## 3 Network Model and Algorithm Objective  
In this study, the wireless network is assumed to be composed of a set of devices represented by a graph \(G=(V,E)\), where \(V\) is the set of network nodes and \(E\) is the set of edges that represents the communication and interfering links. \(N_{k}(i)\) denotes the set of k-hop neighbors of the node \(i\in V\), \(N_{<k}(i)\) represents the set neighbors at distance \(<\)_k-hop_ from \(i\). \(N(i)\) denotes neighbors of node \(i\) at one hop. \(Deg(i)=|N(i)|\) is the degree of node \(i\), i.e., number of nodes in the neighborhood of node \(i\). The distance between the node \(i\) and the node \(j\) is \(Dist(i,j)\) (the number of hops in the shortest path between \(i\) and \(j\)). Nodes use multi-hop communication to relay their data toward their CH. We assume that all devices have the same transmission range (communication links are symmetric). Network devices use a single channel transceiver, i.e., they cannot receive from multiple senders or send and receive at the same time.  
The objective of our proposed solution is to optimize the number of CHs and reduce the waste of energy caused by the long-range communication channels and decrease the risk of congestion. More precisely, we aim to find the smallest set \(C\) of selected cluster heads \(CH_{n}\) (\(n\in\{1,2,...,|C|\}\)) using a two-hop neighbors connectivity metric, while respecting a particular maximum hops constraint \(k\) between each node and its cluster head. Considering that \(M_{n}\) is the set of nodes that belong to the cluster coordinated by the cluster head \(CH_{n}\). The objective of the proposed mechanism is formulated as follows:  
\[C=(Min\mid\bigcup CH_{n}\subseteq V\mid)\ \wedge\ (\ \forall i\in M_{n},\ \forall\ CH_{n}\in V):\]  
\[Dist(i,CH_{n})\ \text{is Minimized}\]  
where  
\[M_{n}=\{\forall\ i\in V,\ \forall\ CH_{n}\in V\mid Dist(i,CH_{n})\leq k\}\]  
\[\bigcup_{n=1}^{|C|}M_{n}=V\wedge\ \ (\forall\ n\
eq m:M_{n}\cap M_{m}=\emptyset) \tag{1}\]  
\begin{table}
\begin{tabular}{l l l l l l l l} \hline
**Algorithm** & **Topology** & **Number of CHs** & **Intra Clustering** & **Inter Clustering** & **Load Balancing** & **Energy Consideration** & **Benchmarks** \\ \hline LEACH [43] & Distributed & Undetermined & Single-hop & Single-hop & No & No & MTE \\ ET-LEACH [44] & Distributed & Undetermined & Single-hop & Single-hop & No & Yes (CH+election) & LEACH, TEEN \\ FL-LEACH [45] & Distributed & Determined & Single-hop & Single-hop & Yes & No & LACH \\ [PERSON] et al. [47] & Centralized & Undetermined & Single-hop & Single-hop & Yes & No & Not specified \\ MH-LEACH [46] & Distributed & Undetermined & Multi-hop & Single-hop & No & Yes (Multi-Hop transmission) & LEACH \\ E-PEGASIS [48] & Centralized & Determined & Multi-hop & Single-hop & No & No & PEGASIS, \\ EE3C [50] & Centralized & Undetermined & Multi-hop & Multi-hop & Yes & No & LBEERA \\ K-ECDS [49] & Distributed & Undetermined & Multi-hop & Multi-hop & No & No & LEDS, HEED \\ [PERSON] et al. [14] & Centralized & Undetermined & Multi-hop & Multi-hop & Yes & Yes (CH+rotation) & EEUC, EUCA \\ Turgut [19] & Distributed & Undetermined & Multi-hop & Single-hop & Yes & No & Not specified \\ Mezghani [34] & Distributed & Undetermined & Multi-hop & Single-hop & Yes & Yes (intra-cluster routing) & MTE, HEED, APTEEN, EDC, THC, VCA \\ \hline \end{tabular}
\end{table}
Table 2: Clustering algorithms properties comparison.  
The communication over multi-hop short-range is usually more energy efficient than directly transmitting in a single-hop long-range communication [51]. Larger intra-cluster hops extends the coverage of the CHs, which reduces the number of elected CHs and the energy dissipation. However, in case of dense network, the interference rate will also increase, which expands the waste of energy due to collusion problems that may occur.