# Application of Graph Theory for Blockchain Technologies
## 5 Graph Theory Application in Blockchain Systems
### NFT Manipulation Mapping
#### 5.5.4 Self-Trading  
Detecting self-trading involves identifying subgraphs where transactions form closed loops back to the original node, potentially through multiple intermediary nodes, indicating self-trading activity [PERSON] et al. [98]. By examining these closed loops within the transaction graph, irregularities suggestive of self-trading can be uncovered, enabling the identification of such manipulative behaviors in NFT transactions. This type of wash trade can be detected by finding Strongly Connected Components (SCCs) in the transaction graph. **Assumption 25**. :
* _Graph Representation. The NFT marketplace is modeled as a directed graph_ \(G=(V,E)\)_, where_
* _V represents individual trading accounts and_
* _E represents transactions between these accounts. Each transaction_ \(e_{ij}\in E\) _from account_ \(i\) _to account_ \(j\) _is associated with attributes such as transaction value_ \(v_{ij}\)_, timestamp_ \(t_{ij}\)_, and the specific NFT_ \(n_{ij}\) _involved._ _Account Linkage. A set of accounts_ \(A\) _is controlled by a single entity engaging in self-trading. These accounts may engage in transactions_ \(T_{A}\) _amongst themselves to simulate genuine trading activity._ _Transaction Analysis. Transactions that contribute to self-trading are characterized by the transfer of NFTs within set_ \(A\) _without introducing external market participants._  
**Application Statement 25**. :
* _Self-trading in the NFT marketplace can be identified by analyzing graph_ \(G\) _for closed transaction loops within the subset of accounts_ \(A\)_, indicating artificial trading activity aimed at inflating perceived market activity and NFT prices. Figure 25: Cross trading. **Proof. **  
1. Identify Linked Accounts. Utilize community detection algorithms or heuristic analysis to identify subsets of accounts \(A\subseteq V\) that exhibit patterns of closed-loop transactions indicative of single-entity control. 2. Detect Closed Transaction Loops. For each account \(a\in A\), trace outgoing transactions \(e_{ab}\in E\) where \(b\in A\), forming closed loops that start and end at the same account or within \(A\). 3. Analyze Transaction Attributes. Examine attributes \(v_{ij}\), \(t_{ij}\), and \(n_{ij}\) of transactions within closed loops for patterns such as repetitive trading of the same NFT, closely timed buy-sell actions, and transaction values inconsistent with market norms. 4. Quantify Artificial Activity. Aggregate the volume of transactions and the frequency of NFT trades within \(A\) to quantify the scale of self-trading activity. 5. Statistical Significance. Assess the statistical deviation of transaction patterns within \(A\) from those of the broader market to confirm the non-random nature of self-trading behavior. Figure 26 visualization represents a directed graph simulating self-trading within the NFT marketplace. Nodes (in skyblue) correspond to individual trading accounts within the marketplace. A subset of these accounts (\(A\) = 1, 2, 3) is controlled by a single entity engaged in self-trading to simulate genuine trading activity. Directed edges symbolize transactions of NFTs between these controlled accounts. The edges are labeled with the specific NFT involved in the transaction, illustrating the artificial circulation of NFTs within the subset \(A\). This model effectively demonstrates how self-trading can be identified through the analysis of closed transaction loops among linked accounts, thereby artificially inflating trading volume and NFT prices. Through the application of graph theory, the phenomenon of self-trading in the NFT marketplace can be effectively detected and quantified. By modeling the marketplace as a directed graph and identifying patterns of closed transaction loops among linked accounts, it is possible to uncover attempts to artificially inflate trading volume and NFT. Figure 26: Self-trading simulation.