## 3 Concepts and Models
### Network Modeling  
Network modeling plays a pivotal role in the research, design, and operation of networks. Scientific network modeling can assist us in gaining a profound understanding of network structures and relationships, accurately unveiling the intrinsic patterns and characteristics of networks. The relationship between networks and graph theory is inseparable. Graph theory offers a concise and powerful method for representing various complex relationships within networks, whether they are computer networks [20], satellite networks [21], transportation networks [22], social networks [23], or networks from other domains. In networks, nodes can correspond to vertices in graph theory, while links can correspond to edges. If the time-varying satellite-UAV-integrated networks can be abstracted as a graph, we can leverage the rich tools and methods of graph theory to analyze and understand the network's structure, characteristics, and behavior. The topological relations of the satellite-UAV-integrated networks can be represented by the geometric representation \(\mathcal{G}=(\mathcal{V},\mathcal{E})\) in graph theory. Where \(\mathcal{G}\) denotes the topological model, and \(\mathcal{V}\) denotes the set of nodes in the topological model with size \(\mathcal{N}\). \(\mathcal{E}\) denotes the set of edges in the topological model with size \(\mathcal{M}\). The unique \(\mathcal{G}\) can be determined by combining \(\mathcal{V}\) and \(\mathcal{E}\). \(\mathcal{G}\) records the information of all nodes and all edges, which is unique and irreducible. The process of abstracting a time-varying dynamic satellite network into a graph structure is shown in Figure 2.  
The delay in satellite-UAV-integrated networks mainly comes from four sources: uplink delay, downlink delay, transmission delay between satellites, and internal calculations and processing delays. Now, suppose there is a satellite-UAV-integrated network with \(N\) satellites and drones. A routing path \(\mathcal{P}\) is made up by an order of \(M\) nodes \(v_{1},v_{2},\ldots,v_{M-1},v_{M}\), and the internal processing delay of node \(v_{i}\) is \(In_{i}\). Then, the total delay for routing path \(\mathcal{P}\) in the time slice \(s\) is  
\[delay(s,v_{1},v_{M})=\sum_{i=1}^{M}In_{i}+\sum_{i=1}^{M-1}distance(s,v_{i},v_{ i+1})+T_{up}++T_{down} \tag{1}\]  
For simplicity, the distance between node pairs replaces the transmission delay between nodes. \(distance(s,v_{i},v_{i+1})\) is the transmission delay between node \(v_{i}\) and \(v_{i+1}\) in time slice \(s\), assuming that the internal delay \(In_{i}\) of all satellites in the network is almost the same and is \(T_{in}\). Further, considering that the delay is the same for uplink \(T_{up}\) and downlink \(T_{down}\) for routing path \(\mathcal{P}\) with the same start and end points, the equation can be simplified as follows:  
\[delay(s,v_{1},v_{M})=MT_{in}+\sum_{i=1}^{M-1}distance(s,v_{i},v_{i+1}) \tag{2}\]  
For old satellites, their computing power is usually low, resulting in the internal delay being far greater than the transmission delay of the signal. This means \(In_{i}\gg distance(v_{i},v_{j})\), so the model can be simplified as  
\[delay(s,v_{1},v_{M})\approx MT_{in} \tag{3}\]  
Figure 2: The time-varying dynamic satellite–UAV-integrated network is abstracted into a graph structure. In a more abstract view, the network forms a graph with undirected edges, in which the distances between node pairs in graph theory are equal to the transmission delays between satellites. The delay only depends on the number of nodes on the routing path \(\mathcal{P}\). For this condition, an unweighted constellation model with an \(N\times N\) edge matrix \(\mathcal{V}_{i,j}\) can be set for further calculation. In a time slice, \(\mathcal{V}_{i,j}=1\) if node \(v_{i}\) and \(v_{j}\) are mutually visible, and \(\mathcal{V}_{i,j}=\infty\) if invisible.