## 5 Results
### Case 2: Maximizing Security and Scalability  
Given \(S(f)=1\) for \(f_{max}<f\) and scalability \((C=1\), _i.e_, \(T\geq D_{desired}\) and \(L\leq L_{max})\), the goal is to achieve maximum decentralization \((D=D_{max}=1)\). To maximize security and scalability, techniques like sharding or layered architectures [33] can be implemented to enhance scalability. In such systems, nodes handle subsets of transactions or participate in smaller consensus groups. However, maintaining security requires ensuring that each shard or layer remains secure against adversaries controlling up to \(f_{max}\) fraction of resources. The impact on decentralization is significant. A sharded network is divided into \(S\) shards, each containing \(n_{s}=\frac{N}{S}\) nodes. In Figure 12 above, we have a network divided into three shards, each containing four nodes, and the shards are interconnected to maintain communication across the network. Nodes participate in shard-specific consensus rather than global consensus. The effective decentralization \(D_{effective}\) can be represented as the following, where \(S\) is the number of shards, \(n_{s}\) the number of nodes per shard, and \(p_{i,j}\) is the participation level of node \(i\) in shard \(j\). \[D_{effective}=\frac{1}{S}\sum_{j=1}^{S}\frac{1}{n_{s}}\sum_{i=1}^{n_{s}}p_{i,j} \tag{40}\]  
As \(S\) increases, \(n_{s}\) decreases, which reduces participation per shard. This reduction in node participation has security implications, as the probability of an adversary compromising a shard \(P_{compromise}\) increases. The minimum threshold for the number of shards \(S\) largely depends on the acceptable security margin and the configuration of malicious participants, as well as the total number of nodes \(N\). For example, consider a network of 100 nodes divided into 10 shards (10 nodes per shard). This configuration generally provides stronger security and decentralization than a network of 10 nodes divided into 10 shards (one node per shard). In the latter scenario, a single node per shard effectively gives that node full control over its shard, eliminating any meaningful security guarantees. Additionally, the choice of consensus protocol influences the minimum shard size. For instance, under Practical Byzantine Fault Tolerance (PBFT) or similar Byzantine-fault-tolerant protocols, ensuring that each shard contains at least 3-4 nodes helps maintain the ability to identify and tolerate malicious actors. If the shard size falls below this threshold, the security and reliability of consensus operations are severely diminished. We start with a blockchain network that is split into multiple shards. Suppose the network has \(N\) nodes  
Figure 12: A network of 12 nodes sharded into three shards. divided into \(S\) shards, each containing \(n_{s}=\frac{N}{S}\) nodes. Among the total \(N\) nodes, a fraction \(f\) is controlled by and adversary. This means that the adversary controls \(f\times N\) nodes in the entire network. We define \(f_{max}\) as the maximum fraction of nodes in a shard that can be under adversarial control before the shard is considered compromised. In other words, if the adversary controls more than \(f_{max}\times n_{s}\) nodes within a single shard, that shard's security is effectively broken. \[t=f_{max}\times n_{s} \tag{41}\]  
This value \(t\) is the threshold number of adversarial nodes above which the shard is deemed compromised. We model the process of selecting which nodes end up in a given shard as random. Each node in the network has a probability \(f\) of being adversarial-controlled. We operate under the assumption that node distribution and adversarial control occur independently. The number of adversarial nodes in a shard is a random variable. The scenario fits well with a binomial distribution, which describes the probability of having exactly \(k\) successes in \(n_{s}\) independent Bernoulli trials, each with success probability \(f\).