# A Robust and Energy-Efficient Weighted Clustering Algorithm on Mobile Ad Hoc Sensor Networks +
## 3 Description of RE2 WCA Algorithm
### Mobility Measurement  
MOBIC [19] is a mobility-aware clustering scheme. Instead of using the node's ID, MOBIC adopts the relative metric to measure the mobility of the nodes. However, if mobile nodes move in an arbitrary manner, the diversity with other nodes is dominant. Even if some nodes have similar motion but a few others move differently. It still results in dramatic increase in the variance. It shows the variance is greatly influenced by the diverse motion. In the paper, a group mobility protocol is adopted to measure whether a node has possibility to be elected as cluster head. The higher group mobility of one node implies it has the similar mobility pattern with the majority of its neighbors [14]. Thus, it suits to be elected as \(CH\). Figure 1 shows the derivation process and motion model of the nodes. Based on record of last movement, the linear displacement of each node can be calculated in real time. The linear displacement of each node will be figured out as soon as nodes move in the network. Then, the relative direction and speed ratio are considered to calculated the spatial dependency between nodes. In addition, the history cache can be updated. As is shown in Figure 1, the displacement of node \({}_{i}\) is lower than \(D_{thr}\) and consequently this record will not be updated into the history cache. From moment \(T\) to \(t^{\prime}\), the displacement \(D^{\prime}\) of node \({}_{j}\) is greater than \(D_{thr}\) and consequently the history cache is updated. In addition, from moment \(t^{\prime}\) to \(t\), movement of node \({}_{j}\)\(D\) is still greater than the \(D_{thr}\), hence the record will be entered into the history cache. Hence, only when node moves, the calculation process will be implement and not all nodes need to save their record into cache due to the cluster radius threshold. Nodes which have the frequent and great displacement will consume more energy for their mobility. Therefore, it is small probability of these nodes to be elected as CHs. Within a cluster or a group, the \(CH\) needs to be rotated for the homogenization of energy. Here, we define the spatial dependency with \(CH\) as \(CHSD\) which demonstrates that each round cluster members need to calculate the spatial dependency with current \(CH\). In order to describe the spatial dependency with \(CH\) (\(CHSD\)), the following measurements will be demonstrated. Table 1 gives the detailed explanation of parameters used in Figure 1.  
According to Algorithm 1, relative direction (RD) and speed radio (SR) at one moment \(t\) between one node \(i\) and cluster head are measured respectively as \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Explanation** \\ \hline \(D_{thr}\) & Cluster radius \\ \(D\) & Distance of displacement \\ \(\varphi\) & Angel of direction \\ \(dx\) & Lateral displacement \\ \(dy\) & Longitudinal displacement \\ \(\hat{S}(t)\) & Speed of node at one moment t \\ \(\theta(t)\) & Direction of node at one moment t \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters Explanation. Figure 1: Nodes displacement schematic. \[RD(i,ch,t)=\cos\bigl{(}\hat{\theta}_{i}(t)-\hat{\theta}_{ch}(t)\bigr{)} \tag{5}\] \[SR(i,ch,t)=\frac{\min\bigl{(}\hat{S}_{i}(t),\hat{S}_{ch}(t)\bigr{)} }{\max\bigl{(}\hat{S}_{i}(t),\hat{S}_{ch}(t)\bigr{)}} \tag{6}\]  
Hence, the _CHSD_ can be calculated by:  
\[CHSD(i,ch,t)=RD(i,ch,t)\times SR(i,ch,t) \tag{7}\]  
The range of _CHSD_ is between [-1, 1]. If _CHSD_ is very close to 1, it indicates the nearly same motion with _CH_ otherwise the less similarity with _CH_. _CH_ has the feature of maximum similarity and lower mobility and it is elected in the last round. However, in the current round, _CH_ is a very ordinary node and it still needs to calculate the speed, the angle. But nodes now will not compare with each other, instead, with _CH_. It is the improvement compared with original measurement. It dramatically decreases the complexity of calculation and reduces great energy consumption.