# Unmanned Vehicles' Placement Optimisation for Internet of Things and Internet of Unmanned Vehicles
## 5 Towards Algorithm Development: Propagation Concepts, Path Loss and Communication Graph
### Path Loss Graph. Communication Graph  
Further, we introduce the concept of a path loss graph, which is based on the path loss and link budget parameters. Let \(G=(V(G),E(G),\psi_{G})\) be the path loss graph, a finite, non-trivial, directed graph where  
\[V(G) =(\
u_{1},\
u_{2},\dots,\
u_{N})\] \[E(G) =(e_{1},e_{2},\dots,e_{N^{2}})\]  
are the network nodes (UVs, GW, WSNs) and the communication links between every two such nodes. The computation of each weight, \(w\) (Equation (23), involves (1) the path loss computed between every two nodes of the network following the specific propagation model and (2) the link budget available for the technical solution used at a particular time, \(LB\) (Table 1). \[w_{i,j}=\frac{PL_{\
u_{i},\
u_{j}}}{LB}+\zeta \tag{23}\]  
with \(w_{i,j}\in[\zeta,\infty)\), where \(\zeta\) is infinitesimal. The presence of infinitesimal \(\zeta\) is required for implementation purposes, as \(w_{i,j}\
eq 0\) even when \(PL_{\
u_{i},\
u_{j}}=0\). The communication link is said to be ensured if \(PL_{\
u_{i},\
u_{j}}<LB\) and, consequently, \(w_{i,j}>1+\zeta\)  
Further, we introduce the concept of a communication graph, which is a subgraph \(H\) of the path loss graph G, where all the edges with weights higher than \(1+\zeta\) were removed, \(H\subseteq G\), \(E(H)=E(G,w_{i,j}\leq 1+\zeta)\). Therefore, the communication graph only comprises the edges corresponding to the nodes that communicate between each other (\(PL\leq LB\)). We must recall that graph representation does not give visual information about the distances or the position in the space of the nodes. With all these concepts discussed, Table 7 highlights the environment, channels, scenarios and corresponding path loss models employed throughout the rest of the paper. Figure 6: Path loss, link budget and critical distance for USV–USV link, when USV antenna height is \(h_{t}=0.5\) m and the solution in [57] is used. Based on Equation (23) and technical solutions in [57; 58; 59], the communication graph for each solution was determined (Figures 7-9). The positioning of the nodes in the 3D space is kept as in Figure 2 for all four technical solutions, and the height threshold for which we applied the FSPL propagation model for the A2A channel is \(h=60\) m. The graph vertices are the network's nodes. The blue lines are the edges of the graph. If there is a communication link between two nodes, then such an edge (blue line) will connect them. The existence of two edges between each of the two nodes represents a bidirectional link. We mention that the position of the graph's vertices or the length of the edges are not related to the position of the nodes in 3D space. Instead, the layout of the graph was automatically chosen by the simulation tool based on the graph's size and structure. The figures on each edge represent the edges' weight computed based on Equation (23). Based on Figures 7-9, we notice that the scenarios dramatically change as the communication technology cannot ensure the communication link. Thus, we highlight the importance of a positioning algorithm that adapts to the performances of the communication technology employed at a certain time. Table 8 summarises the results obtained in terms of several key performance indicators (KPI), such as: the number of connected nodes, number of edges, minimum and maximum weights. We remark that the communication link cannot be ensured between the gateway and each WSN for any of the technical solutions above. \begin{table}
\begin{tabular}{c c c c} \hline
**Environment** & **Channel** & **Scenario** & **PL Model** \\ \hline \multirow{8}{*}{ Rural} & A2G & UAV-GW & \multirow{2}{*}{3 GPP} \\  & A2G & UAV-WSN & & \\  & A2A & UAV-UAV & & \\  & A2A & UAV-UAV & & \\  & FSPL & (\(h\geq 60\) m) \\ \cline{1-1}  & G2G & WSN-GW & & \\  & S2G & USV-WSN & & \\  & OW & USV-USV & & 2-ray \\  & A2S & UAV-USV & & \\ \hline \end{tabular}
\end{table} 
Table 7: Proposed environment, channels, scenarios and path loss models for aerial, terrestrial and water communications.