## 2 Cluster Head Selection
### Assisted Schemes
#### 2.2.1.2 Optimum Number of CHs  
The number of CHs is a determinant factor in the performance of cluster-based algorithms. There are two issues related to this subject, first the optimum number of CHs (Kopt) in a network consisting of \(N\) nodes, and second the variation in number of selected CHs per round around the expected value.  
Regarding the first issue, the optimal value of \(k\) is analytically determined in LEACH using the computation and communication energy models. By setting the derivative of total dissipated energies by CH in receiving signals, aggregating data packets and transmission of the aggregated data to the BS with respect to \(k\) to zero, the optimum number of CHs in an M\(\times\)M area consisting of \(N\) nodes is achieved in Equation (11):  
\[K_{opt}=\frac{\sqrt{N}}{\sqrt{2\,\pi}}\sqrt{\frac{\mathcal{E}_{g}}{\mathcal{E }_{np}}}\frac{M}{d_{Bs}^{2}} \tag{11}\]  
According to the analytical Equation (11) and simulation results reported in [23], the optimal number of CHs for a 100-node network is proposed to be around 3-5. In other words, the desired percentage of CHs in LEACH threshold equation to achieve the optimum network lifetime and performance is around 5 percent. However, it should be noted that the percentage proposed by [23] is calculated based on the direct communication of CHs with the BS and therefore, for the schemes that use multi-hop transmission to transfer the aggregated data of CHs to the BS or the schemes with different energy consumption patterns may vary.  
In LEACH, number of selected CHs in each round is not constant and it varies over a great range. Although the threshold value T(n) in LEACH is proposed to adjust the desired number of CHs per round, its inherent probabilistic nature causes variations in the number of selected CHs per round, and even some rounds may exist where none of the nodes are selected as CH. The consequences of these variations are a significant loss of network stability and performance.  
Although the number of CHs can be easily controlled by the BS in centralized schemes, few researchers address this issue in distributed schemes by supervising the number of selected CHs with assistance of the BS. The authors in [47] address the issue of selection of no CH in a specific round by calling the round invalid and moving the system to the next round without going through the cluster organizing and message transmission phases. The authors of [48] propose splitting the CH selection phase into an initial selection phase and an add-on selection phase. In the proposed scheme, if the number of selected CHs in the initial stage is below some predefined threshold, an add-on stage is called until number of selected CHs exceeds the threshold value. In [49], a semi-centralized scheme is proposed in which the selection of CHs is distributed and self-organized, while the BS controls the number of selected CHs and stops the CH selection phase as the number of selected CHs reaches the predefined optimal value.  
While the number of nodes is usually considered constant, there are cases where new nodes should be added to the network or conversely, some of nodes die with the increase in the number of rounds. In these cases, the number of CHs should be dynamic to ensure network energy efficiency, network robustness and the adaptability of the system to the quality of the network. The variation in the optimum desired numbers of CHs per round due to changes in the number of nodes within the network is highlighted by [50]. This paper proposes that CHs send the number of their cluster members with aggregated data to the BS, whereby the optimum number of CHs in accordance with the updated total number of nodes within the field is calculated for the next round.