## 3 Methods
### Decentralization (D)  
Decentralization refers to the degree to which control and decision making are distributed among all participants in the network. A highly decentralized system prevents any single entity or small group from exerting undue influence, enhancing fairness and resilience. Let \(N\) be the total number of nodes in the network. We introduce the Participation Function to measure decentralization:  
\[D(N)=\frac{1}{N}{\sum_{i=1}^{N}{p_{i}}} \tag{1}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Study**} & **Formal** & **Consensus-** & **L2** & **Addresses Full Trilemma** \\  & **Proof** & **Framework** & **Compatible** & **Addresses Full Trilemma** \\ \hline [PERSON] et al. (2019) [16] & ✗ & ✗ & ✗ & ✗ \\ [PERSON] et al. (2020) [19] & ✗ & ✓ (Conceptual) & ✗ & ✗ \\ [PERSON] et al. (2023) [20] & ✓ & ✗ (PoW Limited) & ✗ & ✗ \\ [PERSON] et al. (2024) [21] & ✗ & ✗ & ✗ & ✗ \\ Our Method & ✓ & ✓ & ✓ & ✓ \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of methodologies and contributions in addressing the Blockchain Trilemma. where \(p_{i}\) is the participation level of the node \(i\), ranging from 0 (no participation) to 1 (full participation). It represents the extent of the node's involvement in network activities, such as transaction validation and consensus processes. In our context, we define this as the number of connections that a node can have; however, this could be generalized for any consensus or system, so we define it as follows. \[p_{i}=\frac{Degree\;of\;Node\;i}{Maximum\;Possible\;Degree} \tag{2}\]  
For maximum decentralization, we consider that every node participates fully and equally. \[p_{i}=1\;\;\forall i\;\epsilon\;[1,N] \tag{3}\]  
Thus, the maximum decentralization value is as follows. \[D_{max}=1 \tag{4}\]  
In an ideal peer-to-peer network, all nodes contribute equally to maintaining the ledger. This scenario reflects maximum decentralization (see Figure 2), as no node holds more influence than another. In another scenario (see Figure 3), in some systems, nodes with more tokens have greater influence. If a few nodes hold a significant portion of the total stake, \(p_{i}\) for those nodes, it increases, while \(p_{i}\) for others decreases, reducing the overall decentralization. Figure 3: Peer-to-peer network with weighted participation. Figure 2: Fully connected peer-to-peer network of eight nodes. Example 1: Fully Decentralized Network  
In this example (Figure 4), we have a network consisting of four nodes \((N=4)\) that are fully connected, meaning that each node communicates directly with every other node. The assumption here is that all nodes participate equally in all network activities such as transaction validation and consensus processes. This implies that the participation level of each node \((p_{i})\) is equal to 1. Therefore, the participation levels for all nodes are as follows. \[p_{1}=1 \tag{5}\]  
\[p_{2}=1 \tag{6}\]  
\[p_{3}=1 \tag{7}\]  
\[p_{4}=1 \tag{8}\]  
To calculate the Participation Function \(D(N)\), we apply the following formula. \[p_{i}=\frac{Degree\;of\;Node\;i}{Maximum\;Possible\;Degree} \tag{2}\]0  
In this case, \(D(N)=1\), which indicates maximum decentralization, since no single node has more influence than another. All nodes participate equally in maintaining the network. Example 2: Network with Uneven Participation Based on Node Degree  
In this example (Figure 5), we have a network consisting of five nodes \((N=5)\), and the participation level of each node \((p_{i})\) is based on its degree, which refers to the number of connections the node has to other nodes. The degree of a node represents its involvement in the network's activities, such as in communication and consensus processes. In this case, nodes have varying degrees of connection. The network is structured as follows:  
* Node 1 is connected to Node 2. * Node 2 is connected to Nodes 1, 3, and 5. * Node 3 is connected to Node 2. * Node 4 is connected to Node 5. * Node 5 is connected to Node 2 and 4.