# Wireless May Benefit Blockchain +
## 5 Numerical Results
### Results and Discussion
#### 5.2.2 Decentralization  
Figure 6 indicates that a higher wireless disconnection rate increases the decentralization level. For PoS, the rate of validators among all the nodes is set to \(\mathsf{R_{v}}=0.2\). For PoC, the rate of node replacement in each round of consensus is set to \(\mathsf{R_{sf}}=0.9\), and the number of rounds between a replacement is \(\Delta_{\text{sfl}}=0\). To wit, \(\mathsf{R_{sfl}}\) gives how many of the current witnesses will be replaced with other ones; and \(\Delta_{\text{sfl}}\) means how frequently the replacement occurs.  
Figure 5: PoC—scalability R versus the number of nodes (\(\mathsf{p_{fail}}=\{1,5,10\}\)%, 10 transactions per block).  
Recall that G closer to 0 gives a higher level of decentralization. (This may be a surprise to some readers: the wireless connection actually helps improve the decentralization in blockchain.) The reason for this result is straightforward. PoS and PoC allow only \"selective\" nodes to participate in a consensus. The intermittent wireless connection hinders such participation. Therefore, the level of \"inequality\" is decreased among nodes participating in consensus. All in all, very interestingly, the intermittence in wireless connectivity contributes to improving the decentralization in a blockchain consensus process.  
Notice from Figure 6 that the rate of cluster (denoted by \(\mathsf{R_{cls}}\)) was varied with PoW only. The rationale is that PoW is particularly susceptible to such clusteredness of node failure transpiration. In contrast, PoS and PoC will be less impacted since they already adopt \"selective\" participation of nodes in a consensus anyway; to wit, if failures occur at non-participating nodes, they have almost zero impact on consensus. We elaborate that the rate of cluster differentiating the PoW curves in Figure 6 is formally written as  
\[r_{\text{cls}}=\frac{\sum_{i=1}^{\mathbb{N}[\mathsf{S_{cls}}]}\lvert\mathbb{R _{cls,i}}\rvert}{\lvert\mathbb{R}^{2}\rvert} \tag{10}\]  
where \(\mathbb{R_{cls,i}}\in\mathbb{S_{cls}}\) denotes the \(i\)th cluster within the entire region \(\mathbb{R}^{2}\); and \(\lvert\cdot\rvert\) gives the area of a two-dimensional space.  
One should also notice that the three consensus mechanisms pose the order of \(\text{PoW}>\text{PoC}>\text{PoS}\) in terms of the decentralization level. This reflects the proportion of chance participating in a consensus: PoS is the most selective according to the number of staked tokens; PoC is less selective as the random wireless connectivity gives a chance of rotating participation in consensus; and PoW is almost not selective by its definition.  
Figure 7 is focused on PoS to highlight the impacts of the validator selectivity \(\mathsf{R_{v}}\) on the decentralization level G. The result is forthright: a lower \(\mathsf{R_{v}}\) (indicating the election of a larger number of validators) yields a lower G (meaning a higher level of decentralization).  
Figure 8 visualizes the impact of wireless disconnection on the decentralization level in PoC. The same phenomenon holds that a higher level of disconnection improves decentralization. The particular focus in the figure is the variation of \(\Delta_{\text{sfl}}\), the number of consensus rounds between participating nodes elections: the decentralization is degraded as the re-election of witnesses occurs less frequently.