# Application of Graph Theory for Blockchain Technologies
## 5 Graph Theory Application in Blockchain Systems
### Analyzing Network Decentralization
#### 5.1.4 Developer Concentration  
Analyzing collaboration graphs and code contribution statistics for blockchain platforms can detect excessive dependencies on a few core developers [69] as depicted in Figure 5. The node in the graph represents developers, lighter shade links for less contribution and darker shades for higher contribution.  
**Assumption 4**.:
* _Graph Model: The collaboration network is modeled as a directed graph_ \(G~{}=~{}(V,E,W)\)_, where:_
* _V represents the set of developers._
* \(E\) _represents collaborative relationships (e.g., working on the same project or module)._
* \(W\) _is a set of weights on the edges, where each weight_ \(w(u,v)\) _quantifies the level of collaboration or contribution from developer_ \(u\) _to developer_ \(v\)_._
* _Degree of a Node: The in-degree_ \(d_{in}(v)\) _and out-degree_ \(d_{out}(v)\) _of a node_ \(v\) _represent the number of contributions received by and made by developer_ \(v\)_, respectively._
* _Strength of a Node: The strength_ \(s(v)\) _of a node is defined as the sum of the weights of the outgoing edges connected to_ \(v\)_, reflecting the total impact of a developer's contributions._
* _Core Developer Identification: A core developer is identified by high out-degree, high strength, or a combination of both, indicating significant influence or dependency in the network._  
**Application Statement 4**.:
* _In a blockchain platform's collaboration network modeled as a weighted directed graph, a skewed distribution of degrees and strengths among developers indicates excessive dependencies on a few core developers._  
**Proof.**  
1. Degree and Strength Analysis: Define the in-degree and out-degree of a node (developer) \(v\) in graph \(G\) as \(d_{in}(v)\) and \(d_{out}(v)\), respectively. These represent the number of contributions received by and made by developer \(v\). The strength \(s(v)\) of a node is the sum of the weights of the edges connected to \(v\), quantifying the impact of a developer's contributions: \(s(v)~{}=~{}\sum_{(u,v)\in E}w(u,v)\).
2. Skewness of Distributions: Analyze the skewness of the degree distributions \(D_{in}~{}=~{}\{d_{in}(v):v\in V\}\) and \(D_{out}~{}=~{}\{d_{out}(v):v\in V\}\), and the strength distribution \(S~{}=~{}\{s(v):v\in V\}\). High skewness indicates a small number of developers (nodes) with significantly higher degrees or strengths.  
Figure 5: Developer Concentration.  
3. Dependency Inference: A core developer can be identified by high values of \(d_{\text{in}}(v)\), \(d_{\text{out}}(v)\), and \(s(v)\). If the skewness in \(D_{\text{in}}\), \(D_{\text{out}}\), or \(S\) is high, it indicates a network structure with excessive dependency on a few developers.
4. Network Health and Robustness: A network's robustness and sustainability can be at risk if it is overly dependent on a few core developers. The temporal evolution of \(D_{\text{in}}\), \(D_{\text{out}}\), and \(S\) can be examined to understand how these dependencies change over time.  
This proof demonstrates that graph theory can be effectively used to analyze developer collaboration networks in blockchain platforms. By examining the degree and strength distributions, one can identify if there are excessive dependencies on a few core developers, which can be critical for understanding the network's health and sustainability.  
Thus, graph metrics enable the ongoing quantification of decentralization levels across various facets. Network visualizations also easily highlight the growth of hubs, clusters and disparities.