## 3 Methodology
### Air-Sea Cross-Domain Communication Network Model
By combining the above formulas with simple trigonometric derivations, the formula for the relative velocity \(v\) can be derived. Since the time interval \(\Delta t\) is very short, and to simplify the calculation, the velocity \(v_{uv}\) can be approximated as constant. Therefore \(d_{13}\approx v_{uv}\cdot\Delta t\), the formula for **v** can be simplified as follows:  
\[v_{uv}=\frac{f\cdot R_{m}\sqrt{2c|f-f^{\prime}|\frac{\sqrt{P_{v _{v}}\mathcal{P}_{\min}}}{f^{2}R_{\max}}-P_{v2}-P_{v3}}}{\sqrt{P_{\min}}}, \tag{4}\]  
To avoid selecting links with a very short lifetime for nodes, the received power at the edge nodes corresponding to the maximum communication radius in the network is set as the power threshold as follows:  
\[P_{threshold}=\frac{P_{s}G_{s}G_{r}\lambda^{2}}{(4\pi)^{2}\Big{(}R_{\max}-2v_{ \max}t_{delay}\Big{)}^{2}L}, \tag{5}\]  
where \(t_{delay}\) represents the communication delay. Therefore, the signal stability between nodes u and v is defined as follows:  
\[S_{uv}=p\cdot\frac{P_{uv}-\max(\min(P_{uv}),P_{threshold})}{P_{s}-\max(\min(P_ {uv}),P_{threshold})}+q\cdot\frac{v_{\max}-v_{uv}}{v_{\max}}, \tag{6}\]  
where \(P_{uv}\) represents the signal power intensity between nodes u and v; \(v_{uv}\) is the relative motion velocity between nodes \(u\) and \(v\); \(P_{s}\) is the transmission power; \(v_{\max}\) is the maximum velocity of the node; and \(p\) and \(q\) are the weight coefficients for power and velocity, satisfying the condition \(p+q=1\). The load \(L_{uv}\) between nodes u and v is defined as the arithmetic average of the message queue lengths of nodes u and v at the current moment. The smaller the load at both ends of the link, the smoother and faster the communication between the nodes, resulting in a lower communication cost between the nodes. Under normal conditions, the smaller the relative motion velocity between two nodes, the stronger the signal intensity and the smaller the link load, making the communication between the nodes simpler and the communication cost lower. Conversely, the communication cost increases. It can be observed that the signal strength is inversely proportional to the link load. The ratio of link load to signal stability is defined as the communication cost, as expressed in the following formula:  
\[w_{uv}=\left\{ \begin{array}{l}\frac{k_{L}L_{uv}}{k_{S}s_{uv}},P_{uv}>P_{ threshold}\\ \inf,P_{uv}\leq P_{threshold}\end{array} \right., \tag{7}\]  
where \(k_{L}\) is the weight of the link load, and \(k_{S}\) is the weight of the signal stability. After calculating the communication cost of the air-sea cross-domain communication network  
Figure 3: The diagram illustrating the relative motion between nodes. and combining the theory of undirected weighted graphs, the network model can be obtained as follows:  
\[W= \begin{bmatrix}w_{11}&w_{12}&\ldots&w_{1n}\\ w_{21}&w_{22}&\ldots&w_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ w_{n1}&w_{n2}&\ldots&w_{nn}\end{bmatrix} , \tag{8}\]  
where \(W\) represents the network communication cost weight matrix, which is an n-order matrix. When \(w_{uv}\) is a real number, it indicates that a link can be established between node \(u\) and node \(v\). The larger the value of \(w_{uv}\), the higher the communication cost. When \(w_{uv}\) is infinite, it means that a link cannot be established between node \(u\) and node \(v\) beyond the maximum communication distance.