# Application of Graph Theory for Blockchain Technologies
## 5 Graph Theory Application in Blockchain Systems
### Evaluating Security and Privacy
#### 5.2.2 Threat Modelling
:
* _For a blockchain network represented by graph_ \(G\)_, a successful DDoS attack is achieved by creating a demand on nodes such that_ \(d(v)>c(v)\) _for a significant subset_ \(V_{DDoS}\subseteq V\)_, leading to service failure for those nodes._  
Proof. :
* Attack Initialization: The attacker distributes a set of malicious nodes \(V_{M}\subseteq V\) or botnets within the network that initiate the attack. * Overwhelming Network Resources: Each malicious node \(m\in V_{M}\) generates a number of transactions \(t(m)\) directed at nodes in \(V\), such that \(\forall v\in V_{DDoS},\sum_{m\in V_{M}}t(m)>c(v)\). * Exploiting Network Control: If combined with a 51% attack, the attacker can prioritize malicious transactions or invalidate legitimate ones, increasing the network's congestion. The attacker can create intentional forks by generating blocks at a rate that overwhelms the network's ability to reach consensus, further contributing to the denial of service. Figure 10: Eclipse attack. * Service Failure: A significant number of nodes \(v\in V_{DDoS}\) are unable to process legitimate transactions, leading to service failure as defined by the inability of the network to perform its intended operations. Figure 11 depicts a highly interconnected network of nodes in a blockchain system. Nodes labeled \"Regular_Node_x\" are honest participants in the blockchain network. Nodes labeled \"Malicious_Node_x\" probably represent attackers or compromised nodes that may be part of a coordinated attack. Edges' dense interconnections shown by the multitude of lines between nodes imply a network where each node is in communication with many others, which is typical for peer-to-peer networks in blockchain systems. The blue color of the \"Regular_Node_x\" suggests they are standard, uncorrupted nodes. The red color of the \"Malicious_Node_x\" nodes implies danger or corruption, indicative of nodes that may be initiating malicious activities like a DDoS attack. The dense web of connections signifies that the network has a high level of redundancy and connectivity, which is usually a strength for resilience and distributed consensus. However, the presence of malicious nodes within this web can be a significant threat. The malicious can flood the network with superfluous requests, transactions, or blocks, attempting to overwhelm the system's computational resources, thereby slowing down or even halting legitimate network operations. The model concludes that a DDoS attack, particularly when coupled with a 51% attack or intentional forks, can cripple a blockchain network's functionality. This highlights the importance of implementing robust transaction validation mechanisms, anti-spam measures, and network capacity planning to mitigate the impact of DDoS and related attacks on blockchain systems. The resilience of blockchain networks against such attacks is crucial for maintaining service continuity and trust in the system. 51% attack in the context of blockchain technology occurs when a single entity gains control of the majority of the network's hash rate, enabling it to unilaterally alter the blockchain's state. This mathematical model, underpinned by graph theory, seeks to encapsulate the dynamics of such an attack. **Assumption 11**. :
* _Consider a blockchain network represented as a weighted directed graph_ \(G=(V,E,w)\) _where:_  
Figure 11: DDoS attack. * _V: Set of nodes, where each node \(v\) represents a miner in the network._
* _E: Set of directed edges representing hash rate contributions from one node to another (in the case of pool mining)._
* _w_(_e_): Weight of edge _e, representing the hash rate directed from one node to another._  
**Application Statement 11**. :
* _In a blockchain network graph_ \(G\)_, an entity controlling a set of nodes_ \(V_{A}\subseteq V\)_, such that the sum of outbound hash rate weights from_ \(V_{A}\) _surpasses 50% of the total network's hash rate, can perform a 51% attack._  
Proof.