# Application of Graph Theory for Blockchain Technologies
## 5 Graph Theory Application in Blockchain Systems
### Simulating Blockchain Networks
#### 5.4.1 Sybil Attacks  
Adding fake identities to subvert consensus is a known risk. Operating Sybil attacks on modeled transaction graphs measures the effectiveness of detection mechanisms [87]. Figure 19 presents a graph-based network modeling and simulation, we can analyze the resilience of blockchain consensus systems against illicit Sybil attacks and benchmark the effectiveness of Sybil detection mechanisms designed to secure network consensus. Light Blue Nodes represent legitimate peers in the blockchain network. Red Nodes represent Sybil nodes inserted into the network by an adversary. These are labeled with 'S' to indicate their Sybil status. Edges' connections between nodes in the network, including those between legitimate peers and Sybil nodes, illustrate how Sybil nodes integrate into the network to subvert consensus mechanisms  
Figure 18: Shaping peer networks. **Assumption 18**. :
* \(G(V,E)\) _represents the transaction or interaction graph of a blockchain network._
* \(S\subseteq V\) _represents the set of Sybil nodes inserted by an adversary._
* \(L\ =\ V\setminus S\) _represents the set of legitimate nodes in the network._  
**Application Statement 18**. :
* _Simulating Sybil attacks by introducing pseudo-nodes (S) into the modeled blockchain transaction graph_ \(G\) _allows for the quantitative evaluation of the effectiveness of Sybil detection mechanisms designed to secure network consensus._  
Proof. : Given a blockchain peer network modeled as an interaction graph \(G(V,E)\), where \(V\) represents the set of nodes in the network, including both legitimate peers and Sybil nodes, and \(E\) represents the set of edges indicating connections between these nodes, our objective is to simulate Sybil attacks and evaluate the effectiveness of Sybil detection mechanisms. 1. Interaction Graph modelling: Define \(G(V,E)\) as the interaction graph of the blockchain network, where nodes represent peers (legitimate and Sybil) and edges represent connections or transactions between these peers. 2. Sybil Node Insertion: An adversary introduces \(n\) Sybil nodes into \(G\), resulting in the updated graph \(G^{\prime}(V^{\prime},E^{\prime})\), where \(V^{\prime}\ =\ V\cup S\) and \(E^{\prime}\) includes edges connecting Sybil nodes to legitimate nodes. Mathematically, for each Sybil node \(s\in S\), connect \(s\) to at least one \(l\in L\) via edge \(e_{sl}\in E^{\prime}\). 3. Community Detection and Sybil Identification: Apply a community detection algorithm to \(G^{\prime}\) aiming to partition \(V^{\prime}\) into disjointed subsets where each subset represents a tightly knit community. Formally, identify partitions \(P_{1},P_{2},\ldots,P_{k}\) such that \(\bigcup_{i=1}^{k}P_{i}\ =\ V^{\prime}\) and \(P_{i}\cap P_{j}\ =\ \emptyset\) for \(i\
eq j\). Detect Sybil groups by analyzing community structures; communities with unusually high edge densities to specific external nodes (potential gateways) are flagged as Sybil. 4. Sybil Detection Algorithm Performance: Evaluate the detection algorithm by calculating the True Positive Rate (TPR) and False Positive Rate (FPR) based on the algorithm's ability to accurately identify inserted Sybil nodes. Define TPR as \(\frac{TP}{TP+TN}\) and FPR as \(\frac{FP}{FP+TN}\), where \(TP\) is true positives, \(FN\) is false negatives, \(FP\) is false positives, and \(TN\) is true negatives. 5. Benchmarking: Benchmark the Sybil detection algorithm's performance across various network topologies and Sybil attack scenarios by varying the number of Sybil nodes (\(n\)) and their connection patterns within \(G^{\prime}\). Figure 19: Sybil attacks. In this modeling of the blockchain network as an interaction graph and simulating Sybil attacks, we quantitatively evaluate the effectiveness of detection mechanisms against such attacks.