## 3 Background
### Virtual Grid Margin Optimization  
According to the assumptions of network model, \(N\) nodes are distributed randomly in a square area of length \(l\). The area of interest is partitioned into grids of equal size, and the nodes within each grid are self-organized into a cluster. The optimization of virtual grid margins impacts greatly the partition of clusters and the whole network's energy consumption. Consider two extreme cases: one is given such that the distance between two neighboring grids equals the perimeter of the entire region, which is to say, all nodes interact with the cluster head in a single-hop communication manner. In this case, the energy of member nodes would be exhausted rapidly due to the large communication radius of the network. The other is given such that the distance between two neighboring grids is excessively small, in this case, the coverage range of a single cluster would become excessively small, and the maintenance cost for each cluster would become excessively high. Moreover, a small distance between cluster heads may lead to disturbances and reduce the validity of data transmission in the network. According to the derivation of the overall energy consumption in the network, the optimal cluster number can be calculated, and the grid margin can be optimized based on this calculated result. In the process of selecting the cluster head, we assume that the temporary cluster head participating competition is denoted by \(\rho+\Delta\rho\), the transmission radius of the broadcast message is denoted by \(R\), and the length of the message is denoted by \(l_{0}\), so that the energy consumed by the network can be calculated by:  
\[ \begin{array}{l}E_{comp}=(\rho+\Delta\rho)[E_{CH-broad}+(N-1)E_{non-CH-broad }]\\ =(\rho+\Delta\rho)(l_{0}E_{elec}+l_{0}e_{up}R^{4})+(N-1)(\rho+\Delta\rho)l_{0}E _{elec}\end{array}  \tag{1}\]  
In which, \(E_{CH-broad}\) is the energy consumption of broadcast message when the node is a cluster head, and \(E_{non-CH-broad}\) is the energy consumption of broadcast message when the node is a non-cluster head. Since the transmission radius of the broadcast message is usually greater than the threshold distance \(d_{0}\) in competition, the energy consumption of the power amplifier should be described by the multiple-path decline signal channel model. During the process of cluster head selection, the cluster head node receives the message from other member nodes. The energy consumed in this process is given as follows:  
\[ \begin{split}& E_{form}=\rho E_{CH-join}+(N-\rho)E_{non-CH-join}\\ &=\rho[l_{1}E_{elec}(\frac{N}{\rho}-1)]+(N-\rho)[l_{1}E_{elec}+l_{ 1}\varepsilon_{fs}d_{toCH}^{2}]\end{split}  \tag{2}\]  
In which, \(E_{CH-join}\) is the energy consumption of a cluster head node that receives the message from other member nodes, and \(E_{non-CH-broad}\) is the energy consumption of a non-cluster head node receives the message from other member nodes. After the completion of cluster head selection, the cluster head receives the collected data from its member nodes in the steady running stage. Let \(l\) denote the bit size of the data packet transmitted in one cycle. The cluster head is responsible for sending data to the base station after the data is collected from the member nodes at the cluster head. In order to guarantee the integrity of all sensor nodes' data and reduce the cost of data recovery, the cluster head should not execute any data fusion task, and the cluster head nodes' data is acquired by utilizing mobile sinks, thus the energy consumed in this stage can be written as:  
\[ \begin{split}& E_{tran}=(N-\rho)E_{non-CH-tran}+\rho(E_{CH- rec}+E_{tran-sink})\\ &=(N-\rho)[lE_{elec}+l\varepsilon_{fs}d_{toCH}^{2}]+\rho[lE_{elec} (\frac{N-\rho}{\rho})+lE_{elec}+l\varepsilon_{mp}d_{toBS}^{4}]\end{split}  \tag{3}\]  
In which, \(E_{tran-sink}\) is the energy consumption of the cluster head node transmits the message to the mobile sinks in one cycle.