# Non-Uniform Clustering Algorithm for UWSNs Based on Energy Equalization
## 3 Underwater Sensor Network Model  
The network scene consists of \(N\) dynamic nodes; each node numbered can be represented as a set \(S=\{s_{1},s_{2},\ldots,s_{N}\}\), and sink can be represented as \(s_{0}\). These nodes are randomly deployed in the 3D scene \(L\times L\times L\). Through the clustering algorithm, cluster heads are selected, and the topology of different clusters is formed based on the selected cluster heads. The member nodes in the cluster transmit data to the cluster head nodes in a single-hop manner. Then, the cluster head nodes fuse the data sent by all the member nodes in the cluster and use other cluster heads as relay nodes and transmit the data to the sink in a multi-hop manner, as shown in Figure 1. The underwater sensor nodes communicate with other nodes in the water medium using an acoustic modem. The sink nodes are equipped with both acoustic and radio frequency (RF) modems and deployed at the center of the water surface. The acoustic modem receives data from the underwater sensor nodes and the RF modem sends data to the satellite, which in turn sends data to the base station on shore. The underwater sensor nodes are all free to move in the horizontal direction at a speed of 1 m/s. The hypotheses of the network can be described as follows:  
* Each node has a unique ID, the transmission range is spherical and is otherwise isomorphic;
* The network operates without considering the sink energy loss;
* Within the communication radius, the nodes can freely adjust the transmission power to save energy consumption;
* The link is symmetric, and the nodes can judge the distance between the two sides based on the received signal intensity.  
Acoustic waves are used as a medium by the energy consumption model for data communication in underwater networks [24]. The energy consumption of a node that sends a data packet is  
\[E_{t}(d)=\lambda_{t}IA(d), \tag{1}\]  
where \(\lambda_{t}\) is the transmit energy consumption coefficient; \(l\) represents the length of the packet; \(d\) stands for the distance the packet needs to be transmitted; and \(A(d)\) refers to the  
Figure 1: UWSN model.  
Each candidate cluster head that has participated in the competition will exit the \(H\)-set and not take part in the competition with other ones. Below is the expression of the remaining candidate cluster heads:  
\[B_{u}=\begin{cases}H,&u=1\\ B_{u-1}\backslash A_{u-1},&u>1\end{cases} \tag{8}\]  
where \(A_{u}\) denotes the set of elements that exits from set \(H\) each time, which can be expressed as  
\[A_{u}=\left\{\begin{array}{ll}h_{m},D_{mi}>R_{m},\;i\in[m+1,M]\\ \{h_{m}\}\cup H_{m}\end{array}\right.. \tag{9}\]  
Finally, the concept of weight is introduced by considering the relationship between a node and all of its cluster heads within its communication radius. In this way, ordinary nodes can choose the optimal cluster head to join. In addition, ordinary nodes measure the weight value relationship with all cluster heads, select the best cluster head to join and become its member nodes. Below is the condition for an ordinary node to join this cluster head:  
\[G_{pq}=\underset{q\in[1,U]}{\arg\min}(w_{pq}^{h}),p=1,2,\ldots,(N-U) \tag{10}\]  
where \(G_{pq}\) denotes node \(s_{p}\) that joins cluster head \(z_{q}\); \(w_{pq}^{h}\) represents the weight between ordinary node \(s_{p}\) and cluster head \(z_{q}\).  
From the above description, it can be seen that reasonable cluster heads have been selected from the network and appropriate node members for each cluster head have been added. By this means, clusters of different sizes are formed based on the weight relationship between nodes and cluster heads.