# Temporal Continuity Expression for Network Topology of Space Information Systems
## 2 Link Relationship Graph--The Time Continuity Model of Network Topology  
In a space information system, the following conditions need to be satisfied for link construction between two satellites. Firstly, the two nodes are visualized [33; 34; 35; 36]; secondly, the distance between the nodes meets the power requirements for the signal transmission and reception [35; 36]; thirdly, the antennas of the satellites adapt to the relative motion relationship between two nodes [37]. The link cannot be effectively established when the relative motion relationship between nodes exceeds the antenna follow-up tracking ability, even if the first two conditions are satisfied. We assume that the tracking ability of the antenna is sufficient to support the demand for link construction, and only consider the impact of the third condition on link quality, which is related to the relative distance relationship and relative motion relationship between nodes.  
The dynamic characteristics of the link relationship graph are composed of two layers. The first layer is the feasibility of link construction, which is a binary function that varies with time. And the second layer is the quality of the link, which is a time-dependent continuous function (or a piecewise continuous function).  
The link relationship graph is a directionless weighted graph \(G=(V,E(\delta(t)))\), in which \(V=\{S_{1},S_{2},\cdots,S_{n}\}\) represents the set of nodes of the space information system, and \(E(\delta(t))=\left\{\delta_{ij}(t)|i,j\in V,i\
eq j\right\}\) represents the set of links (including link quality) established between the nodes.  
A time-related adjacency matrix is used as a mathematical expression of the graph. The matrix is a symmetry matrix \(A\triangleq\left\{\delta_{ij}(t)\right\}_{N\times N}\), wherein the variable \(\delta_{ij}(t)\) represents the link state between the nodes. If there is no link between nodes, then the value of \(\delta_{ij}(t)\) is equal to 0. Otherwise, \(\delta_{ij}(t)=\frac{1}{3}\sum\limits_{k=1}^{3}\sigma_{k}(t)\), which indicates that there is a link between the nodes with the value of link quality equal to \(\frac{1}{3}\sum\limits_{k=1}^{3}\sigma_{k}(t)\). In particular, there is no self-closing link in any node, so \(\delta_{ij}(t)=0\) when \(i=j\).  
The physical layer dynamic characteristics of the space information system are described by the geometric linkable factor \(\sigma_{1}(t)\), a binary function with a value of 0 or 1, which represents the feasibility of inter-satellite link construction in the system cycle. While the transport layer dynamic characteristics indicating the link quality during the connection period between two satellites of the space information system should be characterized by the link distance intensity factor \(\sigma_{2}(t)\) and the relative angular velocity factor of the node \(\sigma_{3}(t)\), both of which are piecewise continuous functions that vary with time. Specifically, \(\sigma_{2}(t)\) indicates the link quality affected by the distance between nodes, and \(\sigma_{3}(t)\) indicates the link quality affected by the relative motion relationship between nodes.