Keywords

1 Introduction

During the last decades we witnessed to the birth of numerous cryptocurrencies, which had a huge impact on the economic systems. Contrary to the well known currencies, a cryptocurrency is not issued by a bank or government, but instead rely on a set of cryptographic tools and distributed consensus protocols used by the users. Therefore, there is no more a central entity, usually a bank, which checks that a payment is valid, but instead all the users of the system have to cooperate such that only valid payments are accepted. Up to date, the most famous cryptocurrency is Bitcoin [17], which is based on the blockchain technology. For the sake of clarity, we use the term Bitcoin, with capital B, to refer to the whole distributed system, and we use the term bitcoin, with lower case b, to refer to the currency unit. Bitcoin came out in early January 2009 as the first functioning cryptocurrency, and over time attracted a lot of interest, both from specialists, coming mainly from computer science and economics fields, and also common people. This widespread interest gathered a very big community, which resulted in the first example of worldwide cryptocurrency economy, thus making it worth to be studied. Contrary to what one may think, the whole history of transactions of Bitcoin is public, meaning that one can read all the transactions issued without even being a Bitcoin user. This attracted a lot of interest in studying the properties of the system from a usage point of view. In fact, we find several studies in literature that try to characterise the system from a purely economic point of view [4, 6, 10]. From computer scientists a lot of effort was put in the study of the transaction graph for various purposes [9, 12, 13, 18]. The most studied topic on Bitcoin is the bitcoin price prediction, which was proven to be a very difficult task. Despite the fact that this problem was tackled with the most various techniques [1, 7, 8, 14], the prediction accuracy is very limited.

In this paper we present a different kind of study, in which we try to find correlations between the bitcoin price and some relevant measures of the transaction graph. Our contribution consists of:

  • A methodological framework for studying the transaction graph in terms of users and the transactions among them;

  • A study of the transaction graph using some efficient techniques in order to determine possible correlation between the bitcoin price and the bitcoin exchanges among users;

  • The detection of particular users, which follow an irregular pattern of acquisition of bitcoin.

With respect to similar studies [2, 10, 19], we make our framework self-contained, meaning that it does not need information coming from outside the Bitcoin blockchain, and focused on very specific time spans which show particular characteristics. Restricting the study on specific time spans, rather than on the whole blockchain, helps in having a focused view, which buries out previously undetected correlations and users with unexpected behaviours.

The rest of this paper is organised as follows: Sect. 2 presents the state of the art regarding the main studies on the Bitcoin transaction and bitcoin price. Section 3 contains the proposed framework from the information stored on the Bitcoin blockchain to the results, while Sect. 4 presents and discusses the obtained results. Concludes the paper Sect. 5 pointing out some possible future works.

2 State of the Art

Blockchain based systems, and Bitcoin in particular, attracted a lot of interest from researchers, especially computer scientists and economists. The Bitcoin blockchain has been studied for the sake of a number of different purposes, including P2P network analysis and the de-anonymization [11, 18], anomaly detection [13], or quantitative analysis [9, 12]. Yet, the most interesting aspects to be studied, from a multidisciplinar point of view, are the ones related to the bitcoin price, that is the amount of dollars one has to pay to buy a bitcoin. In particular bitcoin price prediction has attracted a lot of interest, with techniques coming machine learning [14], time series analysis [1] or theory of signals [7]. We also find studies about bitcoin price volatility [8], and studies that try to determine the factors that drives the bitcoin price, such as supply and demand [4], attractiveness [10], or a combination of them [6]. Some interesting studies also find good correlations between bitcoin price and Google Trends or Wikipedia queries [10], or attempts in speculative trading [2].

However studies in which author try to correlate the bitcoin price with measures from the transaction graph are lacking. In [19] authors study the whole blockchain, up to block at height 317,000 (October 2014), with the aim of trying to find correlations between the Bitcoin graph of transaction and the bitcoin price. In particular, they studied the \(\alpha \) value of the power-law distribution of the transactions, the number of nodes and edges of the transaction graph, the size of the blocks, and lastly the bitcoin gains distribution. The findings show a positive correlation between bitcoin price and number of edges, and number of nodes of the transaction graph, and a similar result for the block size. Instead authors observe no correlation with the \(\alpha \) values of the power laws, and a negative correlation with the bitcoin gains.

3 Bitcoin Rate Study: Methodology

The aim in this paper is to analyse the evolution over time of the bitcoin exchange rate, possibly showing a correlation with other properties of the Bitcoin system. As a major novelty introduced in this paper, we will only focus on some specific time spans. The idea of focusing the study on short time spans comes from the fact that analysing the whole blockchain may not show specific behaviours happening during particular events. Restricting the study to very short time spans will help us in having a more detailed view. Before venturing forward in the description of the methodology used, we introduce some basic concepts to help the reader understand the process.

3.1 Transaction Graph and User Graph

In order to analyse the Bitcoin price variation and the correlation with other characteristics, we need to model the activity of Bitcoin. Within the Bitcoin blockchain, payments are stored in transactions which can have multiple inputs and multiple outputs. Therefore, the most natural modelling tool to model the set of all transactions is an hypergraph. An hypergraph \(H=(X,E)\) is a generalization of a graph in which edges, or hyperedges, have as source and destination sets of nodes. With this model, we can easily model transactions as hyper-edges: the set of input addresses makes the source of the edge, and the set of output addresses makes the destination of the edge. In this way we are able to build the so called Transaction graph. It is important to point out that in the Transaction graph the set X of nodes is made of the Bitcoin addresses of the users, not the users themselves. In fact, users are also encouraged to create more addresses so that their privacy is protected. Analysing the Transaction graph is not trivial for the following reasons:

  • The Transaction graph is an hypergraph, and therefore the well known measures and algorithms must be adapted;

  • The two ends of the edges are Bitcoin addresses, not Bitcoin users, so when we perform the analysis we also must take into account this fact.

It is important for us not to consider the transactions of the Bitcoin system, but actually how users of the Bitcoin system exchange bitcoins. We introduce a different model, called the User graph \(G=(V,E)\) as a multigraph in which the set of vertices V is used to model the users of Bitcoin and the set of edges E is used to model the bitcoin exchanged by the users. In this model, each user \(v\in V\) is associated with one or more addresses and each transaction is associated with one or more edge \(e\in E\) if the User graph.

3.2 From the Transaction to the User Graph

Switching from the Transaction graph to the User graph is not an easy task. The biggest challenges are strictly related to the process used to determine which addresses belong to the same user. In literature there are several heuristics to cluster addresses belonging to the same user based on change analysis [15, 20], on temporal information [16], or other features [5]. The most well known heuristic is the common-input-ownership heuristic [17], formulated by Satoshi Nakamoto himself. This heuristic states that all the inputs of a transaction are likely to belong to the same user. This is because Bitcoin expects that each input of a transaction is signed by the respective owner, and since a private key is needed to sing inputs, it is unlikely that inputs of the same transaction belong to different users. The common-input-ownership is just an heuristic, meaning that even if it makes sense to consider the addresses appearing as input of a transaction belonging to the same user, we have no certainty that this process exactly associates users to their addresses. On the other hand, new techniques are rising, such as CoinJoin, Mixcoin [3], and Blindcoin [21], based on mixing different payments in the same Bitcoin transaction, which cause these heuristics to produce a lot of false positives. However, for what concern this work, we stick to the application of the plain common-input-ownership heuristic [17], as it shows good results while remaining very simple and intuitive.

If we apply recursively the common-input-ownership heuristic to the Transaction graph we observe two effects:

  • The resulting nodes of the hypergraph are the users of Bitcoin;

  • The resulting hyper-edges have a single node as source of the edge.

At this point, it is enough to split the hyper-edges in simple edges to obtain a multigraph where nodes model the users and edges model the bitcoin exchange between the users, or, in other words, the User graph.

3.3 Study of the User Graph

In this paper we are interesting in the study of the User graph, with the particular intent of finding correlation between the bitcoin exchange rate and measures of the graph. For bitcoin exchange rate we mean the number of dollars needed to buy one bitcoin from a reference exchanger. To carry on this study, we firstly detected interesting time spans where the bitcoin exchange rate is extremely volatile. At this point, we built the Transaction graph limited to that time span, and then we created the User graph derived from it.

Before venturing forward in presenting the analyses performed and the intuition behind them, we briefly present and motivate the three time spans chosen. The three spans have two main characteristics:

  • They show high fluctuation of the bitcoin exchange rate;

  • They are shorter than 7 days.

The request of having an high fluctuation of the bitcoin exchange rate during the time span comes from the fact that we expect to detect unusual behaviours in these cases, rather than when the bitcoin exchange rate is stable. Having spans shorter than 7 days help us to concentrate our efforts in time proximity of such events, so that a causality relation between the fluctuation of the bitcon exchange rate and possible unexpected behaviour can be established.

Fig. 1.
figure 1

Bitcoin exchange rate in the three considered time spans

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The first analysed time span starts on the \(7^{th}\) and ends on the \(13^{th}\) of December 2017. As we can see from Fig. 1a, the fluctuation of the bitcoin exchange rate is quite high during all the week. The rate starts just below 13,000$ per bitcoin on the \(7^{th}\), then raises to 17,000$ in just one day. In the following three days we observe the rate continuously rise and fall, often by over 1,000$ per bitcoin, reaching 13,500$ per bitcoin on the \(10^{th}\). In the last few days the bitcoin exchange rate increases once again, reaching the time span maximum of almost 17,500$ per bitcoin on the last day.

The second time span considered starts on the \(16^{th}\) and ends on the \(23^{rd}\) of December 2017, just a few days later than the first one. Figure 1a shows the bitcoin exchange rate in this time span. At the beginning of the time span, a bitcoin is exchanged for 18,000$, and the rate has a steep increase at the beginning of the \(17^{th}\), reaching the all-time high of 19,783.06$ per bitcoin at the end of the same day. Then we observe a gradual decrease of the bitcoin exchange rate, with two big drops on the \(20^{th}\), from 19,000$ to 16,000$ per bitcoin, and on the \(22^{nd}\), from 17,000$ to 12,000$ per bitcoin.

The last time span analysed starts on the \(5^{th}\) and ends on the \(7^{rd}\) of September 2018. This last time span is several months further in time, with respect to the first two, and it is shorter, lasting only three days, but contains a very steep drop, which was of relevant interest for our particular study. The bitcoin exchange rate of this time span is shown in Fig. 1c. As we observe, the exchange rate is much more stable in this time span, except for the two sharp drops. The first one happens on the \(5^{th}\), around midday, and consist of the drop from 7,400$ to 7,000$ per bitcoin, while the second one happens on the \(6^{th}\), around 1 AM, and consist of the drop from 7,000$ to 6,400$ per bitcoin. After this steep two-step drop, we observe that the bitcoin exchange rate remains stable around 6,400$ per bitcoin for the rest of the time span considered. We decided to keep also this two days long tail to possibly observe users acquiring a big amount of bitcoin after the drop of the day before.

4 Bitcoin Exchange Rate Correlation Studies

In this section we present the results of our study in which we try to find correlation between the bitcoin exchange rate and some properties of the User graph. It is important to point out that in this paper we will only focus on empirical correlation found in the various plots we produced.

4.1 Number of Bitcoin Exchanged

Our first idea is to study the correlation between the bitcoin exchange rate and the number of bitcoin exchanged over the same time span. The results we present are aggregated every six hours. Figures 2a, b, and c show the bitcoin exchange rate (black) and the number of bitcoin exchanged (grey) during the three time spans considered. Due to different orders of magnitudes, the bitcoin exchange rate is plotted using the left scale, while the number of bitcoin exchanged is plotted using the right scale. In all three cases we see that the number of bitcoins exchanged follows a periodic pattern of 24 h, which suggests us that this is highly correlated with human life. With respect to the rate, instead, we observe that there is some positive correlation, especially in the two December spans. In detail, in the first December span (Fig. 2a), we see that when the bitcoin exchange rate is high, also the number of exchanged bitcoin is high and vice versa. This correlation is much more clear in the second half of the time span. Also in the second December span (Fig. 2b) we can see this correlation: when the exchange rate is high, the number of bitcoin exchanged ranges between 50,000 and 85,000, while when the exchange rate is low, the number of bitcoin exchanged ranges between 20,000 and 50,000. In the last time span (Fig. 2c), due to its low length, there seems to be not a clear correlation, however we can see that, one day after the steep drop of the bitcoin exchange rate, also the number of bitcoin exchanged seem to drop accordingly.

Fig. 2.
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Correlation graph between the bitcoin exchange rate and the number of bitcoin exchanged in the three considered time spans

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Fig. 3.
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Correlation graph between the bitcoin exchange rate and the number of nodes and edges in the User graph in the three considered time spans

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4.2 Number of Transactions and Number of Users

Having seen that there is a certain degree of correlation between the bitcoin exchange rate and the number of bitcoin exchanged, we analyse the User graph to check if there is a similar correlation found in Sect. 4.1, but this time with the number of nodes and edges in the user graph. Nodes and edges in the User graph correspond to the number of users involved in at least one transaction (nodes) and to the number of transactions (edges). Figures 3a, b, and c show the bitcoin exchange rate (black), and the number of nodes and edges in the User graph (grey) during the three time spans considered. Due to different orders of magnitudes, the bitcoin exchange rate is plotted using the left scale, while the number of nodes and edges are plotted using the right scale. Also the number of nodes and edges in the User graph seem to follow a 24 h recurring pattern, lust like the bitcoin exchange rate, confirming that the Bitcoin ecosystem is highly related to human activities. Concerning possible correlations, what we expected was to find a good degree of correlation, also according to similar studies present in literature. However, what we can see from the three plots is that there is no clear correlation as in all cases the number of nodes and edges in the graph tend to remain much more stable in time. In any case, we observe that during the first two spans, when the bitcoin exchange rate is always higher than 13,000$ per bitcoin, the number of nodes and edges in the User graph fluctuates around 20,000, while in the last span, when the exchange rate is halved, the number of nodes and edges in the User graph is also lower. Finally, we also notice that the number of nodes is similar to the number of edges in the User graph, suggesting us that the degree distribution may be a power law.

Fig. 4.
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Indegree distribution of the User graph in the three considered time spans

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Fig. 5.
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Outdegree distribution of the User graph in the three considered time spans

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This expectation was fully met, as we can see from the distribution of in degree, Fig. 4, and out degree, Fig. 5, for the three time spans in log-log plots. It is also interesting to see that the distribution of both degrees of the September time span, Fig. 4c for the indegree and Fig. 5c for the outdegree, present some possible outliers which are worth to be investigated.

Fig. 6.
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Bitcoin gain distribution in the three considered time spans

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4.3 Bitcoin Gain Distribution

After having detected some possible anomalies in the out degree distribution of the User graph, we tried to further concentrate our effort in discovering the source of this effect. In particular, we study the bitcoin gain distribution, that is the number of bitcoin gained by the users. It is important to point out that the gain distribution has nothing to do with the exchange rate, as we are not measuring how many dollars each user gained. Measuring the gain distribution sums up to count, for each user separately, the number of bitcoin on the incoming edges and subtracting the number of bitcoin on the outgoing edges, and then plotting the distribution of the obtained values. The distribution of the bitcoin gain, during the three time spans considered, are presented in Figs. 6a, b, and c. Also in this case, the y axis has a logarithmic scale while the x axis has a linear scale. Moreover, for readability reasons, we skipped some values on the x axis because they were empty, and we grouped the distribution values in bins of 1,000 bitcoin each. The three distributions follow a Gaussian law centered on 0 with not much variance, showing us that the vast majority of the users does not gain any bitcoin. Considering how we built the User graph, almost all users that acquire bitcoin usually spend them within one week or less. While in this case we see no clear correlation between the bitcoin exchange rate and the bitcoin gain distribution, as the three plots do not show any particular difference, we observe that in all the cases we have an outlier. The outlier is located in all three cases on the right side of the distribution, meaning that he gained bitcoin, rather than giving them away. The address of the outlier is different in each case, meaning that it is not the same user. Moreover, we observe that the amount of bitcoin gained is sensibly higher compared to all the other users: more than 250,000 bitcoin in the two December spans, and 65,000 in the September span.

Fig. 7.
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Bitcoin outlier gain in the three considered time spans

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4.4 Gain Outliers

As final step of this analysis, we study the bitcoin gain outliers. The aim is to see if they adopt a particular pattern to acquire bitcoin, and if the pattern is correlated in some way to the bitcoin exchange rate. Figures 7a, b, and c show the bitcoin exchange rate (black), and the number of bitcoin gained by the outlier (grey) during the three time spans considered. Each plot shows only the outlier found in the same time span, considering the outliers as different users. Due to different orders of magnitudes, the bitcoin exchange rate is plotted using the left scale, while the number of bitcoin gained by the outlier is plotted using the right scale. From these plots we observe that the outliers tend to acquire most of the bitcoin in a very short time span. In fact, in the first December span, Fig. 7a, 70,000 bitcoins, roughly 22% of the total amount of bitcoin acquired by the outlier in the whole time span, are acquired between 6 AM and midday on the \(9^{th}\) of December. We have a similar situation, although less highlighted, in the second December span, Fig. 7b, with 25,000 bitcoin in six hours, out of almost 300,000 acquired in the whole span. Also in the third span, Fig. 7c, we observe that the outlier gathered almost 28% of the total amount of bitcoin in just six hours. Anyway, while there is a clear pattern with which the outliers acquire bitcoin, there seems to be not a clear strategy. In fact, in the first and third spans we observe that the peak is very close to the lowest point of the bitcoin exchange rate. This is not true in the second time span, in which the peak is instead close to the highest point of the bitcoin exchange rate. However, in the second case, the bitcoin acquired by the outlier are more evenly distributed over time, which, joint to the fact that the exchanged rate reached its all time high value, makes us think that there was high uncertainty whether the exchange rate would grow even further or not.

5 Conclusion and Future Works

In this paper we tackled the problem of finding correlations between the bitcoin exchange rate and measures on the User graph. We proposed a methodological framework that, starting from the transactions stored on the Bitcoin blockchain, let us study the User graph in a temporal way, focusing only on specific time spans. In detail, the framework consist of starting from the Transaction graph, an hypergraph where nodes are sets of Bitcoin addresses and edges are the Bitcoin transactions. On the Transaction graph, the common-input-ownership [17] heuristic is applied so that in the resulting graph nodes can be identified with the users of Bitcoin. The resulting graph, called User graph, is a multigraph where each node correspond to a Bitcoin user, and each edge models a bitcoin exchange. We studied several topological measures of the User graph built on limited time spans, which let us identify previously undetected correlations which may also help in the price prediction task. We, moreover, detected some users with unusual behaviour which stockpile bitcoin with unusual patterns.

As future works, we plan to deepen our studies in three directions. At first, we want to replicate our studies on more time spans, possibly characterising them in few categories based on the bitcoin exchange rate, such as “big increase”, “big decrease”, “stability”, or “high volatility”. A second direction to follow is the one of analysing the User graph in more detail, studying more measures, using more advanced techniques or at a finer granularity, but also different heuristics used to detect the users of Bitcoin. One last direction to follow is to study more in depth the outliers and, in particular, use some advanced de-anonymization technique to discover if the user corresponds to a person or an entity, or find out more of the piling up of bitcoin by these users.