# Analyzing the Bitcoin Network: The First Four Years
## 4 Methods
### Network Metrics
#### 4.2.4 Centrality  
Centrality measures the importance, influence or power of a node in the network and is widely applied in social network analysis. Important metrics have been introduced by [PERSON] [37]: degree centrality, betweenness centrality, and closeness centrality. Betweenness and closeness centrality count only geodesic paths, assuming that messages or transactions in a network flow only along the shortest possible paths. The eigenvector measure [38] counts walks, which assumes that trajectories can also revisit nodes and edges multiple times [39].  
The degree centrality is based on the degree, _i.e._, the number of links or direct connections that one node \(i\) has. To compare the degree centrality among networks of different size, one has to normalize by dividing the measure by the maximum possible number of adjacent connections, \(N-\ 1\) (Equation (4), below).  
\[\begin{array}{c}C_{D}\left(n_{i}\right)=\sum\limits_{i}^{N}a_{ij}\\ C_{D}\left(n_{i}\right)=\frac{\sum\
olimits_{i}^{N}a_{ij}}{N-1}\end{array} \tag{4}\]  
Nodes with higher degree centrality \(C_{D}\left(n_{i}\right)\) or connections are more central to the network structure and tend to have more influence on others [37].  
The betweenness centrality is based on the number of shortest paths passing through a node. Nodes with high betweenness play a central role in connecting different groups in a network. In Equation (5) \(g_{jk}\left(i\right)\) is all geodesics linking node \(j\) and node \(k\) which pass through node \(i\), and \(g_{jk}\) is the geodesic distance between node \(j\) and \(k\).  
\[\begin{array}{c}C_{B}\left(n_{i}\right)=\sum\limits_{j<k}^{N}\frac{g_{jk} \left(i\right)}{g_{jk}}\\ C_{B}\left(n_{i}\right)=\frac{\sum\
olimits_{j<k}^{N}\frac{g_{jk} \left(i\right)}{g_{jk}}}{\frac{1}{2}N\left(N-1\right)}\end{array} \tag{5}\]  
In social networks, nodes with high betweenness are the brokers and connectors that bring other groups in the network together. Nodes with the highest betweenness centrality measure result in the largest increase in a typical distance between others when they are removed [40]. The normalized version of the formula is shown in Equation (5) (bottom). It is normalized by the maximum number of pairs of nodes excluding the node itself [41].  
The closeness centrality emphasizes the distance of a node to all other nodes in the network by focusing on the geodesic distance from each node to all others. Closeness centrality can be regarded as measure of how long it will take information to spread from a given node to others in the network. In Equation (6), \(C_{c}\left(n_{i}\right)\) is the closeness centrality and calculated by the sum of the inverse distances \(d\left(n_{i},n_{j}\right)\) between two nodes in the network [40].  
\[\begin{split} C_{c}\left(n_{i}\right)=\sum_{i}^{N}\frac{1}{d \left(n_{i},n_{j}\right)}\\ C_{c}\left(n_{i}\right)=\left[\sum_{i}^{N}\frac{d\left(n_{i},n_{ j}\right)}{N-1}\right]^{-1}\end{split} \tag{6}\]  
The node decentrality or inverse centrality grows when nodes are far apart, and centrality in this context means closeness. Information or other goods originated in the most central position of the network would spread throughout the network in minimum time. The most central node in the network is that with the minimum costs or time for communicating with all others. To remove the impact of the network size for comparability the formula is adjusted as seen in Equation (6) (bottom) [37].  
The introduced centrality measures can be used to examine important hubs (degree) and brokers (betweenness) as well as to assess how efficiently information flows (closeness) within the network.