## 4 Mathematical Model
Where, \(Ps\) of any node \(i\) belonging to \(V_{Me}\) as \(CC\) node is:  
\[\forall\,i\in V_{Me}:Ps\propto R_{e}(i) \tag{13}\]  
Hence, Selection Parameter for \(CC\) nodes, (\(SP_{CC}\)), is a function of nodes' residual energy:  
\[SP_{CC}=f(R_{e}) \tag{14}\]  
\(SP\) for the \(CC\) at time \(t\) is  
\[SP_{CC}(t)=R_{e}(t) \tag{15}\]  
The larger the value of \(SP_{CC}(t)\) for any particular node belonging to \(V_{Me}\), the greater the chances of its inclusion in \(CC\) node set and vice versa. \(CC\) node set recalculation is performed at regular time interval, \(t=120\) s, so that high energy nodes are selected as \(CC\) nodes and a balanced energy consumption is involved in the network throughout the network operation. **Rule 10:** All \(CC\) and non-\(CC\) medium depth vertices in each \(R_{k}\) forward their data towards non-\(DS\) low depth vertices directly or in a cooperative manner (depending on node availability). \[\forall\,i\in V_{Me}\mid i\in CC_{k},R=j\in V_{Me}\mid j\
otin CC_{k}\, \&\,D=l\in V_{Lo}\mid l\
otin DS_{k} \tag{16}\]  
\[\forall\,i\in V_{Me}\mid i\
otin CC_{k},R=j\in V_{Me}\mid j\
otin CC_{k}\, \&\,D=l\in V_{Lo}\mid l\
otin DS_{k} \tag{17}\]  
Similarly, all non-\(DS\) low depth vertices in any sub-graph \(k\) forward their data towards MSs directly or in a cooperative manner (depending on node availability). For any particular \(R_{k}\);  
\[\forall\,i\in V_{Lo}\mid i\in DS_{k},D=MS\,\&\,R=j\in V_{Lo}\mid j\
otin DS_{k} \tag{18}\]  
\[\forall\,i\in V_{Lo}\mid i\
otin DS_{k},D=MS\,\&\,R=j\in V_{Lo}\mid j\
otin DS_{k} \tag{19}\]  
where \(D\) and \(R\) specify destination and relay nodes. For this specific scenario, \(D\) and \(R\) are used instead of \(DS\) and \(CC\) due to the reason that \(D\) and \(R\) do not belong to \(DS\) or \(CC\) node set. \(DS\) recalculation is performed at regular time interval, \(t=120\) s, in order to achieve balanced energy consumption throughout the network operation. The nodes require residual energy information in DS re-calculation. However, in contrast to the existing iAMCTD, the information exchange in our proposed work is limited to one-hop neighbors only. In this way, the overhead is reduced. The flow chart in Figure 4 shows the selection of relay and destination nodes which is then followed by cooperative routing. All nodes find their one-hop neighbors at regular time intervals, \(t_{o}\) and define their \(d_{th}\) by following rule 4. If a node does not possess any neighboring node it goes in the sleep mode in which it does not perform transmission/reception. After selection of \(d_{th}\), neighboring nodes are differentiated among \(N_{ind_{th}}\) and \(N_{outd_{th}}\) according to rule 5. Nodes belonging to \(V_{Hi}\) use rules 6-9 for the selection of \(CC\) and \(DS\) nodes. \(CC\) node belongs to \(N_{indth}\) and possesses the highest \(R_{e}\), whereas, destination node is one that belongs to \(N_{outd_{th}}\) and possesses the highest \(R_{e}\) and least depth. Nodes belonging to \(V_{Me}\) and \(V_{Lo}\) follow rule 10 for selection of relay and destination nodes. This is because all the nodes that belong to \(V_{Me}\) and \(V_{Lo}\) might not act as \(CC\) or \(DS\) nodes, however, all of them have to forward their data to mobile sink and they forward it by following rule 10. If the neighbor finder node belongs to \(V_{Me}\) of any particular sub-graph, then it selects any non-\(CC\) node as a relay andnon-\(DS\) node as a destination node. On the other hand, if the neighbor finder node belongs to \(V_{Lo}\) of any particular sub-graph, then it selects any non-\(DS\) node as a relay, MS as a destination node and forwards data cooperatively. Now, let us consider a special case of a node which does not find its neighbors. How will it communicate? Well, the chances of the occurrence of this special case are very rare due to two reasons: (i) nodes are densely deployed; and (ii) all the nodes uniformly consume energy.