## 5 Results and Analysis
### Connectivity  
Connectivity (CN) is defined as the ratio of the number of nodes involved in valid uplink communication (\(N_{C}\)) to the total number of nodes (\(N_{T}\)) in the network, shown as Equation (10). Note that a node involved in valid uplink communication means that it has at least one path to reach the BS:  
\[CN=\frac{N_{{}_{C}}}{N_{{}_{T}}} \tag{10}\]  
Figure 9 shows the results of connectivity versus time of three protocols. RRP outperforms the simple-link protocol and DRS under each simulated scanning angle. Simple-link tries to construct an uplink forest for data communication, so each node in the network has and only has one parent to forward sensed data. This protocol works without any reconfiguration scheme. Once a node runs out of energy as time goes by, it is expected that the network has a dramatic drop of connectivity since the dead node will cause its entire offspring to lose their communication with the BS. However, in DRS, nodes are isolated from each other after the phase of routing establishment. Similarly, in RRP, the dying nodes are also separated from other nodes when a SOS message has been sent out. The death of a node in both DRS and RRP does not influence the uplink communication of other nodes. Therefore, the connectivity of both DRS and RRP presents a continuous dropping trend while the simple-link shows a discrete dropping trend.  
Using DRS, it is observed that the network forms a star topology with the BS at the center after the phase of routing establishment. Sensor nodes use direct communication with the BS, which is different from RRP that includes both multi-hops and direct communication. To illustrate the difference of  
Figure 8: _Cont._energy dissipated in DRS and RRP, we consider the energy expended in transmitting a single _k_-bit message from a GS, say _S\({}_{i}\)_, located a distance \(d\) from the BS (shown in Figure 10) using a direct communication approach, then we have:  
\[E_{direct}=k\times\varepsilon\times\varphi\times d^{2} \tag{11}\]  
The energy required for multi-hops routing is:  
\[E_{multi-hops}=nE_{,xx}(k,r,\varphi)+nE_{,xx}(k)+E_{,xx}(k) \tag{12}\]  
Then, the direct communication requires less energy than multi-hop communication if \(E_{direct}<E_{multi-hops}\). According to Equations (3-5), we have:  
Figure 10: Comparison of direct and multi-hops communication.  
Figure 9: Connectivity _vs._ time under different scanning angles.  
\[d<\sqrt{n(r^{2}+\frac{E_{R}}{\varepsilon\times\varphi})+\frac{E_{CCR}}{\varepsilon \times\varphi}} \tag{13}\]  
Equation (13) implies that whether or not direct communication can reserve more energy than multi-hop communication closely relies on \(d\) and \(n\) when the network is set up. In DRS, the further a node is away from the BS, the faster it consumes energy. Those nodes located around the bound area generally die quickly. On the contrary, in RRP, most nodes in the network are just a few hops away from the CH, which can slow down their energy consumption rate. Besides, if a node becomes a DN, then its ANs will seek another short path back to the BS by reconfiguration. They will be oriented with the BS if and only if no qualified candidate in their circle area is available. Such strategy reduces the probability of direct transmission that may consume more energy if Equation (13) is not satisfied.  
Interestingly, the advantage of RRP over DRS in connectivity tends to get smaller as the scanning angle increases. Though in the case of larger scanning angle, a node may have higher probability to find a shorter path to the BS, the energy needed per transmission also increases. Actually, in our simulation scenario, which has sparse node density, the expanded scanning angle has speeded up the energy consumption as well as the start of reconfiguration in RRP. A node is likely to have lower probability to find a qualified candidate in its circle area, and in turn it uses direct communication with the BS, removing the gap between DRS and RRP.