## 3 Methodology  
### Air-Sea Cross-Domain Communication Network Model  
This paper assumes that there are only three kinds of nodes in the air-sea cross-domain communication network. The nodes above the water are unmanned aerial vehicles (UAVs), the nodes on the water are unmanned surface vessels (USVs), and the nodes under the water are unmanned underwater vehicles (UUVs). The whole network moves in formation mode. Within each spatial domain, each communication entity communicates with its nearest neighbor node. For instance, in the airspace, each UAV node communicates with its nearest neighbor node. Between spatial domains, only surface nodes can communicate with both aerial and underwater nodes. Aerial nodes can only communicate with surface nodes and cannot communicate with underwater nodes. Similarly, underwater nodes can only communicate with surface nodes and cannot communicate with aerial nodes. It is assumed that all nodes of the same type have the same structure and motion capability. The movement mode of the whole network is the reference group movement model. The network structure is shown in Figure 1. The air-sea cross-domain communication network is modeled by graph theory. The nodes and links in the network are mapped to vertices and edges in graph theory, and the problem of link reconnection is transformed into the problem of edge reconnection in graph theory. It is assumed that the overall movement of the network is consistent, all nodes move within the same speed range, and the transmission power is consistent. The network structure is mapped by graph theory, and the sequence of UAV1-10, USV1-10 and UUV1-10 is sorted and mapped as shown in Figure 2.  
Figure 1: Schematic diagram of air-sea cross-domain communication network. Considering the multi-layer architecture, long distance, and time-varying topology of the air-sea cross-domain communication network, a topology model of the air-sea cross-domain communication network is established based on the communication connections between network nodes per unit time. By using graph theory [28], the network is abstractly mapped into a graph G (V, E, W), where V represents the set of nodes, E represents the set of edges, and W represents the communication weight matrix. \(e_{ij}\) represents the edge between node \(i\) and node \(j\); \(w_{ij}\) represents the communication weight between node \(i\) and node \(j\). The communication cost weight is determined by signal stability and load. The signal stability is defined as the weighted sum of the power intensity and relative speed between nodes. A higher signal power between nodes indicates better link quality between them. The interpolation of the power at adjacent moments is used as the condition for solving the relative motion between nodes. As shown in Figure 3, at time \(t\), the relative positions of nodes u and v are 1 and 2, and after the movement \(\Delta t\), the relative positions of nodes u and v change to 1 and 3 with a relative velocity of \(v_{uv}\). The power of node v at position 2 is \(P_{v2}\) with a frequency of \(f_{v2}\); at position 3, the power is \(P_{v3}\) with a frequency of \(f_{v3}\). When the received power of the node is \(P_{\text{min}}\), the corresponding maximum communication distance between the transmitting and receiving nodes is \(R_{\text{max}}\). Therefore, the distance between node \(v\) at positions 2 and 3 and node u is as follows:  
\[d_{12}=\sqrt{\frac{P_{v2}}{P_{\text{min}}}}R_{\text{max}}, \tag{1}\]  
\[d_{13}=\sqrt{\frac{P_{v3}}{P_{\text{min}}}}R_{\text{max}}, \tag{2}\]  
Assuming that all nodes in the network transmit at the same frequency, denoted as \(f\), then \(\Delta t=\frac{1}{f}\), and the frequency of the carrier received by node \(v\) from node \(u\) is \(f^{\prime}\). According to the Doppler effect, it can be inferred that  
\[v_{uv}=\frac{c|f-f^{\prime}|}{f\cos\angle 2}, \tag{3}\]  
Figure 2: Map of air–sea cross-domain communication network.