## 2 Public Keys and Economically Active Entities
To quantify the activity on the Bitcoin network, Figure 6b,c shows (over time) the number of entities that were active within the last day, and within the last 10, 30, and 100 days. Both plots show the same data, one with a linear and one with a logarithmic \(y\)-axis. We also show the exchange rate between USD and bitcoins in Figure 6a. Comparing Figures 1 and 6c, one can conclude that the number of active entities can be several orders of magnitude lower than the total number of entities. This needs to be taken into account when estimating the size of the anonymity set of Bitcoin by computing the number of active economic entities. For example, reducing the activity period from 100 days to just one day reduces the size of the Bitcoin anonymity set by a factor of 32 (averaged over the last year). Figure 4: Slope of regression line \(\Gamma\) for entity-size distribution over time. Error bars indicate goodness-of-fit, computed as the root of summed squared residuals between data and a fitted linear model. Figure 5: Total number of entities active for a given amount of days. Figure 6: Price per bitcoin and activity of entities over time. (**a**) Exchange rate in USD at Mt. Gox (weighted average) [14]; (**b**) Entity activity over time (linear scale); (**c**) Entity activity over time (logarithmic scale). Furthermore, as evident from Figure 6c, since Bitcoin's nascent days the ratio between the number of active entities for various time scales is constant. Therefore, the overall pattern of usage has become stationary and the above argument on the reduction of the anonymity set will likely hold for the future. In contrast--until the end of July 2011--we observe a strong correlation between the number of active entities and the exchange rate, see Figure 7. Since then, the exchange started to slowly rise again, now hovering at around 10 to 15 USD, whereas the activity on the network surpasses the peak activity during the hype in June 2011. Typically a higher exchange rate is explained by speculation and not pure monetary usage, which implies that speculation is good for anonymity. Looking at the number of transactions over time (depicted in Figure 8), this observation can be confirmed. During the Bitcoin hype the number of transactions peaked and then fell again. After this incident, the number of transactions surpassed the pre-hype level, which is consistent with the number of active entities. Besides the total number of transactions, the figure also depicts the number of transactions where one or more of the 100 most-used public keys were not involved, either as input or output. As both curves started to spread at about January 2012, this means there are some addresses (entities) that account for a large number of transactions on the Bitcoin network. Two of those entities can easily be named: one is the Deepbit mining pool, the other is a gambling service called SatoshiDice, which was announced on 24 April 2012 and gained great popularity within weeks. The way this service works introduces many transactions on the network, which explains most of the separation of the two lines in the figure. Figure 7: Price per bitcoin and active entities. Data points for exchange rates above 14 USD were omitted because for higher exchange rates not enough data points are available (see Figure 6a), _i.e._, an exchange rate of 14 USD or more was reached for 52 days only.