# Temporal Continuity Expression for Network Topology of Space Information Systems
## 1 Introduction
For example, a framework [24] was presented to evaluate the structural performance of dynamic networks using a graph sequence with an unchanged structure in each graph, and a temporal graph-based model [25], in which the network topology in each slot can be regarded as fixed, was adopted to capture the time-varying topology of LEO satellite networks. Since the changes in satellite state and network topology during a short time slot exert little influence on data transmission, an assumption that the topology remains stable in each time slot was raised in refs. [26; 27; 28] wherein the satellite state and network topology are considered static during a time slot, i.e., the total time can be divided into multiple time slots, each of which has a fixed duration. Considering that the existing topology optimization algorithms for radio links is difficult to adapt to the constraints of optical inter-satellite links (OISLs) since the communication mechanism of an OISLs is quite different from that of a radio link, a topological optimization algorithm, known as multi-objective discrete binary particle swarm optimization, was researched by [PERSON] et al. [29] to optimize the topology of OISLs for future navigation satellites. However, they divided the orbit period of satellites into several static topological periods for adaptation to the dynamic characteristics of satellites, which means no dynamic property existing in the topological sub-cycle. Using the visibility matrix as a mathematical representation of the graph, a visibility matrix sequence can be formed by sorting the visibility matrix at each moment, which reflects the dynamic changes in the network topology [30]. In the visibility matrix, the value of each element depends on whether the distance between the two satellites satisfies the visibility distance constraint, i.e., it indicates the existence of a link when it is 1, while there is no link when it is 0. The static routing algorithm, utilizing the predictability and periodicity of satellite networks to deal with the dynamics of the topology of satellite routing, holds the opinion that the status of the network topology is unchanged during a certain period of time. Moreover, in the process of dealing with the dynamic topology of satellite networks, only two states, i.e., on and off [31], are assigned to the satellite topology. However, due to the influence of the dynamic characteristics of satellite networks, the state of inter-satellite links varies with the movement of satellites. If the inter-satellite link model is only characterized by on-off without considering the link attributes, the ISL quality of the network topology cannot be quantified [32]. Thus, [PERSON] et al. [32] proposed a novel time-varying topology model, a weighted time-space evolution graph, to describe the continuous changing process of topology between snapshots and reflect the quality of routing paths by the weight of link utility. But the real-time nature of satellite topology changes still cannot be mapped to this dynamic topology. The methods above generally decompose the system cycle into multiple time periods, and use a certain static network structure to refer to the network topology in each time period. With a time-series connecting the network topology of each time period, the network topology of the system cycle can be formed. The adjacency matrix is used as a mathematical description method to express the network topology. It is obvious that the description of dynamic characteristics is discrete in the time dimension, and cannot reflect the dynamic characteristics of the system network topology in each time period. In the corresponding adjacency matrix, each element is a binary function with a value of 0 or 1. The shortcoming of those methods is that the established network topology of the space information system cannot evaluate the trend of link quality with time, which results in a non-optimal solution of route planning when considering the constraints of link transmission quality.