## 3 Methods
### Scalability (C)  
Scalability (C) refers to the blockchain's ability to handle an increasing volume of transactions efficiently, without significant degradation in performance or excessive consumption of resources. Scalability is a critical property for blockchains to support widespread adoption and practical applications. To understand scalability, we use two key metrics: throughput and latency.  
* Throughput (\(T\)): This represents the number of transactions processed per second (TPS). High throughput is essential for handling a large volume of transactions in real-time applications.
* Latency (\(L\)): This is the time it takes for a transaction to be confirmed and finalized in the network. Low latency is crucial to ensuring that transactions are processed quickly, providing a smooth user experience.  
However, simply defining throughput and latency is not enough; for scalability to be meaningful, we must also introduce thresholds that determine the desired performance:  
* Throughput Threshold (\(T_{desired}\)): This is the minimum TPS required for the blockchain to be practical in real-world applications. If throughput is below this threshold, the system may struggle to meet user demands.
* Latency Threshold (\(L_{max}\)): This defines the maximum acceptable confirmation time for transactions to ensure a good user experience. If latency exceeds this threshold, the system may appear too slow for end-users.  
Given these metrics, we can formally define the scalability function as follows.  
\[C(T,L)=\begin{cases}1&if\ T\geq T_{desired}\ and\ L\leq L_{max}\\ 0&otherwise\end{cases} \tag{34}\]  
This function indicates that scalability is achieved when the system's throughput meets or exceeds \(T_{desired}\), while simultaneously maintaining latency below or equal to \(L_{max}\). If either of these conditions is not met, the system is considered non-scalable.  
To better understand how throughput and latency interact in determining scalability, consider the graph in Figure 6. The figure includes a performance frontier (black line) that represents the best achievable trade-offs between throughput (\(T\)) and latency (\(L\)) under various system configurations. The vertical red dashed line marks the desired throughput threshold (\(T_{desired}\)), and the horizontal blue dashed line indicates the maximum acceptable latency (\(L_{max}\)). The shaded green region to the right of \(T_{desired}\) and below \(L_{max}\) highlights  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Consensus Protocol** & \(f_{max}\) & **Notes** \\ \hline Proof-of-Work (PoW) & \(\frac{1}{2}\) & Requires \textgreater{} 50\% honest hashing power \\ Proof-of-Stake (PoS) & \(\frac{1}{2}\) & Requires \textgreater{} 50\% honest stake ownership \\ PBFT & \(\frac{1}{3}\) & Tolerates up to 1/3 faulty nodes \\ Delegated PoS (DPoS) & Varies & Security depends on the number and trustworthiness of delegates \\ \hline \hline \end{tabular}
\end{table}
Table 2: Consensus protocols and security thresholds.  
where both conditions are met, signaling the system's scalable operating range. This depiction not only shows that the system can sustain an acceptable latency while achieving the desired throughput, but also demonstrates how close the current configuration is to entering and maintaining that ideal scalability zone.