## 2 A Brief Discussion on LEACH, EESR and Sub-Cluster LEACH Protocols  
### LEACH Protocol  
On the one hand, LEACH protocol is a hierarchical protocol that executes on the basis of clusters. A cluster represents the grouping of various sensor nodes deployed at a specific location and it is set on the basis of some defined parameters for specific applications. The cluster consists of a limited number of sensors that disseminate their data to a BS. The whole idea behind it is to reduce the data traffic towards the BS and enhance the network lifetime. Therefore, LEACH supports the clustering approach to provide an energy efficient environment. The cluster is just a clubbing of neighboring sensor nodes sensing and aggregating data. The leader sensor of each cluster is known as cluster head (CH) and other existing nodes in a particular cluster are known as non-cluster head (N-CH) nodes. Therefore, all the nodes monitor their data in a specific time-span, which is regulated by Time Division Multiplexing Access (TDMA). Then the sensed data are disseminated to their cluster head, in a process known as data aggregation. After completion of data aggregation, all collected data are first compressed and then forwarded to the BS either directly, if the CH node is 1 hop distant from BS, and otherwise via intermediate nodes. This whole process reduces the energy consumption rate to a great extent. For every sensor, its election as CH in LEACH protocol can be calculated with certain probability denoted as \(P_{i}\) (\(\mathbb{S}_{n}\)) at the beginning of round r + 1. \[E_{n}\ [\#\ CH\ ]\ =\ \sum_{i=1}^{N}P_{i}\ \ (\ \mathbb{S}_{n}\ )\ *\ 1\ =\ n \tag{1}\]  
where \(N\) denotes total nodes, \(P\) denotes the probability of a node to be selected as \(CH\), \(E\) denotes the energy and \(\mathbb{S}_{n}\) denotes a sensor node. For selection as CH node, the calculation of probability for each sensor node \(i\) is evaluated in Equation (2):  
\[P_{i}\ \ (\ \mathbb{S}_{n}\ )\ =\ \left\{ \begin{array}{ll}\frac{n}{N-n\ *(r\mod N )}&:C_{i}\ (\mathbb{S}_{n})=1\\ 0&:C_{i}\ (\mathbb{S}_{n})=0\end{array} \right. \tag{2}\]  
In Equation (2), \(r\) denotes rounds accomplished, and \(C_{i}\) (\(\mathbb{S}_{n}\)) = 0 is denoted as CH. If the remaining energy is higher, then only one node can be selected as the CH node. It is calculated that after \(\frac{N}{N}\) rounds, all sensor nodes are selected as the CH node once in their lifetimes. The term \(\sum_{i=1}^{N}C_{i}\ (t)=1\) helps to check the eligibility of a node to be elected as the CH node. \[E_{n}[\sum_{i=1}^{N}C_{i}\ (\mathbb{S}_{n})]\ =\ \ N-n*(r\mod\ \ \frac{N}{n}) \tag{3}\]  
Equation (4) shows the energy distribution. \[ \begin{array}{lcl}E[\#CH]&=&\sum_{i=1}^{N}P_{i}\ (\mathbb{S}_{n})\ *\ 1\\ &=&(N-n\ *(r\mod\ \frac{N}{n}))\ *\frac{k}{N-n\ *(r\mod\ \frac{N}{n})}\\ &=&n\end{array}  \tag{4}\]  
The above expression is based on the assumption that all nodes have the same energy level. Now for CH selection based on different energy levels, Equation (5) is considered. \[P_{i}\ (\mathbb{S}_{n})\ =\ \frac{E_{i}(\mathbb{S}_{n})}{E_{total}(\mathbb{S}_{n })}n \tag{5}\]  
\[E_{total}(\mathbb{S}_{n})\ =\ \sum_{i=1}^{N}E_{i}(\mathbb{S}_{n}) \tag{6}\]  
Hence, from Equation (7), the total higher energy of CH node can be calculated. \[ \begin{array}{lcl}E[\#CH]&=&\sum_{i=1}^{N}P_{i}(\mathbb{S}_{n})\ *1\\ &=&(\frac{E_{1}(\mathbb{S}_{n})}{E_{total}(\mathbb{S}_{n})}+........+\frac{E_{N }(\mathbb{S}_{n})}{E_{total}(\mathbb{S}_{n})})n\\ &=&k\end{array}  \tag{7}\]  
Additionally, the expected energy can be calculated by Equation (8) for \(nr\) as CH nodes and \(N-nr\) as N-CH nodes. \[E[E_{tot}]\ =\ E_{0}\ (N-\ nr)\ +\ (E_{0}\ -\ E_{CH})(nr) \tag{8}\]  
From Equation (5):  
\[ \begin{array}{lcl}E[P_{i}(S_{n})]&=&\frac{E_{0}n}{(N-nr)+(E_{0}-E_{CH})nr}&: C_{i}(S_{n})=1\\ \frac{(E_{0}-E_{CH})n}{E_{0}(N-nr)+(E_{0}-E_{CH})nr}&:C_{i}(S_{n})=0\end{array}  \tag{9}\]  
Since \(E_{0}\ >(\ E_{0}-\ E_{CH})\), this expression can be simplified:  
\[