# Application of Graph Theory for Blockchain Technologies
## 5 Graph Theory Application in Blockchain Systems
### NFT Manipulation Mapping
#### 5.5.2 Coordinated Trading  
By examining the timestamps and correlation of transactions among a subset of nodes, one can detect abnormal spikes in activity that suggest coordination. Clustering algorithms can help identify groups with unusually synchronized trading patterns [96].  
**Assumption 23**.:
* _Let_ \(G=(V,E)\) _be a directed graph where_
* \(V\) _represents traders in the NFT market, and_
* \(E\) _represents transactions of NFTs between traders. Each transaction_ \(e_{ij}\in E\) _from trader_ \(i\) _to trader_ \(j\) _is associated with attributes like transaction value_ \(v_{ij}\)_, timestamp_ \(t_{ij}\)_, and the specific NFT_ \(n_{ij}\) _involved._
* _Define a subset_ \(S\subseteq V\) _where traders are involved in coordinated trading, engaging in transactions within a narrowly defined time frame_ \(T\)_._  
**Application Statement 23**.:
* _In the NFT marketplace represented by graph_ \(G\)_, the subset_ \(S\) _of traders engaging in coordinated trading activities can be identified by analyzing the temporal clustering of transactions_ \(E_{S}\) _within_ \(T\)_, which leads to an abnormal surge in trading volume and price movement for the involved NFTs._  
**Proof.**  
* Temporal Clustering. For each NFT \(n_{ij}\) involved in transactions \(E_{S}\) within the subset \(S\), cluster transactions based on their timestamps \(t_{ij}\). Identify clusters where the number of transactions exceeds a threshold within time frame \(T\), indicative of coordinated trading.
* Volume and Price Analysis. Calculate the total transaction volume \(V_{S}\) and average price change \(\Delta P_{S}\) for NFTs involved in identified clusters. Compare \(V_{S}\) and \(\Delta P_{S}\) with the network-wide averages \(V_{avg}\) and \(\Delta P_{avg}\) to identify significant deviations.
* Statistical Significance. Use statistical tests to evaluate the likelihood that observed surges in volume and price movement are due to chance. Significant deviations from the averages suggest the presence of coordinated trading.  
Figure 24 visualization represents a directed graph of NFT market transactions, highlighting the dynamics of coordinated trading. In the graph, Skyblue Nodes represent all traders participating in the NFT market. Red Nodes indicate the subset of traders involved in coordinated trading within a defined time frame, showcasing potential collusion to manipulate market perception. Edges symbolize transactions between traders, annotated with NFT identifiers, transaction values, and timestamps, providing a comprehensive view of market activity. This model allows us to identify clusters of coordinated trading activity (red nodes) and analyze the temporal clustering of transactions within the narrowly defined time frame \(T\). By examining these patterns and comparing the transaction volume and price changes of involved NFTs to network-wide averages, we can effectively detect and quantify instances of market manipulation.  
The ability to visualize and mathematically analyze such trading activities highlights the importance of graph-based analysis in enhancing market transparency, integrity, and fairness. Through these methodologies, stakeholders can better understand market dynamics, implement measures to mitigate manipulative practices and foster a more trustworthy NFT ecosystem.