# A Robust and Energy-Efficient Weighted Clustering Algorithm on Mobile Ad Hoc Sensor Networks +
## 3 Description of RE2 WCA Algorithm
### Weight Model and Hybrid Clustering Algorithm  
After the initialization of the network, one node within cluster announces the information of its cluster head. The cluster head needs to be rotated to prolong the survivability and lifespan of the network. Therefore, we put forward the weight probability average. The high weight value of node implies it has more possibility to be selected as cluster head. Hence, a weight model is proposed and its measurement is given by Equation (8):  
\[weight(i)=C_{prob}\times\left[a\times\frac{E_{i}^{residual}}{E_{i}^{initial}} +b\times CHSD_{map}(i)\right] \tag{8}\]  
where \(a\) and \(b\) are weighted coefficient index of residual energy and group mobility feature respectively; _CHSDmap_(_i_) is the spatial dependency with _CH_ at one moment after linear mapping from [-1, 1] to [0, 1]; _weight(i)_ denotes one node probability to be _CH_; _CPprob_ is an initial iteration probability. It is obvious that the range of _weight(i)_ value is limited with (0, 1). Additionally, the summation of \(a\) and \(b\) is set as 1. Generally, \(a>b\) is rendered. The adequate residual energy of the node implies that it can manage a cluster better than other nodes. In addition, the high group mobility of _CHSD_ demonstrates that one of node within cluster has a high correlation with its cluster head. Hence, in order to make the times of re-clustering due to nodes' mobility, the node with a high _CHSD_ value is elected as the cluster head in the group. In addition, we use _CPprob_ to limit the announcement of cluster head. Through calculating the weight value above, it is concluded that the nodes with high group mobility and residual energy is more suitable to be the cluster head. Through limiting iteration of cluster head, a _CH_ selection protocol is put forward. In order to prolong the lifetime of the network, we regard _average minimum reachability power_ (_AMRP_) as the cost of election. Compared with the other types of cost, such as _node degree_ or _1/node degree_, the _AMRP_ is a compromise between network density and load balancing. It has been proved that HEED has fewer iterations than General Cluster (GC) Scheme [34]. In this paper, we consider a greater minimum iteration threshold \(p_{thr}\). Due to the mobility, the radius of cluster or group will be greater. However, with the increase of the radius, the iterations of GC are dramatically increased but HEED has to stop iterating for the \(p_{thr}\). Considering mobility of nodes, increasing \(p_{thr}\) appropriately will reduce the iterations so that save energy. Equation (9) demonstrates the calculation:  
\[CH_{prob}=\max(weight(i),p_{\min}) \tag{9}\]where \(CH_{prob}\) represents the probability of each node to be \(CH\). Through a limited iteration, \(CH\) will be elected. In every iteration, the \(CH_{prob}\) will be doubled and until the cluster formation accomplished or finished beyond the threshold. The iteration times can be calculated:  
\[N_{iter}\leq\left\lceil\log_{2}\frac{1}{p_{\min}}\right\rceil+1 \tag{10}\]  
Therefore, \(N_{iter}\approx O(1)\). If \(p_{thr}\) sets to \(10^{-4}\), there is 15 iterations to end up. Actually, the greater number of iterations will consume much energy, thus it is necessary to select an appropriate iteration threshold. We define the ratio of residual energy of \(CH\) and average energy of cluster is \(ER\) (energy ratio) as Equation (11) described:  
\[ER=\frac{E_{ch}^{residual}}{E_{k}^{new}} \tag{11}\]  
In addition, \(p_{min}=\max\left\{ER,p_{thr}\right\}\) is chosen. For Equation (11), if residual energy of cluster head is equal to the average energy of cluster approximately, \(ER\) is about 1. Therefore, the times of iteration is limited within 2. Considering an extreme case, if when the residual energy of \(CH\) is close to \(10^{-3}\) J (a totally unsuitable residual energy) and cluster average energy is 1 J, then \(ER=10^{-3}\). Thus, if we set initial \(p_{thr}=10^{-2}\), then the iteration times will be limited within 7 rather than 10.