# Application of Graph Theory for Blockchain Technologies
## 5 Graph Theory Application in Blockchain Systems
### Evaluating Security and Privacy
#### 5.2.2 Threat Modelling
From the graph Figure 7, we can infer that the network is under a DNS attack, where certain DNS queries from users are being intercepted and redirected to malicious DNS servers. This redirection can lead to a range of malicious activities, such as phishing, spreading malware, or data interception. The graph visually demonstrates the extent of the DNS attack, showing which users are potentially affected by the compromised DNS servers. DNS attacks have the potential to disrupt the normal functionality of a blockchain network by isolating users and miners from the real network. This graph theory-based proof highlights the critical need for secure DNS protocols and the implementation of robust, decentralized resolution mechanisms within blockchain infrastructures to safeguard against such vulnerabilities. The integrity and reliability of blockchain communications rely on the ability to resist such isolation attacks, reinforcing the network's foundational premise of decentralized security. Phishing attacks. Graph theory offers a systematic way to model and analyze the dynamics of phishing attacks in a network, especially in the context of blockchain where key compromise can lead to significant security breaches. This is a graph theory-based mathematical model to understand phishing for key compromise. **Assumption 7**. :
* _Let us construct a graph_ \(G=(V,E)\) _to model the blockchain network where:_
* _V: Set of nodes, representing users of the blockchain network._
* _E: Set of edges representing trust relationships or communication channels between users._
* _Each node_ \(v\) _has an associated trust level_ \(t(v)\)_, representing the user's vulnerability to phishing attacks, with a higher trust level indicating greater susceptibility._  
**Application Statement 7**. :
* _A phishing attack within a blockchain network graph_ \(G\) _can result in private key theft if the attacker can successfully deceive a subset of nodes_ \(V_{p}\subseteq V\)_, where nodes in_ \(V_{p}\) _have a high trust level_ \(t(v)\)_._  
Proof. :
* Targeting High Trust Nodes: The attacker identifies nodes with high trust levels \(V_{p}\) by analyzing the graph for nodes with the most edges, indicating a high interaction with other nodes, and therefore, higher visibility for phishing attacks. * Trust Exploitation: Through phishing techniques, the attacker attempts to compromise nodes in \(V_{p}\) by exploiting their trust levels. Let \(t_{thresh}\) be the threshold above which the nodes are highly susceptible. Then, \(V_{p}~{}=~{}\{v\in V|t(v)>t_{thresh}\}\). * Key Compromise: The attacker crafts deceptive messages that are sent across edges \(E_{p}\subseteq E\) to nodes in \(V_{p}\). If a node \(v\in V_{p}\) is deceived, its private key is considered compromised, transitioning its status to \(s(v)=C\). * Asset Theft: Using the compromised private keys, the attacker gains access to the nodes' assets and can perform unauthorized transactions, leading to theft. Figure 7: DNS attack. The critical importance of user awareness and secure communication within blockchain networks. The trust levels within a network, as modeled by graph \(G\), are directly correlated to the vulnerability of users to such attacks. Network Representation. Let \(G=(V,E)\) be a directed graph where \(V\) represents the set of nodes (participants) in the network, and \(E\) represents the set of directed edges (communications or transactions) between these nodes. Node Attributes. Each node \(v\in V\) has several attributes:  
* Key Status \(k(v)\): Indicates whether the private key of the node is compromised (\(C\)) or secure (\(S\)). * Phishing Susceptibility \(p(v)\): A probabilistic measure of the node's vulnerability to phishing attacks, ranging from 0 (immune) to 1 (highly susceptible). Edge Dynamics. Edges in \(E\) represent potential channels through which phishing attacks can be propagated or information can be exchanged, such as email communications, social media interactions, or blockchain transactions. Attack Dynamics.