiency of these networks. With continuous advancements in satellite technology and the scaling up of satellite networks, the construction of large-scale satellite constellations has emerged as a significant trend in the communication industry. Representative projects like SpaceX's Starlink satellite system aim to provide global high-speed Internet coverage by deploying a multitude of low-Earth-orbit (LEO) satellites [1] However, compared to traditional ground networks, satellite networks face a series of unique challenges, including signal transmission delays and unstable communication links, which not only affect the communication quality but also limit the application and efficiency of satellite networks. In response to these challenges, research into topology control strategies is particularly crucial. These strategies involve adjusting and optimizing the structure and manner of communication links within satellite networks to enhance the network stability, reliability, and flexibility [2]. Traditional inter-satellite topologies often rely on fixed links connecting consecutive satellites in the same orbit and two adjacent satellites in different orbits [2]. Nevertheless, such configurations exhibit limitations in flexibility and adaptability, potentially resulting in low link utilization and poor network resilience [3]. Reference [4] analyzes the inter-satellite transmission characteristics of low-Earth-orbit (LEO) mega-constellations such as Starlink. It introduces the commonly used Walker constellation and inter-satellite link concepts in LEO satellite networks [4]. The study proposes a method for estimating the number of inter-satellite hops directly based on user and destination geographical locations. Additionally, it investigates the impact of Starlink constellation parameters, link endpoints' geographical positions, and link establishment methods on the inter-satellite transmission paths. Another article [5] discusses the networking techniques and research status of low-Earth-orbit (LEO) mega-constellations, with a focus on analyzing the challenges posed by the constellation scale and complexity expansion for key networking technologies. It also examines the applicability of various existing solutions in the context of broadband networks within large-scale constellations. Another study [6] discusses the conditions for the permanent visibility of inter-satellite links in optimizing multi-layer satellite network configurations, which is crucial for understanding the stability and persistence of inter-satellite links. However, in the optimization of multi-layer satellite network configurations, this study may overlook the dynamic variations encountered in practical applications when considering permanent inter-satellite link visibility conditions. The research in the article by [7] analyzes the inter-satellite geometric characteristics and spatial parameters of standard Walker constellation satellites, designing an inter-satellite fixed link topology structure capable of full network coverage with a low number of hops. This effectively reduces the number of inter-satellite links, enhancing the system's implementability and reliability. However, it does not take into account the limitations of laser communication in inter-satellite links, such as signal attenuation issues [8].