# Application of Graph Theory for Blockchain Technologies
## 5 Graph Theory Application in Blockchain Systems
### Evaluating Security and Privacy
#### 5.2.2 Threat Modelling
For all \((u,v)\in E_{L}\), if \(v\in V_{LC}\), then \(V_{FC}\) is in \(R(u,v)\). * Spatial Partitioning Effect: The network experiences spatial partitioning where \(V_{LC}\) only receives blockchain data from \(V_{FC}\). The integrity of the data received by \(V_{LC}\) is compromised, and the attacker can control the blockchain view of \(V_{LC}\). * Control over Lightweight Nodes: The attacker manipulates blockchain data flowing to \(V_{LC}\), effectively controlling these nodes. Figure 8: Phishing Attacks. Figure 9 graph represents a simulated blockchain network and illustrates the impact of BGP hijacking on network participants. The graph contains two types of nodes, full nodes and lightweight nodes, each playing a different role in the blockchain network. Full nodes, labeled as \"Full_Node_x\", are responsible for maintaining a complete copy of the blockchain ledger and validating transactions. Lightweight nodes, labeled as \"Lightweight_Node_x\", rely on full nodes for transaction validation and network information. The color of the nodes indicates their security status. Green nodes are secure and operating as intended. Red nodes have been compromised through BGP hijacking, indicating that an attacker controls them. The attacker can manipulate the data that flow through these nodes, potentially leading to incorrect transaction validations or false information being fed to the rest of the network. The orange nodes in the graph represent a state that is between the secure represented in green and compromised in red statuses. Edges connecting the nodes represent communication paths. In the context of a BGP hijack, these paths may be altered to route through compromised nodes, which is a critical aspect of the attack as it can lead to network partitioning and isolation of certain nodes from the rest of the network. The graph organizes nodes in layers, with full nodes being more centrally located, representing their pivotal role in the network's operation. The lightweight nodes are positioned around the periphery, demonstrating their reliance on the full nodes for information. The compromised full nodes (central red nodes) effectively create a false sub-network that can mislead the connected lightweight nodes. This spatial partitioning demonstrates the potential of a BGP hijack to disrupt blockchain network operations by isolating and controlling segments of the network. The simulation aims to visualize how BGP hijacks can lead to significant vulnerabilities within blockchain infrastructures, particularly for those nodes that do not hold a full copy of the blockchain and depend on others for accurate information. In the context of BGP hijacking and blockchain, an orange node indicates a full node that is under attack but not fully compromised, or it could signify a node whose status is uncertain or in the process of being verified. BGP hijacking can lead to a spatial partitioning of a blockchain network by compromising the integrity of full nodes and, by extension, controlling associated lightweight nodes. The model highlights the critical importance of securing full nodes against such routing attacks, as they are pivotal in ensuring the correct operation of lightweight nodes within the blockchain infrastructure. This proof highlights the need for robust, secure  
Figure 9: BGP hijacking. routing protocols and practices within blockchain networks to prevent such disruptive spatial partitioning. Eclipse attack is characterized by the isolation of honest nodes in a blockchain network. The adversary strategically places compromised nodes to intercept or eclipse the victim's connections, controlling their view of the network and potentially leading to double-spending or other fraudulent activities. **Assumption 9**.