# Analyzing the Bitcoin Network: The First Four Years
## 6 Analysis and Results
### Network Analysis
#### 6.2.1 Degree Distribution and Power Law of the Bitcoin Network  
The degree distribution captures the structure of the network in terms of the individual connectivity and is expected to follow a power law distribution since the Bitcoin network is considered as real world network such as the World Wide Web or social networks. To conduct the analysis within NetworkX the package \"powerlaw\" is required to calculate the slope coefficient \(a\), and plot the probability density function (PDF) and complementary cumulative distribution function (CCDF) [46]. With this analysis one can examine how the power law evolves over time, differs between particular businesses or countries. A power law distribution (\(P\left(k\right)\ \sim\ k^{-a}\) ) with a slope coefficient of \(2\leqslant a\leqslant\ 3\) indicates a scale-free network, which is often found in real world networks. Figure 21 shows the development of the PDF and CCDF distribution and the associated power law fit over time. The PDF requires logarithmic binning (default in \"powerlaw\" package) to account for the heavy tail in the distribution and a smooth visualization. The CCDF distribution does not use binning; thus, all information of the distribution is included [46]. The plot from the Bitcoin network in 2010 shows a good fit by the power law to the PDF as well as the CCDF distribution. With increasing activity in the network in 2011 the slope \(\alpha\) converges to a theoretically almost ideal value of 2.569, although the power law does not show a good fit for the CCDF distribution. This effect reduces over time with more user activity in the network and trade or exchange patterns. The development of the slope coefficient \(\alpha\) for several snapshots over time is depicted in Figure 22.  
Figure 21: Degree Distribution (probability density function (PDF), complementary cumulative distribution function (CCDF)) over Time. The same statistic for the business categories: gambling, exchanges, and mining indicate strong heavy tails when considering only particular business categories. As mentioned earlier, the categories are mainly driven by SatoshiDice (gambling), Mt.Gox (exchanges), and Deepbit (mining). These nodes have abnormal high degrees and lead therefore to the strong heavy tails. When looking at the plot (Businesses) in Figure 23 for the Bitcoin economy excluding the three major businesses one can see a good power law fit to the PDF and CCDF distribution. This might be explained by characteristics that are close to a real world economy or other social networks with various types of businesses and common behaviors by the participants. However, the slope coefficient \(\alpha\) for different business categories in Figure 24 indicates the existence of a scale-free network for all business categories except the wallets business. Despite the strong heavy tails, the businesses exchanges and mining have a very good slope coefficient \(\alpha\) with 2.495 and 2.517, respectively. Figure 23: Degree Distribution (PDF, CCDF) for different Business Categories. Figure 22: Development of Power Law Fit \(\alpha\) over Time. The degree distribution on country level aggregations reveals again the existence of a scale-free network (Figure 25). The major Bitcoin markets Germany and the U.S. show a good power law fit with a slope coefficient \(\alpha\) of 2.132 and 2.281, respectively. The plots for Russia and Brazil, the representatives for emerging markets, illustrate that the power law does not fit the PDF and CCDF distribution as good as in the case for the developed countries. This cannot be seen as a general case, since Sweden has a similar power law fit to the degree distributions like Russia. In both countries the business distribution is dominated by the mining sector. Hence, the relationship between the degree distribution (CCDF) for the mining business and for countries such as Russia or Sweden, indicating that a dominant business category such as mining in Russia have influence on the distribution on country level.