## 3 Results and Discussion
### Simulation for Data Accuracy in Randomly Placed Non-Hierarchical Bidirectional Graph
#### 3.3.3 Network Lifetime  
In this section, we investigate the network lifetime of a bidirectional graph in which retransmissions are not allowed. When retransmissions do not occur, the energy on acknowledgment frame/packet transmission and reception will be zero. Therefore, the Equations (7) and (8) will reduce to Equations (45) and (46), respectively. \[E_{TX}=X(P_{c}+P_{t}/\eta)(L_{d}/R_{d})) \tag{45}\]  
\[E_{RX}=X(P_{r}L_{d}/R_{d}) \tag{46}\]  
As proved in the complexity section, \(X=j,\ j,\ 3j\) for Average, Proposed, and EBDKF fusion techniques, respectively. \(P_{c}\) also varies depending on the complexity of the information fusion technique. Since the complexity of the three fusion techniques increase in the order of Average, Proposed and EBDKF; \(P_{c}\) is also expected to vary in the same manner. By arguing in this manner and using Equations (45) and (46); it can be derived that for an ad hoc network which retransmissions are not allowed, the network lifetime in decreasing order is Average, Proposed method and EBDKF. We simulate the network given in Figure 3 where \(j\) is approximately 3 to prove the preceding argument as shown in Figure 21. For this experiment also, we use Atmel studio 7 to evaluate the processing cycles to obtain the CPU time as shown in Table 6. \begin{table}
\begin{tabular}{c c c} \hline \hline \(j\) & **Fusion Function** & **Complexity of Fusion** \\ \hline
1 & Average & 2 Additions, 1 division \\
1 & Proposed method & 2 Additions, 5 divisions, 1 multiplication \\
1 & EBDKF & 6 Additions, 4 divisions, 8 multiplications \\
2 & Average & 3 Additions, 1 division \\
2 & Proposed method & 4 Additions, 7 divisions, 1 multiplication \\
2 & EBDKF & 10 Additions, 4 divisions, 8 multiplications \\
3 & Average & 4 Additions, 1 division \\
3 & Proposed Method & 6 Additions, 9 divisions, 1 multiplication \\
3 & EBDKF & 14 Additions, 4 divisions, 8 multiplications \\ n & Average & \((n+1)\) Additions, 1 division \\ n & Proposed Method & 2n Additions, \(3+2n\) divisions, 1 multiplication \\ n & EBDKF & \((4n+2)\) Additions, 4 divisions, 8 multiplications \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Average fusion complexity per node for different information fusion functions in a bidirectional graph. It can be observed that a similar performance with respect to network life time occurs when we employ the proposed statistical method and AVG method by observing the results in Figure 11. When the proposed statistical method or AVG is employed, 12 data gathering cycles are possible prior to losing the first node in the network. This value is much lower with respect to 59 obtained for proposed method in chain-based network as it was observed in Figure 11. The reason for the difference occurs as in the unstructured network considered, a node in average broadcasts 3 data frames/packets in a data cycle while in chain-based network it was only one. When observing Table 6, it can be observed that the fusion complexity of proposed method is three times that of average method. In spite of that, the network lifetime has been similar because of the equal j values and no retransmissions are allowed. The energy consumed for data fusion has been less significant with respect to the energy consumption for transmissions. That is because when combined with the fact that energy for transmitting a single bit is quite expensive than computing a single instruction, the energy consumption for the computational complexity of the proposed statistical method can be neglected. On the other hand, in EBDKF, the first node is died after four data gathering cycles. Further, we can observe that by 97 data gathering cycles, networks employing Average fusion and proposed method had lost 85% of nodes and EBDKF has lost all of its nodes. When observing Table 6 it is very clear that highest computational consumption for fusion belongs to EBDKF method and \(j=9\) are the reasons for having least network lifetime out of fusion techniques.