## 6 Experiments and Discussion
### Discussion  
In this section, we discuss the results presented in Section 6.1 and compare them with those from Section 2 in order to compare the examined algorithms for distributed averaging with their application to distributed network size estimation.  
As shown in [23; 24; 25; 26; 27; 28; 29], \(BG\) outperforms \(RG\) and \(GG\) in terms of numerous various metrics. However, in [26], it is identified that the error of \(BG\) cannot drop below a threshold value whereby \(BG\) is outperformed by \(RG\) and \(GG\) for a higher number of time instances. This fact causes that \(BG\) requires the lowest number of sent messages for consensus achievement (significantly lower than the other algorithms), but its estimation precision is so low that this algorithm cannot be applied to estimating the network size--very low precision is achieved in all our experiments. Thus, a high convergence rate of \(BG\) at the cost of lower estimation precision can be beneficial if the arithmetic mean is estimated. However, in the case of network size estimation, its low precision makes this algorithm inapplicable to estimating the network size. Furthermore, it is shown in many papers that \(GG\) performs better than \(RG\) in numerous scenarios. However, in our analyses, \(GG\) requires the most sent messages for consensus achievement and thus is worse for network size estimation than \(RG\). In \(GG\), many redundant messages are often transmitted as two agents many hops away from one another with zero inner states perform the pairwise averaging operation (3). In our experiments, it is identified that these two algorithms achieve high precision in general and therefore can be used to estimate the network size in real-world systems.  
As shown above, a comparison of \(PS\) and \(PP\) both for distributed averaging is provided in [30; 31; 32; 33]. Generally, it is identified that \(PS\) outperforms \(PP\) except for several scenarios. In addition, the application of these algorithms to estimating network size is discussed in [30; 34; 37; 38; 39; 40; 41; 42], but a comprehensive analysis is provided in none of them. In our research, it is identified that there is no significant difference between these two algorithms (\(PS\) performs a bit better), their precision is high in general, and they require fewer messages for consensus than \(RG\) and \(GG\).  
In addition, our research identifies that the selection of the leader has only a marginal impact on the performance of all the five algorithms. According to the presented results, we can conclude that the best-performing algorithm for network size estimation is \(PS\), which requires the second-lowest number of sent messages for consensus, and its estimation precision is very high (except for cases when \(P\) takes high values in Scenario 1). The best performance of all the four applicable algorithms is achieved for \(P=0.001\) in Scenario 1,where the estimation precision is high, and the number of sent messages is lower than in Scenario 2 or in Scenario 1 with \(P\) = 0.0001. It is further identified that the precision in Scenario 2 is much higher than in Scenario 1 but at the cost of a significant increase in the number of sent messages.