## 5 Results
### Case 2: Maximizing Security and Scalability
Here, a \"success\" is the event that a node is adversarial controlled, thus:  
\[X\ \sim\ Binomial(n_{s},f) \tag{42}\]  
where \(X\) is the number of adversarial nodes in a shard of size \(n_{s}\). The shard is compromised if the number of adversarial nodes \(X\) is at least \(t=f_{max}\times n_{s}\). \[P_{compromise}=P(X\geq f_{max}\times n_{s}) \tag{43}\]  
The probability \(P(X=k)\) of having exactly \(k\) adversarial nodes is given by the binomial probability formula. \[P(X=k)=\binom{n_{s}}{k}f^{k}(1-f)^{n_{s}-k} \tag{44}\]  
Since we want \(X\geq t\), the probability can be represented as above, where:  
* \(f_{max}\): The maximum fraction of nodes that an adversary is allowed to control without compromising the security of the shard. * \(n_{s}\): The number of nodes in each shard. * \(f\): The fraction of nodes controlled by the adversary. * \(k\): The number of nodes controlled by the adversary in the shard. \[P_{compromise}=\sum_{k=f_{max}n_{s}}^{n_{s}}\binom{n_{s}}{k}f^{k}(1-f)^{n_{s}-k} \tag{45}\]  
As \(n_{s}\) decreases, \(P_{compromise}\) increases, making each shard more vulnerable to adversarial control. Furthermore, introducing super-nodes or coordinators to manage inter-shard communication may introduce central control points, reducing overall decentralization. In conclusion, techniques to achieve \(C=1\) and \(S(f)=1\) necessitate reducing effective decentralization \((D<D_{max})\). Therefore, maximizing security and scalability leads to a decrease in decentralization, resulting in a contradiction. The following section presents an example for calculating the effective decentralization \(D_{effective}\) and the probability of shard compromise \(P_{compromise}\) for a network consisting of nine nodes, organized into three shards, with two nodes being malicious (see Figure 13). Parameters of the Graph
* \(N=9\): Total number of nodes. * \(S=3\): Number of shards. * \(n_{s}=\frac{9}{3}=3\): Number of nodes per shard. * Honest nodes: \(p_{i,j}=1\) (full participation). * Malicious nodes (Node 2 and Node 8): \(p_{i,j}=0.5\) (reduced participation). * \(f_{max}=\frac{1}{3}\): Maximum fraction of nodes that can be controlled by the adversary without compromising security. * \(f_{malicious}=\frac{2}{9}\ \approx 0.222\) Participation Level in Each Shard For each shard, we calculate the sum of participation levels of nodes in that shard. Shard 1 (Nodes 1, 2, 3):
* Node 1 (honest): Participation level = 1. * Node 2 (malicious): Participation level = 0.5. * Node 3 (honest): Participation level = 1. * Total participation for Shard 1 = 1 + 0.5 + 1 = 2.5. Shard 2 (Nodes 4, 5, 6): All nodes are honest: Participation level = 1 for each. * Total participation for Shard 2 = 1 + 1 + 1 = 3. Shard 3 (Nodes 7, 8, 9):
* Node 7 and Node 9 (honest): Participation level = 1 for each. * Node 8 (malicious): Participation level = 0.5. * Total participation for Shard 3 = 1 + 0.5 + 1 = 2.5. Average Participation in Each Shard
* Shard 1: Average participation = 2.5/3 = 0.8333. * Shard 2: Average participation = 3/3 = 1.0. * Shard 3: Average participation = 2.5/3 = 0.8333. Effective Decentralization \(\left(D_{effective}\right)\)  
Figure 13: Graph of nine nodes with two malicious nodes. The effective decentralization is calculated by averaging the participation values across all shards. \[D_{effective}=\frac{1}{S}\sum_{j=1}^{S}(\text{{Average Participation in Shard j}}) \tag{46}\]  
\[D_{effective}=\frac{1}{3}\left(\frac{2.5}{3}+1+\frac{2.5}{3}\right)=0.8889 \tag{47}\]  
Probability of Shard Compromise (\(P_{compromise}\))  
The probability that an adversary compromises a shard depends on the number of nodes they control within a shard. We need to calculate the probability that the adversary controls at least \(f_{max}\times n_{s}\) nodes in a shard. Figure 12:0  
The probability is calculated using the binomial distribution formula, as shown in Equation (45) were \(\binom{n_{s}}{k}\) represents the number of ways to choose \(k\) nodes out of \(n_{s}\).