# Authenticity, and Approval Framework for Bus Transportation Based on Blockchain 2.0 Technology
## 6 Methodology
The standardized direct influence matrix, \(\widetilde{X}\), shown in Table 13, is obtained as the result: \[\widetilde{X}=\frac{1}{max_{1\leq i\leq n}}\sum_{j=1}^{n}x_{ij}.\] (2) \begin{table}
\begin{tabular}{c c c c} \hline \multirow{2}{*}{**Number of Users**} & \multicolumn{3}{c}{**Processing Time**} \\ \cline{2-4}  & **Proof of Stake** & **Proof of Stake with GA Algorithms** & **Proof of Stake with Tabu Algorithms** \\ \hline
10 & 0.25 & 0.21 & 0.21 \\ \hline
100 & 1.06 & 0.98 & 0.72 \\ \hline
500 & 1.89 & 1.25 & 1.09 \\ \hline
1000 & 2.098 & 1.568 & 1.23 \\ \hline \end{tabular}
\end{table} 
Table 10: Time complexity of system’s performance with and without stochastic algorithms. Figure 11: Performance analysis of the proposed solution with and without the stochastic algorithms. * The comprehensive influence matrix \(T\), shown in Table 14, is obtained as the next step, according to Equation (3) [2], where \(I\) is the identity matrix of size (\(9\times 9\)): \[T=\widetilde{X}(I-\widetilde{X})^{-1}.\] (3)
* Next, we calculated two important factors from the \(T\) matrix, which were influence degree \(D\) and affected degree \(R\), as shown by Equations (4) and (5) [2]: \[D_{i} =\sum_{j=1}^{n}t_{i,j}\] (4) \[R_{i} =\sum_{i=1}^{n}t_{i,j}.\] (5) Furthermore, centrality and causality were calculated. Centrality was defined as the sum of \(D\) and \(R\), and causality was the difference between \(D\) and \(R\). The resultant comprehensive impact matrix is shown in Table 15, which contains centrality, causality, affected degree, and influence degree information. * The overall influence matrix is required, to determine how the overall influence relationship imitates factors in the system, because the comprehensive influence matrix \(T\) only shows mutual influence between different factors. Equation (6) [2] calculated the influence matrix \(H\), and its values are shown in Table 16: \[H=T+I.\] (6)
* The reachable matrix \(M\), as shown in Table 17, was calculated after applying threshold value \(\gamma\), which was calculated after taking the average of the T matrix. Consequently, M was obtained after some redundant factors were removed, according to Equation (7): \[\gamma =\sum_{i-1}^{n}\sum_{j=1}^{n}\frac{\left[t_{i,j}\right]}{N}\] (7) \[M =\left[m_{ij}\right]_{n\times m}\] (8) \[m_{ij} = \begin{cases}1,&\text{if }\gamma\geq h\\ 0,&\text{if }\gamma\leq h.\end{cases} \] (9)

From Table 15, it can be analyzed that factors \(F_{1}\) (transaction cost), \(F_{7}\) (scalability), and \(F_{8}\) (transparency) created an effect on other factors, as their causality value was greater than zero. Similarly, it can also be deduced that factors \(F_{2}\) (execution time), \(F_{3}\) (memory consumption), \(F_{4}\) (processing time), \(F_{5}\) (security), and \(F_{6}\) (portability) were greatly affected by other factors, as their causality value was less than zero. \(F_{2}\) (execution time) was affected by \(F_{1}\) (transaction cost) in the proposed BC-BAS system. Similarly, it can be observed that \(F_{5}\) (security) was greatly affected by \(F_{8}\) (transparency). Moreover, \(F_{7}\) (scalability) had a significant effect on the system's \(F_{6}\) (portability). Centrality identifies the most important factors in our proposed BC-BAS system. Subsequently, it can be observed that when sorted in descending order, \(F_{2}\) (execution time), \(F_{7}\) (scalability), \(F_{9}\) (reliability), \(F_{3}\) (memory consumption), and \(F_{5}\) (security), this order reflects the importance of these factors in our system. It can also be observed that memory consumption is very critical for our system, as shown in Figure 9d. Moreover, the \(F_{2}\) (execution time), \(F_{4}\) (processing time), \(F_{7}\) (scalability), \(F_{8}\) (transparency), and \(F_{9}\) (reliability) values reflect that these factors are the most influential elements of the system. For example, \(F_{2}\) greatly influences the overall system performance. Therefore, the proposed framework must consider the execution time of the system.