## 3 System Model
### Clustering and Selection of Cluster Heads  
First, we determine the expected distances between CHs and SINK using an integral, then the expected distances between CHs and nodes inside its cluster. Then, the entire predicted energy consumption value is calculated to obtain the ideal number of cluster heads. The node density denotes the tightness of the relationships between the activity nodes and displays the difference between a complete graph and a distribution graph. In the following method, we choose the best CHs by considering node energy consumption, degrees, and distances, as shown in Algorithm 1. Assume that the network contains k clusters and N nodes. Under the free-space model, network's energy consumption is given by Equation (7):  
\[E_{CH}=\left(\left|\frac{N}{k}\right|-1\right)\cdot\cdot E_{\mathrm{lec}}\ +\ \frac{N}{k}\cdot l\cdot E_{\mathrm{da}}\ +\ l\cdot E_{\mathrm{lec}}\ +\ l\cdot e_{fs}\cdot DCH-Sin \tag{7}\]where \(d^{2}_{CH-Sink}\) is the estimated distance between CH and Sink, \(\left|\frac{N}{k}\right|\); \(\left(\left|\frac{N}{k}\right|-1\right)\) is the average number of nodes in each cluster, and \(E_{\text{lec}}\) denotes the amount of energy used by CH to receive \(\left(\left|\frac{N}{k}\right|-1\right)\) bits of data. The energy usage for CH to transmit l bit of data is represented as \(l\cdot E_{\text{lec}}+l\cdot e_{fs}d^{2}_{CH-Sink}\). In a free-space model, the best CH location is (x, y), while the Sink's coordinate is (x, y) (a, b). As a result, the following equation is the expected squared distance between the CH and Sink:  
\[\text{E}\left[d^{2}_{CH-Sink}\right]=\iint\left((a-x)^{2}+(b-y)^{2}\right)d_{ CH}(x,y)\text{d}x\text{d}y=\iint\frac{(a-x)^{2}+(b-y)^{2}}{S}\text{d}x\text{d}y \tag{8}\]  
The \((x,y)\) is the distance between source node and CH, and S is the square area of the node distribution. Consider that each cluster contains l bit of data to be conveyed. Then, Equation (9) is used to determine the energy usage of one cluster in one round:  
\[E_{CH-members}=l\cdot E_{\text{elec}}+l\cdot e_{fs}^{2}_{DCH-members} \tag{9}\]  
Within a cluster, the expected squared distance from CH to the source node is given by Equation (10):  
\[\text{E}\left[d^{2}_{CH-members}\right]=\iint\left(x^{2}+y^{2}\right)d(x,y) \text{d}x\text{d}y=\frac{L^{2}}{2\pi k} \tag{10}\]  
To begin, determine whether node vi is a cluster head by calculating the following:  
\[CH_{\text{Candidate-i}}=\alpha\cdot\left(\frac{E_{\text{Remain-i}}}{E_{\text{ Initial}}}\right)+\beta\cdot\left(\frac{Deg\text{ max}}{Deg_{i}}\right)+\gamma\cdot\left(\frac{Dis_{i-Sink}}{Dis_{\text{max}}- Sink}\right) \tag{11}\]  
where \(\alpha\), \(\beta\), and \(\gamma\) are parameters that satisfy 0 \(<\)\(\alpha\), \(\beta\), \(\gamma\)\(<\) 1, \(\alpha\)\(+\)\(\beta\)\(+\)\(\gamma\) = 1; \(E_{\text{Remain-i}}\) represents the remaining energy of node vi; initial is the initial energy of all nodes; \(Deg_{i}\) is the degree of node vi; Deg max denotes the maximum degree of all nodes; \(Dis_{i-Sink}\) refers to the distance between node vi and Sink; and \(Dis_{\text{max}}-Sink\) is the greatest distance between the source node and Sink. Candidate set \(CH_{\text{Candidate-i}}\) is created by sorting the candidate values from high to low using the aforementioned algorithm. When the network's minimal CR is equal to or greater than 90%, the best CH is chosen from the candidate sets. ```
Need:  N denotes number of alive nodes  V(I, j), I, j = 1, 2.....N  S(a,b)  Neighbor vi(j) r denotes node transmission range  Confirm:  List CH[i] i = 1,...,Not-CH  1. State List CH [j] = NULL  2. Evaluate candidate value of node vi as a CH  3. For I = 1 to N do  4. N = +1  5. If \(n\) is not equal to Not-CH, then  6. For j = i + 1 to N, do  7. If CR \(<\) 90%, then  8. Choose next node vi  9. End if  10. End for

### Adaptive Clustering-Based Locative Data Transmission (ACLDT)

To characterize data evolution in ACLDT, the data transmission method trusts on building an initial identical predictive method at theCH and SN.