# Distributed Average Consensus Algorithms in d-Regular Bipartite Graphs: Comparative Study
## 4 Experiments and Discussion
### Experimental Results and Discussion about Observable Phenomena
CW achieved the highest performance with the highest examined \(\epsilon\), like for \(d=2\) and \(3\). However, GMH achieved the maximum convergence rate with \(\epsilon=0.6\); therefore, with a higher value of the mixing parameter than for \(d=3\). Again, we can see that an increase in \(\epsilon\) resulted in a decrease in the number of iterations for consensus until the maximum convergence rate was achieved, like for \(d=3\). Then, a further increase in \(\epsilon\) caused the performance of GMH to worsen. In addition, for \(d=5\) (see Figure 12 and Table A4 in Appendix A), OW outperformed the concurrent algorithms for each precision of the stopping criterion, like for the two previously analyzed values of \(d\). The second-best-performing algorithm for each precision of the used stopping criterion was paradoxically MH, which was the worst approach for \(d=2\), \(3\), and \(4\), as seen from Figures 9-11. Thus, GMH was the best performing with \(\epsilon=1\) (i.e., MH), which is, theoretically, the slowest initial configuration in non-regular non-bipartite topologies. For the low precision, the next in order was CW, but this algorithm performed the best with \(\epsilon=\) MD 0.8 in this case. The worst algorithm of the applied stopping criterion for the low precision was BC: the fastest algorithm with the Perron matrix. However, for the medium/high precision of the stopping criterion, BC outperformed CW with \(\epsilon=\) MD 0.8. The performance of CW increased with the growth of \(\epsilon\) until the maximum convergence rate was obtained. A further increase in \(\epsilon\) caused the algorithm to decelerate. In the case of GMH, increasing \(\epsilon\) ensured a better performance of the algorithm. Figure 10: Convergence rate expressed as number of iterations required for consensus achievement in 3-regular bipartite graphs and for three precisions of applied stopping criterion. In the largest examined graphs (i.e., \(d=10\), see Figure 13 and Table A5 in Appendix A), OW achieved the highest performance for each precision of the applied stopping criterion again. Here, BC performed well and was the second best performing in these graphs. The third best was CW but with \(\epsilon=\text{MD 0.7}\) in this case. The best-performing configuration of GMH was the configuration with \(\epsilon=1\) (MH), which was, however, outperformed by all the other concurrent approaches. As further seen from the figures, a decrease in \(\epsilon\) caused the performance of GMH to decline. In the case of CW, an increase in \(\epsilon\) ensured acceleration of the algorithm until the maximum convergence rate was achieved (note that this maximum was achieved for lower \(\epsilon\) than for \(d=5\)). Afterward, a further increase in the value of \(\epsilon\) decelerated the algorithm, like in the previous analysis. Figure 11: Convergence rate expressed as number of iterations required for consensus achievement in 4-regular bipartite graphs and for three precisions of applied stopping criterion. Figure 12: Convergence rate expressed as number of iterations required for consensus achievement in 5-regular bipartite graphs and for three precisions of applied stopping criterion. As also seen from Figures 9-13, MD and LD algorithms diverging in regular bipartite graphs were unable to achieve the consensus for each precision of the stopping criterion. One can also observe that an increase in the precision of the stopping criterion caused all the examined algorithms to decelerate. In addition, we see that an increase in \(d\) ensured that the algorithms performed better (the only exceptions were GMH and MH). In this paragraph, we summarize the results from Figures 9-13 for each examined algorithm to identify the best-performing approach in \(d\)-regular bipartite graphs. We can see from the presented results that the performance of MH was the worst in many scenarios. Its high performance was only observed for \(d=5\)--it was the optimal initial configuration of GMH in these graphs (and also for \(d=10\)).