# Application of Graph Theory for Blockchain Technologies
## 5 Graph Theory Application in Blockchain Systems
### NFT Manipulation Mapping
#### 5.5.3 Cross Trading  
Cross-referencing transactions involving the same NFTs across different graphs can reveal cross-trading activities. Matching transactions by time, NFT, and price across platforms would highlight potential cross-trading schemes [97].This approach enables the detection of coordinated trading practices that span multiple platforms, contributing to a more comprehensive understanding of market activities and potentially manipulative behaviors.  
**Assumption 24**.:
* _Let us consider a collection of directed graphs_ \(\{G_{1},G_{2},\ldots,G_{n}\}\) _where each graph_ \(G_{k}=(V_{k},E_{k})\) _represents a transaction network on a distinct NFT trading platform_ \(k\)_. In each graph:_
* \(V_{k}\) _is a set of vertices representing traders on platform_ \(k\)_._
* \(E_{k}\) _is a set of directed edges representing transactions of NFTs between traders on platform_ \(k\)_, with each edge_ \(e_{ij}^{k}\in E_{k}\) _from trader_ \(i\) _to trader_ \(j\) _associated with attributes like transaction value_ \(v_{ij}^{k}\)_, timestamp_ \(t_{ij}^{k}\)_, and the specific NFT_ \(n_{ij}^{k}\) _involved._ _Cross-trading is characterized by transactions involving the same set of traders and NFTs across different platforms_ \(\{G_{1},G_{2},\ldots,G_{n}\}\)_._  
**Application Statement 24**.:
* _In a multi-platform NFT marketplace represented by a collection of graphs_ \(\{G_{1},G_{2},\ldots,G_{n}\}\)_, the presence of identical subsets of traders_ \(S\subseteq V_{1}\cap V_{2}\cap\ldots\cap V_{n}\) _ent-gaging in transactions involving the same NFTs across these platforms is indicative of the cross-trading aimed at artificially inflating perceived demand and price._  
Proof.:
* Identification of Common Traders Across Platforms. Identify subsets of traders \(S\) who are active across multiple platforms, i.e., \(S\subseteq V_{1}\cap V_{2}\cap\ldots\cap V_{n}\).
* Transaction Matching. For each trader \(s\in S\) and for each pair of platforms \((G_{k},G_{l})\), match transactions based on NFT identifiers and approximate timestamps. That is, find pairs of transactions \((e_{ij}^{k},e_{mn}^{l})\) where \(n_{ij}^{k}=n_{mn}^{l}\) and \(|t_{ij}^{k}-t_{mn}^{l}|\) is minimal, suggesting a deliberate attempt to show activity on multiple platforms.
* Volume and Price Analysis. Calculate the aggregated transaction volume \(V_{S}\) and average price \(P_{S}\) for matched transactions across platforms. Compare these with platform-specific and network-wide averages to identify significant deviations.  
Figure 24: Coordinated Trading Visualization.  
4. Statistical Significance. Employ statistical methods to assess whether the observed patterns of cross-platform trading deviate significantly from expected distribution under normal market conditions.  
Figure 25 visualization illustrates the concept of cross-trading across two distinct NFT trading platforms, represented as directed graphs for platforms G1 represented in blue nodes and G2 represented in green nodes. Nodes represent traders participating in the NFT market on each platform. Directed edges symbolize transactions of NFTs between traders. This model enables the identification of cross-trading activities by analyzing transactions involving the same NFTs by the same group of traders across different platforms. By leveraging graph theory, stakeholders can better safeguard the ecosystem against such deceptive strategies, ensuring a more transparent and fair market environment.