# Data Aggregation Based on Overlapping Rate of Sensing Area in Wireless Sensor Networks
## 3 Network Model and Aggregation Rules
### Aggregation Rules  
We have three aggregation rules for the gathering area construction. Although in general, \(E_{R}\) is slightly smaller than \(E_{T}\) which is relevant with the transmission distance, to simplify the explanations of these rules, we assume \(E_{R}=E_{T}=E\) and \(E\cdot d=e\). **Rule 1. **_In gathering areas, the upper-level nodes have priorities over the lower-level nodes to be selected as gathering nodes._  
We evaluate the validity of Rule 1 in an instance gathering area with a upper-level node \(v_{j}\) and a lower-level node \(v_{i}\). Apparently, \(L_{i}=L_{j}+1\geq 2\). Case 1: \(v_{i}\) is the gathering node. After \(v_{j}\) joins the gathering area \(GA_{i}\), \(v_{j}\) as the member node sends \(d\) data to \(v_{i}\). Thus \(d_{j}^{R}=0\), and \(d_{j}^{T}=d_{i}^{R}=d\). After data aggregation, \(v_{i}\) transmits additional \(\gamma\cdot d(0<\gamma=1-OR_{i,j}<1)\) data as well as its own data \(d\). Hence, \(d_{i}^{T}=(1+\gamma)d\). Since these data need to be forwarded to the sink through \(L_{i}\) hops. The energy consumed for delivering aggregated data from the gathering node \(v_{i}\) is \(E_{GN_{i}}=(2L_{i}-1)d_{i}^{T}\cdot E=(2L_{i}-1)(1+\gamma)e\). Therefore, the total energy consumption of those nodes in the gathering area \(GA_{i}\) is computed by  
\[E_{GA_{i}} =d_{j}^{T}\cdot E+d_{i}^{R}\cdot E+E_{GN_{i}}\] \[=[(2L_{i}-1)(1+\gamma)+2]e. \tag{1}\]  
Case 2: \(v_{j}\) is the gathering node. Similarly, the total energy cost of data delivery from those nodes in \(GA_{j}\) to the sink is  
\[E_{GA_{j}} =[(2L_{j}-1)(1+\gamma)+2]e\] \[=[(2L_{i}-3)(1+\gamma)+2]e. \tag{2}\]  
Obviously \(E_{GA_{j}}<E_{GA_{i}}\). Because only replicated data are cleaned in aggregation, no matter which node is the gathering node, the quantities of information after aggregation are the same. As a result of a higher level of the gathering node, Case 1 has more energy consumed than Case 2. Thus selecting the gathering nodes from the upper-level nodes is better than from the lower-level nodes. Figure 3: An instance of wireless sensor network (WSN) with three gathering areas. **Rule 2**. : _After the lower-level nodes complete the construction of gathering areas, the free nodes at the upper level firstly select their gathering nodes from the candidate parent nodes. If no suitable candidate parent node exists, then select from the neighbor nodes._  
Assume that \(v_{k}\) (the current node) is selecting a gathering node. \(v_{j}\) is its candidate parent node, while \(v_{i}\) is a neighbor node of \(v_{k}\). Hence \(L_{k}=L_{i}=L_{j}+1\). Case 1: \(OR_{i,k}\geq OR_{T}\) and \(OR_{j,k}\geq OR_{T}\). In this case, \(v_{k}\) select \(v_{i}\) or \(v_{j}\) to be its gathering node. If \(v_{i}\) is the gathering node of \(v_{k}\), \(v_{i}\) receives data of size \(d\) from \(v_{k}\) and transfers additional \(\alpha\cdot d(0<\alpha=1-OR_{i,k}<1)\) data as well as its own data after aggregation. The total energy cost of transmitting data from \(v_{k}\) to the sink via \(v_{i}\) is  
\[E_{i}^{k}=[(2L_{k}-1)\alpha+2]e. \tag{3}\]  
Otherwise, if \(v_{k}\) chooses \(v_{j}\) as its gathering node, the total energy cost for data delivery from \(v_{k}\) to the sink via \(v_{j}\) is  
\[E_{j}^{k}=[(2L_{k}-3)\beta+2]e, \tag{4}\]  
where \(0<\beta=1-OR_{j,k}<1\). Comparing \(E_{i}^{k}\) and \(E_{j}^{k}\), we know that if \(\alpha\geq\beta\) (in other words, \(OR_{i,k}\leq OR_{j,k}\)), then \(E_{i}^{k}>E_{j}^{k}\), and \(E_{i}^{k}\) is larger than \(E_{j}^{k}\) by at least \(2\beta e\). When \(\alpha<\beta\), considering \(L_{k}\geq 2\), we obtain that if \(E_{i}^{k}\geq E_{j}^{k}\), then \(\dfrac{\beta}{3}\leq\alpha<\beta\). Accordingly, if \(\alpha<\dfrac{\beta}{3}\), \(E_{i}^{k}\) might be smaller than \(E_{j}^{k}\). Note that to guarantee a relatively high data correlation, \(OR_{T}\) is often larger than 0.4 (as discussed in Section 5.3.2), and thus \(0<\alpha,\beta<0.6\). If \(\alpha<\dfrac{\beta}{3}<0.2\) (\(OR_{i,k}>0.8\)), then \(E_{i}^{k}\) has some probability to be smaller than \(E_{j}^{k}\).