## 5 Numerical Simulations
### Simulation Results  
The results of simulations of the matrix type (i) are reported in Table 4 and in Figure 3. The numbers reported in boldface font are calculated from the spanning trees. This happens when \(k=n(n-1)/2-(n-1)=(n-1)(n-2)/2\). Such numbers do not appear in Table 4 if \(n\geq 7\). Due to excessive processing time, we did not compute the average \(CR\) for all \(k>12\) except for some random large \(k\) values shown in Tables 6 and in 7. The first row in Table 4 represents the average \(CR\) of the original matrices when the number of missing entries is null, i.e., \(k=0\). It can be observed that the average \(CR\) decreases across the columns when the number of missing entries rises in each matrix size \(n\). On the contrary, the sequence is increasing across all rows with the exception of \(k=0\). Note that our 'initial complete matrices are, on average, inconsistent. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**Matrix Size \(n\)**} \\ \cline{2-9} \(k\) & **4** & **5** & **6** & **7** & **8** & **9** & **10** \\ \hline
0 & 1.0028 & 1.0007 & 1.0022 & 0.9990 & 0.9988 & 1.0003 & 1.0008 \\
1 & 0.6512 & 0.8350 & 0.9022 & 0.9357 & 0.9556 & 0.9675 & 0.9728 \\
2 & 0.3432 & 0.6682 & 0.8091 & 0.8731 & 0.9117 & 0.9365 & 0.9497 \\
3 & **0.0602** & 0.4983 & 0.7094 & 0.8066 & 0.8651 & 0.8989 & 0.9217 \\
4 & \(-\) & 0.3460 & 0.6062 & 0.7479 & 0.8219 & 0.8667 & 0.8940 \\
5 & \(-\) & 0.1904 & 0.5107 & 0.6744 & 0.7688 & 0.8338 & 0.8695 \\
6 & \(-\) & **0.0530** & 0.4113 & 0.6094 & 0.7296 & 0.8012 & 0.8447 \\
7 & \(-\) & \(-\) & 0.3161 & 0.5429 & 0.6768 & 0.7631 & 0.8209 \\
8 & \(-\) & \(-\) & 0.2168 & 0.4773 & 0.6348 & 0.7292 & 0.7907 \\
9 & \(-\) & \(-\) & 0.1313 & 0.4125 & 0.5855 & 0.6979 & 0.7647 \\
10 & \(-\) & \(-\) & **0.0481** & 0.3445 & 0.5420 & 0.6612 & 0.7369 \\
11 & \(-\) & \(-\) & \(-\) & 0.2811 & 0.4944 & 0.6278 & 0.7128 \\
12 & \(-\) & \(-\) & \(-\) & 0.2144 & 0.4432 & 0.5914 & 0.6845 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Average \(CR\) of 10,000 random matrices from the Saaty scale. Figure 3: Average \(CR\) of 10,000 matrices vs. the number of missing comparisons \(k\) with respect to the matrix size \(n\) corresponding to Table 4.  
The relations between the number of missing comparisons (\(k\)) and their respective average \(CR\) are depicted in Figure 3. As it can be seen from all line graphs, they are decreasing for each \(n\) while the number of missing comparisons (\(k\)) is increasing. Having the same type of simulations work, the average values of \(CR\) obtained from the matrix type (ii) are presented in Table 5 and in Figure 4. The numbers that appear in boldface font are also calculated from the spanning trees. The results are similar to Table 4 and Figure 3 in terms of monotonic values. An interesting simulation result, in this case, is that the average \(CR\) values of the last row in each column are less than [PERSON]'s threshold 0.1, even though the initial complete matrices had a \(CR\), on average, greater than 0.1. In our simulations, we observed the performance of the algorithm by taking the matrix size \(n\) from 4 up to 10, and the number of missing comparisons (\(k\)) up to 12 because of excessive computational time. However, the average CR for some number of missing comparisons (\(k\)) are reported in Tables 6 and 7, in order to examine the efficiency of the algorithm for large \(k\). Since our simulation results are based on connected graphs, the eigenvalue function is strictly convex and therefore the optimal solutions obtained by the proposed algorithm are unique. Moreover, the algorithm provides more consistent PCMs with more incomplete information. This means that when \(k\) is closer to \(n(n-1)/2\). We conclude that the algorithm performs well and is capable of providing an optimal completion for the incomplete PCM up to \(k=(n-1)(n-2)/2\). Furthermore, due to the connectedness of the associated undirected graphs used in our numerical simulations, the optimal solutions obtained by the method are unique (from Theorem 1).