# Temporal Continuity Expression for Network Topology of Space Information Systems
## 1 Introduction  
The space information system consists of a large number of satellite nodes, where nodes transmit information by connecting with each other through the inter-satellite link. Based on the satellite network, the overall efficiency of the system can be enhanced through an effective collaboration within the system, so as to improve the quality of services such as backbone transmission, broadband access, and global connectivity. With the implementation of the mega-constellation plans like Star-link and One-Web [1], and the development of 6G mobile communication networks [2; 3; 4; 5], the complexity of the space information system has been increasing dramatically. The high-speed movement of nodes in the system causes the significant dynamics of the system network topology. Since implementing the main functions of the space information system is based on the network topology, it is of great significance to understand the network topology of the system and learn the rules of operation. For extremely complex networks, such as the Internet of Things, the integration of deep-learning models is becoming a research trend [6]; in the field of satellite-terrestrial integrated networks, most of the current research focuses on multiple access during network transmission [7; 8], secrecy-energy efficiency [9], and optimization of the concurrent accessranging process [10]. Correspondingly, the network topology of the space information system, as an important foundation for the function, has become a research hotspot. Scholars have completed a lot of research for the purpose of route planning, and proposed many description methods, e.g., the time slice method, snapshot sequence, finite state automation, evolutionary graph, etc. A time slice division method [11] based on the variation in interstellar distance was constructed using the number of satellites in each orbit, in which the equidistant time slicing is utilized to construct a virtual topology that enables dynamic simulation and management of the topology to be achieved. [PERSON] et al. [12] divided the whole period into a series of time slices with a fixed value, which means the topology remains unchanged in the time slice, to solve the dynamic problem of the space-ground information network. A routing optimization method for LEO satellite networks with stochastic link failure was proposed in ref. [13], where a discrete-time strategy was used for the satellite network to acquire several static topological graphs in a single cycle. To deal with the dynamic topology of LEO satellite networks, refs. [14; 15] developed a snapshot division method. In this method, the minimum snapshot length is calculated, and a fixed time step that is less than the minimum snapshot length is subsequently set to sample a sequence of time instances, which represents a granular description for the network topology. To reduce the complexity of control management for a large number of satellite nodes, SDN was introduced into satellite networks with multi-domain satellite network snapshot [16; 17] as a division of satellite network topology. [PERSON] et al. [18] divided the satellite network into a series of static networks (snapshots) with stable topology, since the operation period of the satellite constellation is stable and the network topology state also shows periodic changes. Snapshots were also used to describe the topology in refs. [19; 20; 21; 22]. Finite state automation (FSA) was used to model the satellite network by [PERSON] et al. [23], to circumvent the problem of dynamic visibility caused by periodically orbital motions of satellites. They divided the repeat cycle into consecutive equal-length states, while each state was further partitioned into several time slots to ensure sufficient satellite pairs. As an intuitive means of expression, graph sequences are taken to characterize the topological dynamics of the satellite network.