# Application of Graph Theory for Blockchain Technologies
## 5 Graph Theory Application in Blockchain Systems
### Analyzing Network Decentralization
#### 5.1.3. Wealth Concentration  
The degree and strength distribution of transaction graphs can identify the concentration of token ownership and flow among addresses over time [68]. Figure 4 is a representation of the Wealth concentration constructed from actual transaction data and the analysis would focus on identifying nodes (addresses) with a high number and value of incoming and outgoing transactions, indicating the concentration of wealth. Nodes represent individual addresses or wallets in a blockchain network. Directed edges (arrows) signify transactions between these addresses. The width and color intensity of the edges reflect the transaction value (with thicker, darker edges indicating higher values).  
Figure 4: Wealth Concentration.  
Figure 3: Stake Pool.  
**Assumption 3**.: * _Graph Model: The blockchain transaction network is modeled as a weighted directed graph_ \(G~{}=~{}(V,E,W)\)_, where:_ * _V represents addresses (wallets) in the network._ * _E represents transactions between addresses._ * _W is a set of weights on the edges, where each weight_ \(w(u,v)\) _represents the value of the transaction from address_ \(u\) _to address_ \(v\)_._
* _Degree of a Node: The degree of a node (address)_ \(d(v)\) _in_ \(G\) _represents the number of transactions involving that address. The degree is the number of_ _walled transactions._
* _Strength of a Node: The strength of a node_ \(s(v)\) _is defined as the sum of the weights (transaction values) of the edges connected to the node. This represents the total value of transactions for that address._
* _Wealth and Transaction Correlation: An address's wealth is assumed to correlate with both its degree and strength in the graph._  
**Application Statement 3**.: * _In a blockchain transaction network modeled as a weighted directed graph, a skewed degree and strength distribution indicates a concentration of token ownership and transactional flow among a limited number of addresses, suggesting wealth concentration._  
**Proof.**  
1. Degree and Strength Analysis: Analyze the degree distribution \(D~{}=~{}\{d(v):v\in V\}\) and the strength distribution \(S~{}=~{}\{s(v):v\in V\}\) in graph \(G\). A uniform distribution implies decentralized token flow, while a skewed distribution indicates concentration.
2. Skewness of Distributions: Apply skewness metrics to \(D\) and \(S\). High skewness values in \(D\) and \(S\) indicate that a few addresses have much higher degrees and strengths, respectively.
3. Wealth Concentration Inference: Given the correlation between an address's wealth and its degree/strength, the skewness in \(D\) and \(S\) can be interpreted as indicators of wealth concentration. A few addresses (high-degree, high-strength nodes) dominate the total token flow, indicating wealth concentration.
4. Temporal Evolution: By examining the evolution of \(D\) and \(S\) over time, trends in wealth concentration can be identified. Increasing skewness over time suggests increasing wealth concentration.  
This proof demonstrates that by analyzing the degree and strength distributions in a transaction graph of a blockchain network, it is possible to identify trends of wealth concentration. High skewness in these distributions over time indicates an increasing concentration of wealth among fewer addresses, which can have significant implications for the network's decentralization and security.