# Wireless May Benefit Blockchain +
## 4 Analysis on Consensus Performance
### Decentralization  
Now, we proceed to formulating the decentralization part of our proposed analysis framework. We propose to adopt the Gini coefficient (denoted by G) to represent the level of decentralization. It is noteworthy that we modify G in such a way that the \"inequality\" indicates the level of participations being concentrated to a fewer number of nodes, which can also be understood as \"centralized\". We justify the selection of the Gini coefficient as the metric measuring the decentralization level as follows. The Gini coefficient is a widely recognized and intuitive metric for measuring inequality and has been effectively adapted to assess the degree of decentralization in blockchain networks [59]. Its primary strength lies in its ability to represent the distribution of power or resources (e.g., hash rate, stake ownership) across participating nodes, providing a single scalar value that is easy to interpret. There are other metrics that could be considered for the measurement of decentralization. [PERSON]'s index is one of them, which is derived from information theory and usually requires logarithmic computations [60], whereas, the Gini coefficient directly captures the relative dispersion within the network; this simplicity makes it particularly well suited for comparative studies and stakeholder communication. Furthermore, the Gini coefficient is scale-invariant and robust to varying network sizes, ensuring that decentralization assessments remain meaningful even as the network grows [61]. As such, the Gini coefficient was chosen for its broad adoption in prior blockchain research and its ability to succinctly encapsulate inequality, which is central to understanding and comparing the decentralization of blockchain systems. Let us consider an example scenario. Suppose a blockchain that has accomplished 100 consensuses. We divide them up among 10 nodes, giving the first 1, the second 3, etc., so that the \(k\)th person is assigned to the distribution function \(u(k)=2k-1\). Note that as such, in this distribution we are ranking the nodes in ascending order according to the number of transactions whose consensus they have taken part in. Also, suppose that there can be inequality: one node has partaken in only a single consensus, whereas another node has in 19 consensuses. Given this distribution function, we can define the associated Lorenz curve, \(L(x)\), as the graph of the cumulative proportion function. In other words, \(L(x)\) is the proportion of consensuses participated by the poorest \(100x\%\) of the population. In our example, \(L(0.1)=1/100\), \(L(0,2)=4/100\), and in general, \(L(x)=x^{2}\forall x=0,0.1,0.2,\cdots,1\). So, for example, \(L(0.5)=0.25\) indicates that the bottom 50% of the population has participated in 25% of all the consensuses. The top 10% of the population has taken part in \(1-L(0.9)=19/100\) or 19% of all the consensuses. Building on this understanding, one can find a more formal way to write this decentralization analysis. We define the Gini coefficient as half of the relative mean absolute difference, which is equivalent to the definition based on the Lorenz curve [62]. The mean absolute difference is the average absolute difference of all the pairs of items of the population, and the relative mean absolute difference is the mean absolute difference divided by the average, \(\bar{x}\), to normalize for scale. If \(x_{i}\) is the number of node \(i\)'s participations in consensus, and there are \(n\) nodes, then \(\mathsf{G}\) is given by  
\[\mathsf{G} =\frac{\sum_{i=1}^{n}\sum_{j=1}^{n}\bigl{|}x_{i}-x_{j}\bigr{|}} {2\sum_{i=1}^{n}\sum_{j=1}^{n}x_{j}}\] \[=\frac{\sum_{i=1}^{n}\sum_{j=1}^{n}\bigl{|}x_{i}-x_{j}\bigr{|}}{2 n^{2}\bar{x}} \tag{8}\]  
where \(\frac{1}{2}\) exists because the \(\sum_{i}\sum_{j}\) yields two sums for each pair of \(i\) and \(j\) and thus needs to be divided by 2. Further, \(\frac{1}{n^{2}}\) is to compensate the numerator's scale-up by being summed among \(n\times n\) instances.