# A Population-Based Search Approach to Solve Continuous Distributed Constraint Optimization Problems
## 5 Experiment Results
### Statistical Analysis  
To clarify the superiority of our algorithms from a statistical perspective, we use the Wilcoxon signed rank test for statistical analysis on each benchmark problem with 100 agents, and the results are shown in Table 2. It can be clearly seen that our algorithms significantly outperform competing algorithms on three benchmark problems.  
Table 3 lists the average improvement rates of PLSA and PGSA compared to HCMS, PCD, C-CoCoA, and PCD-LD on three benchmark problems. In addition, we calculate the standard deviation of the improvement rates under the same problem and show it next to the average improvement rate, denoted as \(\sigma\). We can observe that the average improvement rates of our algorithms compared to HCMS and PCD are stable and more than 20%. Compared to C-CoCoA, the average improvement rates of PLSA and PGSA are not stable on different benchmark problems, since C-CoCoA has a better performance on simple graph structures. The average improvement rates of PLSA and PGSA are stable compared to PCD-LD, which are, on average, about 4% and 7%, respectively.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \multirow{2}{*}{**Type**} & \multirow{2}{*}{**Ours**} & \multicolumn{3}{c}{**HCMS**} & \multicolumn{3}{c}{**PCD**} & \multicolumn{3}{c}{**C-CoCoA**} & \multicolumn{3}{c}{**PCD-LD**} \\ \cline{3-13}  & & \(R^{+}/R^{-}\) & \(w^{+}/w^{-}\) & \(p\) & \(R^{+}/R^{-}\) & \(w^{+}/w^{-}\) & \(p\) & \(R^{+}/R^{-}\) & \(w^{+}/w^{-}\) & \(p\) & \(R^{+}/R^{-}\) & \(w^{+}/w^{-}\) & \(p\) \\ \hline \multirow{2}{*}{Sparse} & PLSA & 30/0 & 465/0 & 0.000 & 30/0 & 465/0 & 0.000 & 30/0 & 465/0 & 0.000 & 28/2 & 441/24 & 0.000 \\  & PGSA & 30/0 & 465/0 & 0.000 & 30/0 & 465/0 & 0.000 & 30/0 & 465/0 & 0.000 & 29/1 & 464/1 & 0.000 \\ \hline \multirow{2}{*}{Dense} & PLSA & 30/0 & 465/0 & 0.000 & 30/0 & 465/0 & 0.000 & 30/0 & 465/0 & 0.000 & 30/0 & 465/0 & 0.000 \\  & PGSA & 30/0 & 465/0 & 0.000 & 30/0 & 465/0 & 0.000 & 30/0 & 465/0 & 0.000 & 30/0 & 465/0 & 0.000 \\ \hline \multirow{2}{*}{Scale-free} & PLSA & 30/0 & 465/0 & 0.000 & 30/0 & 465/0 & 0.000 & 28/2 & 462/3 & 0.000 & 25/5 & 414/51 & 0.000 \\  & PGSA & 30/0 & 465/0 & 0.000 & 30/0 & 465/0 & 0.000 & 30/0 & 465/0 & 0.000 & 29/1 & 464/1 & 0.000 \\ \hline \multirow{2}{*}{Small-world} & PLSA & 30/0 & 465/0 & 0.000 & 30/0 & 465/0 & 0.000 & 29/1 & 463/2 & 0.000 & 22/8 & 375/90 & 0.003 \\  & PGSA & 30/0 & 465/0 & 0.000 & 30/0 & 465/0 & 0.000 & 30/0 & 465/0 & 0.000 & 30/0 & 465/0 & 0.000 \\ \hline \end{tabular}
\end{table}
Table 2: Wilcoxon signed rank test results of solution quality over 30 independent runs of our algorithms and competing algorithms with the significance level \(a_{sl}=0.05\), where \(R^{+}\) and \(w^{-}\), respectively, represent the times and rank sum. The solution quality of our algorithms is better than that of competing algorithms; \(R^{-}\) and \(w^{-}\) mean the opposite, respectively. In addition, \(p\) represents the \(p\)-value.  
Figure 8: The solution quality of PLSA and PGSA against the competing algorithms with different numbers of agents. (**a**) Sparse random graphs. (**b**) Sense random graphs. (**c**) Scale-free networks. (**d**) Small-world networks.