## 4 Description of MRP
### Phase I: Cluster Formation  
The conventional hierarchical routing algorithms [11, 12, 27] do not fit for monitoring burst events in reactive WSNs. For example, though there is no event happening, each node will still have to flood control packets to periodically reconstruct clusters. Clustering is not related to an event, _i.e._, nodes sensing the event locate in different clusters, which will reduce the data aggregation efficiency. If the clusters are reconstructed, an event may happen, and result in the event not being detected. In order to overcome the disadvantages [11, 12, 27], MRP adopts a dynamic clustering algorithm, _i.e._, nodes having information about an event taking place nearby will join clustering. The clustering algorithm obeys the rules as follows:  
* Nodes located in the event area can sense the distance to the event according to _RSS_. * Nodes know the residual energy of neighbor nodes in the event area. * If \(RSS_{i}\geq\) Threshold Value [29], node \(i\) locates in the event area. (\(RSS_{i}\) is the received signal strength of node \(i\))  
**Theorem 2**: _When a CH locates in the center of the event area, the sum of energy consumption for transmitting data is the least in the cluster._  
_Prove:_ There is \(m\) nodes distributed in a cluster. Node \(i\) locates the center of the event area with coordinate (0, 0). From equation (1), energy consumption of communication between nodes is directly proportional to \(d^{\prime}\). Typically, we consider \(\gamma\) to be 2. We have  
\[D_{i}=\sum_{k=1,k,\
eq i}^{m}(x_{k}^{2}+y_{k}^{2}) \tag{13}\]  
where \(D_{i}\) is the sum of square distance between node \(i\) and other nodes in the same cluster. (Node \(i\) is the CH.) If node \(j\) is selected as the CH with coordinate (\(x_{j}\), \(y_{j}\)), we have  
\[D_{j}=\sum_{k=1,k,\
eq i}^{m}(x_{k}-x_{j})^{2}+(y_{k}-y_{j})^{2} \tag{14}\]  
We calculate the expectation of (13), (14), respectively, as follows  
\[E(D_{i})=E(\sum_{k=1,k,\
eq i}^{m}(x_{k}^{2}+y_{k}^{2})) \tag{15}\]  
\[E(D_{j})=E(\sum_{k=1,k,\
eq i}^{m}(x_{k}-x_{j})^{2}+(y_{k}-y_{j})^{2}) \tag{16}\]According to (15), (16), we have  
\[E(D_{i})=E[(x_{j}^{2}+y_{j}^{2})+\sum_{k=1,k=i(,j)}^{m}(x_{k}^{2}+y_{k}^{2})]=E(x_ {j}^{2}+y_{j}^{2})+(m-2)\times E(x^{2}+y^{2}) \tag{17}\]  
\[E(D_{j})=E[(x_{j}^{2}+y_{j}^{2})+\sum_{k=1,k=i(,j)}^{m}(x_{k}-x_{j})^{2}+(y_{k}- y_{j})^{2}] \tag{18}\]  
where \(x\) and \(y\) are the coordinates of a generic node. From (18), we have  
\[ \begin{split} E(D_{j})=E[(x_{j}^{2}+y_{j}^{2})]+E[\sum_{k=1,k=i(, j)}^{m}(x_{k}-x_{j})^{2}+(y_{k}-y_{j})^{2}]\\ =E[(x_{j}^{2}+y_{j}^{2})]+(m-2)\times C(x,y)\end{split}  \tag{19}\]  
\(C(x,y)\) is given by  
\[ \begin{split} C(x,y)=& E((x_{k}-x_{j})^{2}+(y_{k}- y_{j})^{2})=E(x_{k}^{2}+x_{j}^{2}-2x_{k}x_{j}+y_{k}^{2}+y_{j}^{2}-2y_{k}x_{j})\\ =& 2(E(x^{2})+E(y^{2}))-2E^{2}(x)-2E^{2}(y)\end{split}  \tag{20}\]  
where \(E(x)=0\), \(E(y)=0\). From (19), (20), we have  
\[ \begin{split} E(D_{j})=& E[(x_{j}^{2}+y_{j}^{2})+2 \sum_{k=1,k=i(,j)}^{m}(x_{k})^{2}+(y_{k})^{2}]\\ =& E(x_{j}^{2}+y_{j}^{2})+2(m-2)\times E(x^{2}+y^{2} )\end{split}  \tag{21}\]  
From (17), (21), we have  
\[E(D_{i})<E(D_{j}),\text{ if }2<m \tag{22}\]  
According to the radio model described above, we can infer that the energy consumption is associated with radio distance, _i.e._, the shorter the radio distance, the smaller of energy consumption would be. Considering (22), we can draw a conclusion that the CH located in the center of event area consumes the least energy for transmitting data. In order to prolong the network lifetime, we can describe the goal of clustering: maximizing the working time of the cluster, while minimizing the energy consumption in the cluster, which can be formulated as follows:  
\[\max T_{C},\min\sum_{(i,j\in\mathbb{K})}E_{s_{C}} \tag{23}\]  
where \(T_{C}\) is the working time of a cluster. \(E_{S\epsilon}\) is the sum of energy consumption in a cluster. Since the CH takes on a lot of work in a cluster, the residual energy need be considered when selecting a CH.