## The System Model  
### Network Model  
In the grid-based development field, there are great many randomly distributed sensor nodes. The network topology including all sensor nodes and the BS builds a graph \(\mathcal{G}\left(\mathcal{V},\mathcal{E}\right)\), where \(\mathcal{V}=\{v_{0},v_{1},\cdots,v_{i},\cdots,v_{n}\}\) is the node set, \(v_{0}\) is the BS, and \(\mathcal{E}=\{e_{1},\cdots,e_{j},\cdots,e_{m}\}\) is the communication link set of sensor nodes. Then, we make some reasonable and realistic assumptions regarding the WSNs [21; 22]:  
1. The BS and all sensor nodes are stationary after deployment, and are equipped with a Global Positioning System (GPS) unit. Hence, these nodes are location-aware. 2. All properties for each sensor node are identical, while the BS is manually maintained and has enough energy to support continuous operations, with its energy denoted as \(\varepsilon_{0}\). 3. Provided with sufficient energy, each node can control the transmission power according to the distance between the transmitter and the receiver. 4. The amount of transmission data is exactly equal for each sensor node of valid routes. At the transmission round \(t\), the residual energy of all sensor nodes is denoted as \(\Psi(t)=[\varepsilon_{0},\varepsilon_{1}(t),\cdots,\varepsilon_{i}(t),\cdots, \varepsilon_{n}(t)]^{T}\), while their corresponding initial node energy is denoted as \(\Psi(0)=[\varepsilon_{0},\varepsilon_{1}(0),\cdots,\varepsilon_{i}(0),\cdots, \varepsilon_{n}(0)]^{T}\), where \(t\) is the time scale in terms of transmission rounds of the WSNs [21]. The neighboring communication link between two sensor nodes is determined by the communication radius \(r_{s}\) that is defined as follows [22]:  
\[r_{s}=\left[1-\frac{d_{\max}-d\left(v_{i},v_{0}\right)}{\mu\left(d_{\max}-d_{ \min}\right)}\right]r_{\max}\;,\quad i\in\{1,2,\cdots,n\} \tag{1}\]  
where \(r_{max}\) is the maximum communication radius and is defined as the width of the sensing field, and \(d_{max}\) and \(d_{min}\) represent the maximum and minimum distances between the sensor node and the BS, respectively. The distance \(d\left(v_{i},v_{0}\right)\) denotes the distance between \(v_{i}\) and the BS \(v_{0}\), and the variable \(\mu\) can be adjusted according to the real environment. Extensive experiments show that a smaller value of \(\mu\) easily damages the network connectivity, and that a larger value of \(\mu\) is prone to cause packet collisions among sensor nodes. According to the mathematical analysis of Equation (1), for the variable \(\mu\) there exists a point \(\mu^{*}\) at which the increasing trend of \(r_{s}\) changes dramatically. The communication radius \(r_{s}\) increases rapidly before the point \(\mu^{*}\), while it increases slowly after the point \(\mu^{*}\). In order to minimize packet collisions among the sensor nodes and to guarantee the transmission quality of WSNs as much as possible [22], we conduct diverse experiments for different values of \(\mu\) within the numerical region \([\mu^{*}-\Delta\mu,\;\mu^{*}+\Delta\mu]\) to obtain the optimal communication radius \(r_{s}^{*}\), where \(\Delta\mu=0.5\) is an offset boundary of the trim parameter that can be adjusted. For the sensor node \(v_{i}\), its neighboring nodes' set comprises sensor nodes within the optimal communication radius \(r_{s}^{*}\). Meanwhile, the node residual energy must accommodate the basic requirement of effective communication; otherwise, the sensor node is defined as a \"dead node\"and is excluded from its corresponding neighboring nodes' sets. The neighboring communication link between \(v_{i}\) and \(v_{j}\) with the distance \(d(v_{i},v_{j})\) is denoted by  
\[m_{ij}=\left\{ \begin{array}{ll}1;\ d(v_{i},v_{j})\leq r_{s}^{*},\ \varepsilon_{i}(t)> \varepsilon,\ \varepsilon_{j}(t)>\varepsilon\\ 0;\ \text{otherwise}\end{array} \right. \tag{2}\]  
where \(i,j\in\{0,1,2,\cdots,n\}\), and \(\varepsilon\) is the minimum residual node energy that pledges communication.