# On Ad Hoc Communication in Industrial Environments
## 5 Examination of the Industrial Ad Hoc Channels
### Network Connectedness  
The network connectedness is the average percentile of neighboring nodes (encountered nodes with which direct data exchange is possible) [33]. A connection can be established if an encounter is registered. It is assumed that two nodes could communicate for the time of 1 dt after an encounter was registered. Within a network of \(n\) nodes, \(n-1\) connections are simultaneously possible for any node. Hence, nodes cannot connect to themselves. The network connectedness of the network at time \(t\) is then defined as  
\[N(t)=\frac{1}{N}\sum_{i=1}^{N}\frac{1}{n-1}\left(\sum_{j=1}^{N}C(i,j,t)\right)-1 \tag{8}\]  
where \(C\) is the NEP, as described in Section 4, and \(i\) and \(j\) iterate over \(N\) nodes (transmitter and receiver) in the examined network. When de-normalized and averaged over the time of the recording, the network connectedness is equal to the previously used metric \(N_{\mathcal{O}}=\frac{1}{T}\int_{0}^{T}nN(t)dt=N_{\mathcal{O}}^{TP}\). Figure 1 illustrates the meaning of different network connectedness values. A network connectedness \(\mathcal{C}_{N}=0\) describes a network where no nodes are connected. The network connectedness increases once connections are available. If all nodes can reach all other nodes the network connectedness reaches 1.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Parameter Name**} & \multirow{2}{*}{**Unit**} & **Reference** & \
\begin{tabular}{c} **Static** \\ **Industry** \\ **Test** \\ \end{tabular} &
\begin{tabular}{c} **Mobile** \\ **Industry** \\ **Test** \\ \end{tabular} & \multirow{2}{*}{**Description**} \\ \hline \(dt\) & s & 0.2 s & 0.2 s & 0.2 s & Time resolution of the NEP \\ \(T\) & s & \textgreater{}11,800 s & \textgreater{}8800 s & \textgreater{}10,400 s & Run time of measurement \\ \(N\) & & 6 & 7 & 8 & Number of nodes \\ Mobility & Type & None & Group & AGV & Type of mobility \\ Environment & Type & Office & Industry & Industry & Environment description \\ \hline \hline \end{tabular}
\end{table}
Table 1: Measurement parameter description and values for measurements.  
Figure 1: Illustration of network connectedness and probability for unidirectional channels with three example networks.  
The network connectedness of the three examined measurement configurations are displayed in Figure 2. On average, the network in the reference measurement is fully connected. This means all nodes can communicate with all other nodes. Short-lived variations occur due to interference between the nodes and interference with other wireless communication systems within the same spectrum. The second measurement set shows slightly lower connectedness within the network. This indicates that the industrial environment might contain more sources for interference than the office environment. The variations in network connectedness indicate spatial correlation. The NEP of mobile nodes in the industrial environment exhibits the lowest network connectedness and even higher variations in connectedness as the static measurement in the industrial environment. In Figure 2 it can be seen that the effect of mobility is far more pronounced than the one of interference.