## 4 Mathematical Model
In this way, nodes just require one-hop neighbor information instead of complete network information. \(d_{th}\) plays an important role in selection of destination and relay nodes. Neighbors of a node are distinguished as potential \(DS\) and Cooperative Connector (\(CC\)) nodes on the basis of \(d_{th}\) as shown in Figure 3. Destination and relay nodes belong to \(DS\) and \(CC\) node sets. As shown in Figure 2(a), number of nodes are present within the transmission range of source node. Due to this high node density its \(d_{th}\) is high (\(d_{th}^{h}\)). As node density within the transmission range decreases, \(d_{th}\) also decreases. \(d_{th}^{m}\) and \(d_{th}^{l}\) show the decreased \(d_{th}\) in Figure 2(b),c.  
**Rule 5:** Neighbor set of an \(i_{th}\) node is represented as  
\[N(i)=N_{outd_{th}}(i)\cup N_{ind_{th}}(i) \tag{6}\]  
where  
* \(N_{outd_{th}}(i)\) represents neighbors of an \(i_{th}\) node that lie outside the boundary specified by \(d_{th}\) as shown in Figure 3 as potential \(DS\) nodes. Figure 3: Depth thresholds based on number of neighbors  
* \(N_{ind_{th}}(i)\) represents neighbors of an \(i_{th}\) node that lie inside the boundary specified by \(d_{th}\) as shown in Figure 3 as potential \(CC\) nodes. \[N_{outd_{th}}\in V_{Lo},N_{ind_{th}}\in V_{Me}\] (7)  
Equation (7) signifies that \(N_{outd_{th}}(i)\) and \(N_{ind_{th}}(i)\) are low and medium depth vertices in any \(R_{k}\), respectively. **Rule 6:** For each \(R_{k}\), the potential \(DS\) nodes are  
\[DS_{k}\subset N_{outd_{th}}(V_{Hi})\ \forall\ V_{Hi}\in k \tag{8}\]  
Initially, for any particular \(R_{k}\), lowest depth one-hop neighbors (\(N_{outd_{th}}\)) of vertices represented by \(V_{Hi}\) are considered eligible for inclusion in the \(DS\). Smaller the size of \(DS\), lesser is the message overhead and network routing efficiency is also improved. In order to reduce the size of \(DS\), nodes are further pruned, using their residual energy as a pruning parameter. After pruning in each \(R_{k}\), lowest depth nodes possessing highest energy are present in \(DS\). **Rule 7:** The lowest depth and highest energy \(DS\) nodes are the destination nodes for cooperative routing. Whereas, Probability of selection (\(Ps\)) of any node \(i\) belonging to \(V_{Lo}\) as \(DS\) node is  
\[\forall i\in V_{Lo}:\left\{ \begin{array}{c}Ps\propto R_{e}(i)\\ \\ Ps\propto 1/D(i)\end{array} \right. \tag{9}\]  
Hence, Selection Parameter for \(DS\) nodes, (\(SP_{DS}\)), is a function of nodes' depth and residual energy:  
\[SP_{DS}=f(D,Re) \tag{10}\]  
\(SP\) for \(DS\) nodes at time \(t\), \(SP_{DS}(t)\) is given as:  
\[SP_{DS}(t)=R_{e}(t)/D(t) \tag{11}\]  
where  
\(\mathbf{R_{e}(t)}\): Residual energy of any \(DS\) node at time \(t\) in joules  
\(\mathbf{D(t)}\): Depth of any \(DS\) node at time \(t\) in meters  
The higher the value of \(SP_{DS}(t)\) for any particular node, the greater the chance of its inclusion in \(DS\) and vice versa. \(DS\) node set recalculation is performed at regular time interval, \(t=120\) s, so that the nodes having minimum depth and high energy are selected as \(DS\) nodes. In this way, chances of data reception at MS are increased and a balanced energy consumption is involved in the network throughout the network operation. **Rule 8:** The highest energy one hop neighbors (belonging to \(N_{ind_{th}}\)) of vertices represented by \(V_{Hi}\) are considered eligible to be used as \(CCs\). \(CCs\) are the relay nodes used in cooperative routing. They connect source nodes with the \(DS\) nodes via an alternate path. For each \(R_{k}\), \(CC\) nodes are:  
\[CC_{k}\subset N_{ind_{th}}(V_{Hi})\ \forall\ V_{Hi}\in k \tag{12}\]  
**Rule 9:** The highest energy nodes are the \(CCs\) for cooperative routing.