## 2 System Model
### Encounter Probability Analysis Model  
As mentioned above, there are three different types of node cooperation in the considered integrated UAV-ICV networks. In this paper, when communication node \(i\), which carries a message copy, moves into the communication range of the next-hop relay node \(j\), communication nodes \(i\) and \(j\) are referred to as a communication node pair, denoted as \(\varphi_{x}=(i,\,j)\). All communication nodes in integrated UAV-ICV networks can be represented by a set \(\varphi\), \(\varphi=\{\varphi_{1},...,\varphi_{x},...,\varphi_{n}\}\). Based on the different node types, there are nine distinct states of communication node pairs in integrated UAV-ICV networks, namely, (Node(\(i_{1}\)), Node(\(i_{1}\))), (Node(\(i_{1}\)), Node(\(i_{2}\))), (Node(\(i_{1}\)), Node(\(i_{3}\))), (Node(\(i_{2}\)), Node(\(i_{1}\))), (Node(\(i_{2}\)), Node(\(i_{2}\))), (Node(\(i_{3}\))), (Node(\(i_{3}\)), Node(\(i_{1}\))), (Node(\(i_{2}\)), Node(\(i_{2}\))), (Node(\(i_{3}\))), (Node(\(i_{3}\)), Node(\(i_{1}\))), (Node(\(i_{3}\)), Node(\(i_{2}\))), and (Node(\(i_{3}\)), Node(\(i_{3}\))). Therefore, depending on the node type, three different behaviors are generated: (1) forwarding all message copies of the nodes; (2) forwarding message copies of some nodes; (3) refusing to forward message copies of all other nodes. In this situation, we need to consider both the reception and packet loss of message copies to evaluate the behavior of communication nodes. When communication node \(i\) moves into the communication range of the next-hop relay node \(j\), the probability that node \(j\) successfully receives the message copy from node \(i\) mainly depends on the size of the message copy, the number of copies, and the cache size of node \(j\). Let \(t\) be the current time and \(\Delta t\) be the duration of contact between nodes \(i\) and \(j\). By using (4), this paper derives the cumulative distribution function (CDF) \(F(X)\) of the random variable \(X\) as \[F(X)=\int_{-\infty}^{x}f(t)\mathrm{d}t, \tag{4}\]  
where \(f(t)\) is the probability density function (PDF) of successfully receiving a message copy for a pair of communication nodes. We have \(f(t)\geq 0\), \(t\in(-\infty,\ +\infty)\), and \(\int_{-\infty}^{+\infty}f(t)\mathrm{d}t=1\). Similarly, the PDF \(f(\Delta t)\) of \(\Delta t\) is given by  
\[f(\Delta t)=\lambda_{i,j}\times\exp\left(-\lambda_{i,j}\Delta t \sum\limits_{k=1}^{n}X_{k}(t)\right), \tag{5}\]  
where \(\lambda_{i,j}\) is the encounter probability between nodes \(i\) and \(j\), and \(X_{k}(t)\) is an index that represents whether the node buffer is full, \(X_{k}(t)\in\{0,\ 1\}\). In addition, \(X_{k}\) can be further expressed as  
\[X_{k}=\left\{ \begin{array}{ll}1,&L_{u}(i)\leq L_{r}(j)+\sum\limits_{z(m)}L_ {m}(j),\\ 0,&L_{u}(i)>L_{r}(j)+\sum\limits_{z(m)}L_{m}(j),\end{array} \right. \tag{6}\]  
where \(L_{u}(i)\) is the size of the message copy that node \(i\) needs to transmit; \(L_{r}(j)\) is the remaining cache size of node \(j\); \(L_{m}(j)\) is the size of the message copy cached by node \(j\); \(z(m)\) is the priority function, which is used to indicate that the priority of the message copies cached in node \(j\) is lower than the number of message copies transmitted by node \(i\). According to the above content, we can calculate the probability \(P_{\text{suc}}(i,\ j)\) of a pair of communication node \((i,\ j)\) successfully receiving message copies within \(\Delta t\) time, which can be expressed as  
\[P_{\text{suc}}(i,\ j) =F(t,\ \Delta t)\] \[=\int_{t}^{t+\Delta t}f(t,\ \Delta t)\mathrm{d}\Delta t \tag{7}\] \[=\int_{t}^{T_{\text{max}}}\lambda_{i,j}\Delta t\sum\limits_{k=1} ^{n}X_{k}(t)\mathrm{d}\Delta t,\]  
where \(T_{\text{max}}\) is the TTL of the message copies. After the relay node successfully receives the message copy, it may refuse to forward the message copy to non-destination nodes due to selfish behavior and drop this message copy.