# Design of Multi-Modal Ship Mobile Ad Hoc Network under the Guidance of an Autonomous Ship
## 4 Connectivity Properties of Ship MANET
### Link Existence Possibility and Link Holding Time Calculation  
During the voyage, the communication link can be maintained only when the ships are within the communication range; otherwise, the communication link will be interrupted. The possibility of link existence reflects whether a node can form an inter-ship link with other nodes in the network under the expected encounter scenario. It can be seen that this probability can be used as one of the references for the design of the clustering routing algorithm. The link hold time calculates how long an inter-ship link in the network can be maintained, that is, the two nodes are within the communication range of each other. This result is used as a time threshold, and the interaction process related to this node must be completed within the existence period of this link.  
In the \"approaching then leaving\" scenario, as shown in Figure 7. In the figure, the red circle represents the communication range of Ship 1, the blue circle represents the communication range of Ship 2, the black dashed arrow represents the communication radius of the ship, the green dashed line represents the relative motion line of the two ships, the green dashed arrow represents the relative direction of motion and the purple solid line represents the distance between the two ships. It is assumed that at time \(t_{0}\), the target ship is at \(T\), the relative motion speed is \(V_{r}\), and the communication radius of Ship 1 and Ship 2 is \(R_{1}\) and \(R_{2}\), respectively. If the two ships both keep course and speed, it can be seen that when Ship 2 sails to A (at time \(t_{1}\)), the two ships enter the communication range of each other and sail to B (at time \(t_{2}\)), the two ships leave each other's communication range. After time \(t_{2}\), such as Ship 2 said to C at time \(t_{3}\), both ships were no longer within their communication range of each other. Then for ship 2, its connectivity probability at \(T\):  
\[P_{c}=\frac{|AB|}{|TA|+|AB|} \tag{15}\]  
It can be seen that when \(|TA|\to 0\), the connectivity probability \(P_{c}\to 1\).  
When Ship 2 moves along \(TA\), its connectivity probability gradually increases within a certain time range.  
When Ship 2 is between \(AB\), \(P_{c}=1\).  
Figure 7: Link calculation in approaching then leaving scenario.  
When Ship 2 is at \(C\),  
\[P_{c}=\frac{|AB|}{|AB|+|BC|} \tag{16}\]  
It can be seen that the connectivity probability is gradually reduced when advancing along \(BC\).  
When \(|BC|\rightarrow+\infty\), the connectivity probability \(P_{c}\to 0\).  
And retention time:  
\[T_{m}=\frac{|AB|}{|V_{r}|} \tag{17}\]  
The \"leaving scenario\" was shown as Figure 8. In the figure, the red circle represents the communication range of Ship 1, the blue circle represents the communication range of Ship 2, the black dashed arrow represents the communication radius of the ship, the green dashed green dashed line represents the relative motion line of the two ships, the green dashed arrow represents the relative direction of motion and the purple solid line represents the distance between the two ships. At the initial time \(t_{0}\), the target ship is at \(T\), and the relative motion speed is \(V_{r}\). Similarly, the communication radius of Ship 1 and Ship 2 is \(R_{1}\) and \(R_{2}\), respectively. Ship 2 proceeds along the current relative motion line. At time \(t_{2}\), Ship 1 and Ship 2 are still connected, so at B at the time \(t_{3}\), it is a connected boundary.  
When Ship 2 fails between \(T\) and \(B\), there is \(P_{c}=1\).  
After leaving B, the connectivity probability gradually becomes smaller and finally has \(P_{c}\to 0\).