## 4 Proposed Approach  
In this section, we present our distributed k-hop intra-clustering approach for wireless networks named Distributed Clustering based 2-Hop Connectivity (DC2 HC). The clustering approach consists of grouping nodes with high connectivity into k-hop clusters. As cluster heads consume more energy compared with other nodes, the number of CHs should be reduced [49]. Therefore, DC2 HC aims to produce an optimized number of clusters while maintaining the network coverage, reducing the number of isolated nodes and providing more extended network durability. The algorithm adopts a weight based mechanism during the cluster heads election process. This latter consists of selecting as a CH node with the maximum weight in its k-hop neighborhood. Multi-hop spanning trees [52] are formed inside each cluster where CMs use the Received Signal Strength Indicator (RSSI) to select their parents in the routing path leading to the corresponding CH. Therefore, clusters will be constructed with a topology that consumes less energy. Nodes weight combines three metrics:  
1. **Two-hop connectivity ratio (TCR)**: this parameter represents the connectivity ratio of a node relative to its neighborhood. The \(TCR\) value of a given node \(i\) is calculated using Equation (2). Each node computes the average connectivity within its two hop neighborhood (\(N_{\leq 2}(i)\)) using \(\Phi_{i}\), then compares the obtained value with the local degree to define the \(TCR_{i}\). A negative \(TCR_{i}\) value (\(|N(i)|-\Phi_{i}\leq 0\)) reflects the low connectivity proportion of the node \(i\) relatively to its surrounding environment. Higher \(TCR_{i}\) value means that node \(i\) is surrounded by a large number of neighbors and these neighbors are well connected with many other nodes, thus \(i\) is a suitable CH candidate to maintain network connectivity. Therefore, it covers the largest number of nodes within the maximum hop constraint \(k\) and generates more fault tolerant and stable cluster topology. Indeed, in case of potential CH failure, the neighborhood of this node is well connected and the replacement of the current CH does not affect the cluster performance. \[\Phi_{i}=\frac{\left[\sum\limits_{j=1}^{|N_{\leq 2}(i)|}|N(j)|\ /\ j\in N_{\leq 2}(i) \right]+|N(i)|}{|N_{\leq 2}(i)|+1}\] \[TCR_{i}=|N(i)|-\Phi_{i}\] (2)
2. **Residual energy (\(E^{ratio}\))**: the remaining energy of network nodes is introduced in the CH election process. The ratio of remaining energy of a node \(i\) is computed as: \[E^{ratio}_{i}=\frac{E^{residual}_{i}}{E^{init}_{i}}\] (3) where \(E^{init}_{i}\) is the initial energy of the current device and \(E^{residual}_{i}\) is its remaining energy. 3. **Communication link quality (RSSI)**: DC2 HC uses Radio Signal Strength Indicator (RSSI) as a metric to measure the quality of communications. The RSSI value (the received transmission power \(P_{r}\)) can be represented by the Log Distance Path Loss Model [53] as follows: \[P_{r}(d)(\mathrm{dBm})=P_{t}(\mathrm{dBm})-10\times\alpha\ log(d)-X_{\sigma}\] (4)\(P_{t}\) represents the power of the transmitter's radio signal in dBm. The distance \(d\) between the sender and the receiver is measured in meter. \(\alpha\) is the path loss exponent that depends on the environmental conditions (\(\alpha=2\) in the free space propagation model). \(X_{\sigma}\) is a Gaussian random variable used in case of shadowing effect. Otherwise, it equals zero. Therefore, the weight \(W_{i}\) of a node \(i\), based on the previous parameters, is computed as follow:  
\[W_{i}=\alpha\times TCR+\beta\times E_{i}^{ratio}+\gamma\times RSSI \tag{5}\]  
where \(\alpha,\beta,\gamma\) represent the weighted coefficients of the corresponding metrics with the constraint \(\alpha+\beta+\gamma=1\). The contribution and importance of each parameter, relatively to the others, is indicated by its corresponding weighted coefficient. They are adjusted according to the system requirements and the network environment.