## 4 Theoretical Analyses
### Balancing the Average Energy Consumption of Nodes in Different Units  
For any sensor node in unit \(i\) (\(1\leqslant i\leqslant\) m), if Equation (18) is satisfied, this node can work in an energy neutral state. Then we explore how to maximize network information gathering under the energy neutrality constraint. Assuming that the information can be continuously collected by each sensor node in the network, then the following question is to determine the information transmission cycle of a sensor node based on its available energy. For a CH in the unit closer to BS, once received data from the CH farther from BS, it should immediately forward this data if it has enough energy, otherwise, it discards this data. This is reasonable in terms of the following two aspects: First, real-time transmission of data is more meaningful than non-real-time transmission; Second, as sensor nodes in the unit closer to BS have heavier relay traffic and consume energy much faster than the ones in the unit farther from BS, then if a CH has not enough energy to forward the received data, it still has not enough energy to forward this data over time because it should give priority to forward the newly received data after it harvested some energy. In addition, then, if \(T_{j}\)\(<T_{i}\) (\(j>i\)), packet loss problem will occur since CHs in the unit closer to BS have lower data transmission rate and can not timely forward data from the CHs in the unit farther from BS. In addition, if \(T_{j}\)\(>T_{i}\) (\(j>i\)), the network information gathering can not be maximized as the nodes in the unit farther from BS have the potential to enhance their data transmission rate. Thus, to solve these problems, the constraint \(T_{i}=T_{j}(i+j)\) should be satisfied. In addition, according to Equation (18), the average energy consumption of nodes in different units should be balanced to guarantee this constraint. That is  
\[Eav_{i+1}=Eav_{i}\left(i=1,2,\cdots,m-1\right) \tag{19}\]  
where \(Eav_{i}\) and \(Eav_{i+1}\) are the average energy consumption of nodes per cycle in unit \(i\) and \(i+1\), respectively. Combining Equations (6), (7), (9), (10), (16) and (19), we have  
\[\left(\psi+\phi\frac{1}{c_{i+1}c_{i}}\right)\left(c_{i}-c_{i+1}\right)=\omega \tag{20}\]where \(c_{i}\) and \(c_{i+1}\) are the number of clusters in unit \(i\) and \(i+1\); and parameters \(\varphi\), \(\psi\) and \(\omega\) are shown respectively as follows:  
\[\psi=\frac{2 mqE_{dec}}{\rho S} \tag{21}\]  
\[\phi=q\varepsilon_{fs}\frac{4S}{9\pi m} \tag{22}\]  
\[\omega=\left(m-i+1\right)aq\varepsilon_{fs}d_{i}^{2}-\left(m-i\right)aq \varepsilon_{fs}d_{i+1}^{2}+2 aqE_{dec} \tag{23}\]  
where \(m\) is the number of units in the network; \(q\) is data packet length of each node; \(\rho\) is node density; \(S\) is sensor field size; \(a\) is data aggregation ratio and \(d_{i}\) is the average distance between a CH in unit \(i\) and its routing head or BS. According to Equations (11) and (14), we can have  
\[d_{i+1}<d_{i}\left(i=1,2,\cdots,m-1\right) \tag{24}\]  
which implies that the CHs in the unit closer to BS have longer inter-cluster communication distance. Thus, we have \(\omega\)\(>0\) according to Equation (23). In addition, based on Equations (20-22), we have  
\[c_{i+1}<c_{i}\left(i=1,2,\cdots,m-1\right) \tag{25}\]  
which shows that for the purpose of balancing the average energy consumption of nodes in different units, clusters in the unit closer to BS should strictly more than that in the unit farther from BS. As a result, if Equation (20) is met, the average energy consumption of nodes in different units can be balanced. In addition, then the constraint \(T_{i}\)= \(T_{j}\)(\(i+j\)) can be satisfied. Let \(T\) denote the network data transmission cycle, then we have  
\[T_{i}=T_{j}=T\left(i,j=1,2,\cdots,m\right) \tag{26}\]  
According to Equations (11), (14) and (21)-(23), to calculate \(\varphi\), \(\psi\) and \(\omega\) in Equation (20), the number of units \(m\) should be determined.