Bitcoin Transaction Networks: An Overview of Recent Results

Cryptocurrencies are distributed systems that allow exchanges of native (and non-) tokens between participants. The availability of the complete historical bookkeeping opens up an unprecedented possibility: that of understanding the evolution of a cryptocurrency's network structure while gaining useful insights into the relationships between users' behavior and cryptocurrency pricing in exchange markets. In this article we review some recent results concerning the structural properties of the Bitcoin Transaction Networks, a generic name referring to a set of three different constructs: the Bitcoin Address Network, the Bitcoin User Network, and the Bitcoin Lightning Network. The picture that emerges is of a system growing over time, which becomes increasingly sparse and whose mesoscopic structural organization is characterized by the presence of an increasingly significant core-periphery structure. Such a peculiar topology is accompanied by a highly uneven distribution of bitcoins, a result suggesting that Bitcoin is becoming an increasingly centralized system at different levels.


INTRODUCTION
A cryptocurrency is an online payment system for which the storage and the verification of transactions -therefore, the safeguard of the system consistency itself -are decentralised, i.e. do not require the presence of a trusted third party.This result can be achieved by securing financial transactions through a clever combination of cryptographic technologies [1].
Bitcoin, the first and most popular cryptocurrency, was introduced in 2008 by Satoshi Nakamoto [2]: it consists of a decentralised peer-to-peer network to which users connect to exchange the property of the account units of the system, i.e. perform bitcoins transactions.Each transaction becomes part of a publicly available ledger, the blockchain, after having been validated by the so-called miners, i.e. users that verify the validity of issued transactions according to the consensus rules that are part of the Bitcoin protocol [3,4].A new block, containing transactions that where known to the miner since the last block, is 'mined' -on average -every 10 minutes, thereby adding new transactions to the blockchain.Thus, these transactions are 'confirmed', in turn enabling users to spend the bitcoins they received through them 1 .The cryptography protocols Bitcoin rests upon aim at preventing the so-called double-spending problem, i.e. the possibility for the same digital token to be spent more than once in absence of a central party that guarantees the validity of the transactions [1,2]: remarkably, the transaction-verification mechanism Bitcoin relies on allows its entire transaction history to be openly accessible, a feature that, in turn, allows researchers to analyse it in different network representation.
The gain in popularity of Bitcoin led its community to face new problems such as the lack of scalability of the transaction-verification method: The first one concerns the (relatively low) maximum number of transactions that can be verified per second -especially if compared with the mainstream competitors such as centralised payment networks.The large concentration of mining power in mining pools (implying a decrease in decentralisation of verification in the network) and the tendency of users to hoard.In order to solve the aforementioned problems, threatening the overall functioning of Bitcoin as a medium of exchange, new instruments were adopted.Proposed in 2015 [5], the Bitcoin Lightning Network (BLN) is a 'Layer 2' protocol that can operate on top of Blockchain-based (Bitcoin-like) cryptocurrencies by creating bilateral channels for off-chain payments which are, then, settled concurrently on the blockchain once the channel gets closed.As both the transaction fees and the blockchain confirmation are no longer required, the network is spared from avoidable burden; moreover, the key features of Bitcoin, i.e. its decentralised architecture, its political organisation and its wealth distribution are no longer sacrificed, while the circulation of the native assets is enhanced.
Although Bitcoin is almost ten years old, researchers have started to investigate the Bitcoin structural properties only recently: in [6], the authors consider the network of transactions between addresses at the weekly time scale, showing the emergence of power-law distributions and that the number of incoming transactions reflects the wealth of nodes; in [7], the authors consider the network of transactions between users at the macroscale, in order to check for its small-worldness; in [8], the authors investigate the network of international Bitcoin flows, identifying socioeconomic factors that drive its adoption across countries.Several studies also focus on the problem of Bitcoin users de-anonymisation [9][10][11][12][13] while others analyse the interplay between social interactions and the movements of the Bitcoin price [14,15].
With the present work, we aim at summing up the results of three papers [16][17][18].In [16], the authors analyse the local properties of two different Bitcoin representations, i.e. the Bitcoin Address Network (BAN) and the Bitcoin User Network (BUN) and inspect the presence of correlations between (exogenous) price movements and (endogenous) changes in the topological structure of the aforementioned networks.In [17], the mesocale structure of the BUN is under scrutiny: particular attention is devoted to the identification of the best network model able to describe it; besides, the same exercise as above is carried out, i.e. the comparison between the evolution of purely structural properties and the appearance of price bubbles in a cyclical fashion.Lastly, in [18], the authors inspect the evolution of the BLN topology, pointing out that it is becoming an increasingly centralised system and that the 'capital' is becoming increasingly unevenly distributed.

DATA
As previously said, Bitcoin relies on a decentralised public ledger, the blockchain, that records all transactions among Bitcoin users.A transaction is a set of input and output addresses: the output addresses that are 'unspent ', i.e. not yet recorded on the ledger as input addresses, can be claimed, and therefore spent, only by the owner of the corresponding cryptographic key.This is the reason why one speaks of pseudonimity: an observer of the blockchain can see all unspent addresses but cannot link them to the actual owners.

The Bitcoin Address Network (BAN)
The BAN is the simplest network that can be constructed from the blockchain records: from a technical point of view, it is a directed, weighted graph whose nodes represent addresses; the direction and the weight of links are provided by the input-output relationships defining the transactions recorded on the blockchain.The BAN has been considered across a period of 9 years, i.e. from 9th Janury 2009 to 18th December 2017 at the end of which the data set consists of 304.111.529addresses, exchanging a total number of transactions amounting at 283.028.575.In terms of traded volume, the transactions between addresses amount at 4.432.597.496bitcoins.

The Bitcoin User Network (BUN)
Since the same owner may control several addresses [11], one can define a network of 'users' whose nodes are clusters of addresses.These clusters are derived by implementing different heuristics, provided by the state-of-theart literature [9,10,19,20].The 'user networks' we obtain should not be considered as a perfect representation of the actual networks of users but, rather, an attempt to group addresses while minimising the presence of false positives.The BUN has been considered across the same period of the BAN (i.e. of 9 years, from 9th Janury 2009 to 18th December 2017) at the end of which the data set consists of 16.749.939users, exchanging a total number of transactions amounting at 224.620.265.In terms of traded volume, the transactions between users amount at 3.114.359.679bitcoins.

The Bitcoin Lightning Network (BLN)
The BLN is constructed in a fashion that is similar to way the BAN is defined: it is a directed, weighted graph whose nodes are the addresses exchanging bitcoins on the 'Layer 2'.Three different representations of the BLN have been studied so far, i.e. the daily one, the weekly one and the daily-block one: while a daily/weekly snapshot includes all channels that were found to be active during that day/week, a daily-block snapshot consists of all channels that were found to be active at the time the first block of the day was released (hence, the transactions considered for the daily-block representation are a subset of the ones constituting the daily representation).The BLN was considered across a period of 18 months, i.e. from 14th January 2018 to 13th July 2019, at the end of which the network consists of 8.216 users, 122.517 active channels and 2.732,5 transacted bitcoins.

Notation
Although the information about the magnitude of transactions is available, the BAN and the BUN have been analysed as binary, directed networks; as such, they are completely specified by their binary, asymmetric adjacency matrices A (t) BAN and A (t) BUN , at time t.The generic entry a (t) ij is equal to 1 if at least one transaction between address (user) i and address (user) j takes place, i.e. bitcoins are transferred from address (user) i to address (user) j, during the time snapshot t and 0 otherwise.The BLN, on the other hand, is a weighted, undirected network, represented by a symmetric matrix W (t) BLN whose generic entry w ji indicates the total amount of money exchanged between i and j, across all channels, at time t; here, we will mainly focus on its binary projection B (t) BLN , whose generic entry reads b

The Bitcoin Address and User networks
Let us start by reviewing the results concerning the BAN and the BUN at the weekly time scale.Similar results are observed for the BAN and the BUN at the daily time scale [16].
Basic statistics.Let us start by commenting one the evolution of some basic statistics characterising the BAN and the BUN.As fig. 1 shows, both the number of nodes N and the number of links L = i j( =i) a ij increase steadily in time, irrespectively from the considered representation; the link density d = L N (N −1) , however, decreases, meaning that the system becomes sparser.The dependence of d from N can be better specified from a mathematical point of view, upon noticing that the average degree N is constant over time [16]; hence, it follows that L ∝ N and d ∼ N −1 .As a general comment, notice how the basic statistics have started to evolve in a more stationary fashion since middle 2011 (see also [12]).
Degree distributions.Generally speaking, both out-and in-degrees are characterised by heavy-tailed distributions, indicating that a large number of low-connected nodes co-exists with few hubs whose degree is several order of magnitudes larger.A visual inspection of the functional form of the degrees distributions suggests the latter ones to follow a power-law [16,21]; to test this hypothesis the authors in [16] employed an algorithm based on a double Kolmogorov-Smirnov statistical test [22?]: what emerges is that the hypothesis above cannot be rejected, at a 0.05 confidence level, for almost half of the considered snapshots.
Of particular interest is the evolution of the out-degrees standard deviation, especially for what concerns its informativeness about exogenous events.As an example, let us consider the failure in February 2014 of Mt.Gox, a quasi-monopolist exchange market at the time.Such an event deeply affected the overall Bitcoin structure: the percentage of snapshots for which the null hypothesis (i.e. the out-degrees distribution follows a power-law) can be rejected amounts at 50% before February 2014 while it drops to 25% afterwards.
The authors in [16] also argue that the presence of heavy-tailed distributions may be explained by a mechanism similar to the preferential attachment one: new, or occasional, users 'preferentially' connect to already well-connected nodes (exchange markets, utility providers, etc.), thus leading to the formation of super-hubs.Elsewhere it has been argued that the related mechanism known as 'fittest-gets-richer' or 'good-gets-righer' [23] may be also at work, the computational resources of a node playing the role of its fitness [7].
Bitcoin structure VS Bitcoin price.The result concerning the evolution of the out-degrees distribution suggests that the Bitcoin network structure indeed brings the signature of exogenous events.Since Bitcoin is a cryptocurrency, a natural step is that of studying the presence of correlations between purely topological quantities and its price.
The simplest analysis to carry out is that of scattering the network size and the network link density versus the Bitcoin price (in USD).As shown in fig.2, a clear trend appears, indicating that the price and the size N (the link density d) are overall positively (negatively) correlated throughout the entire Bitcoin history.Notice, however, the trend inversion that can be appreciated immediately after the Mt.Gox failure: it is a consequence of the prolonged price decrease observed in 2014-2016, during which the network size has increased of (almost) one order of magnitude.
To further confirm the presence of a double regime, the authors in [16] have inspected the correlation between the moments of out-degrees distribution and the Bitcoin price over time.To this aim, the so-called Price and its Moving Average (RPMA) indicator has been adopted, defined as Although the network size increases, it becomes sparser (irrespectively from the considered representation).Similar results are observed for the BAN and the BUN at the daily time scale.See also [16].
RPMA t = 100 log 10 P t with τ representing a tunable temporal parameter.As shown in [16], the standard deviation and the kurtosis diverge as the network size grows larger than the value observed in correspondence of the Mt.Gox failure, thus confirming the 'two regimes hypothesis'.Moreover, as fig. 2 shows, larger values of the aforementioned moments (observed after the Mt.Gox failure) correspond to price drops, while temporal snapshots corresponding to smaller values of the same quantities seem to be characterised by price increases.A multivariate Granger test [24] has been also carried out to unveil possible lagged correlations hidden in the data (see fig. 5 in [16]).To this aim, data have been split in two sub-samples, i.e. 2010-2013 and 2014-2017, and the number of nodes N , the number of links L and the higher moments of the empirical (out-and in-) degrees distributions have been put in relation with the log-returns of the Bitcoin price (in USD), within each sub-sample.To sum up, when the BUN is considered at the weekly time scale, a positive feedback loop occurs between N and the price log-returns, whereas at the daily time scale a price increase predicts an increase of the number of nodes N but the viceversa is no longer true.The causality structure is consistent within the two sub-samples.
Analysis of the BUN mesoscale structure.Let us now revise the results concerning the mesoscale structure of the BUN.A recently proposed method [25] based on the surprise score function was adopted by the authors of [17] to assess the statistical significance of a peculiar mesoscale organization, known as core-periphery structure.According to the interpretation proposed in [25], revealing the core-periphery structure by minimising the surprise means individuating the partition that is least likely to be explained by the null model known as Random Graph Model (RGM) with respect to the null model known as Stochastic Block Model (SBM) -see also Appendix A. As fig. 3 shows, a core-periphery See also [16].
structure is indeed present: more precisely, during the biennium 2014-2015 the core size amounts at 30% of the total network size; after 2016, instead, it seems to shrink back to 2010-2013 values.According to the interpretation provided above, this finding, in turn, means that the BUN is indeed characterised by subgraphs with very different link densities, an evidence that cannot be explained by a model defined by just one global parameter as the one characterising the RGM.
A deeper inspection of the BUN core-periphery structure reveals it to be even richer: in fact, the core portion of the BUN is, actually, the strongly-connected component (SCC) of a bow-tie structure whose remaining portions (e.g. the IN and OUT components) compose the BUN periphery [17].More specifically, while the SCC is the set of nodes that are mutually reachable (i.e. a directed path from any node to any other node, within the SCC, exists), the IN and OUT components are respectively defined as the set of nodes from which the SCC can be reached and the set of nodes that can be reached from the SCC.Hence, the picture provided by the evolution of the core-periphery structure can be further refined as follows: since 2016 both the SCC and the OUT-component shrink while the IN-component becomes the dominant portion of the network [17].
An additional analysis, aimed at better quantifying the extent to which a generic, purely topological quantity X and the Bitcoin price are related, can be carried out by plotting the evolution of the temporal z-score where T is the mean over a sample of values covering the period T before time t -in our case, the year before t -and s X = X 2 − X 2 is the corresponding standard deviation.For example, the choice X = σ[k out ] allows price drawdowns to be revealed and, in some cases, anticipated [16]: in the triennium 2010-2012, and after 2017, the price grows as z (t) σ[k out ] increases while drawdowns appear in periods during which z (t) σ[k out ] decreases.Other possible choices are X = N core and X = r, i.e. the number of core nodes and the network reciprocity, defined as r = i j( =i) aij aji i j( =i) aij , i.e. as the percentage of links having a 'partner' pointing in the opposite direction.The evolution of the temporal z-score for the two aforementioned quantities is shown in fig. 4. Overall, the two trends show some similarities, being characterised by peaks in correspondence of the so-called bubbles, i.e. periods of 'unsustainable' price growth [26]: interestingly, such periods are characterised by values of the inspected topological quantities which are significant also in a statistical sense, as the value of the corresponding temporal z-score proves (in fact, z (t) ≥ 2 in both cases).Moreover, peaks are also revealed in the triennium 2014-2016, thus signalling some kind of 'activity' missed by purely financial indicators (e.g. the RPMA).

The Bitcoin Lightning network
Let us now move to review the results concerning the BLN.In what follows we will focus on the daily-block snapshot representation.
Basic statistics.As observed for the BAN and the BUN, both the number of nodes N and the number of links L = i j(>i) b ij of the BLN increase steadily in time while it becomes sparser.Interestingly, however, the evolution of the BLN link density seems to point out the presence of two regimes: as fig. 5 shows, during the first phase (i.e.N ≤ 10 3 ) L increases linearly in N and the link density is well described by the functional dependence d ∼ N −1 ; afterwards, the link density decrease slows down, seemingly indicating that L has started to grow in a super-linear FIG. 4. Evolution of the temporal z-score for the number of core nodes (top panel) and for the reciprocity (bottom panel), for the BUN weekly representation.Shaded areas individuate the so-called bubbles, i.e. periods of price increase according to [26].Additionally, each dot representing an observation is coloured according to the value of the Ratio between the current Price and its Moving Average (RPMA) indicator.Overall, the two trends show some similarities as peaks are clearly visible in correspondence of the so-called bubbles, identified by the shaded areas (see also [26]).Interestingly, these values are significant in a statistical sense, as the temporal z-scores reach values z (t) ≥ 2. See also [17].
fashion with respect to N .
Analysis of the BLN mesoscale structure.In order to inspect the evolution of the BLN 'centralisation', the authors in [18] have considered two different sets of quantities.First, they have computed the Gini index for four centrality measures, i.e. the degree, closeness, betweenness and eigenvector centrality (respectively indicated with the symbols i , e c i -see also Appendix B) and plotted it versus the number of nodes.Interestingly, G c increases for three measures out of four, pointing out that the values of centrality are more and more unevenly distributed [18].
Additionally, they have also computed the so called centralisation indices, encoding the comparison between the structure of a given network and that of a reference network, i.e. the 'most centralised' structure -see also Appendix B. For what concerns the degree, closeness and betweenness centrality, it is the star graph: for what concerns the eigenvector index, the star graph does not represent the maximally-centralised structure, although it is kept for the sake of homogeneity with the other quantities.The evolution of the centralisation indices indicates that the BLN is not evolving towards a star graph (indeed, a too simplistic picture to faithfully describe the BLN topology) but towards a suitable generalization of it, i.e. the core-periphery structure [18] (see also later).
The authors in [18] have also benchmarked the observations concerning the evolution of the centrality and the centralisation indices with the predictions, for the same quantities, output by the maximum-entropy null model known for the BLN daily-block snapshot representation.As for the BAN and the BUN, the position d ∼ N −1 well describes the link density dependence on N , at least for the snapshots for which N ≤ 10 3 .See also [18].
as Undirected Binary Configuration Model (UBCM -see also Appendix C).To this aim, they have explicitly sampled the ensembles of networks induced by the UBCM [27,28] and compared the ensemble average of the quantities of interest with the corresponding empirical values.From a merely technical point of view, the authors adopt an iterative, reduced algorithm to solve the system of equations defining the UBCM, i.e.
a choice allowing them to solve it within tens of seconds even for configurations with millions of nodes [17] -see also Appendix C. As fig.6 shows, such a comparison reveals that the UBCM tends to overestimate the values of the Gini index for the degree, the closeness and the betweenness centrality and to underestimate its values for the eigenvector centrality.This seems to point out a non-trivial (i.e.not reproducible by just enforcing the degrees) tendency of well-connected nodes to establish connections among themselves -likely, with nodes having a smaller degree attached to them: such a disassortative structure could explain the less-than-expected level of unevenness characterising the other centrality measures, as each of the nodes behaving as the 'leaves' of the hubs would basically have the same values of degree, closeness and betweenness centrality.
For what concerns the analysis of the centralisation indices, fig.6 shows that the UBCM underestimates both the betweenness-and the eigenvector-centralisation indices: in other words, a tendency to centralisation 'survives' even after the information encoded into the degrees is properly accounted for, letting the picture of a network characterised by some kind of more-than-expected 'star-likeness' emerge.This observation can be better formalised by analysing the BLN mesoscale structure via the optimization of surprise: as observed for the BUN, a core-periphery structural organization, whose statistical significance increases over time, indeed emerges [18] (see also fig. 7).
In [17], the authors have also adapted the iterative, reduced algorithm cited above to the resolution of the Directed Binary Configuration Model (DBCM -see also Appendix C).
A quick look at the weighted structure of the BLN.Having a quick look at the weighted structure of the BLN leads to two notable observations: both the total amount of exchanged bitcoins and the unevenness of their distribution increase.This trend is confirmed by the evolution the Gini coefficient whose value reaches 0.9 for the last snapshots of our data set.On average, across the entire period, about the 10% (50%) of nodes holds the 80% (99%) of the bitcoins at stake in the network [18].

DISCUSSION
The public availability of the complete Bitcoin transaction history allows to quantify the evolution of structural quantities that characterise different Bitcoin Transaction Networks, and to inspect the inter-dependency between them and the dynamics of Bitcoin price -an analysis that intends, also, to gain insight on the behaviour of Bitcoin Bottom panels: comparison between the observed degree-, closeness-, betweenness-and eigenvector-centralisation measures and their expected value computed under the UBCM.Once the information contained into the degree sequence is properly accounted for, a (residual) tendency to centralisation is still visible, letting the picture of a network characterised by some kind of more-than-expected 'star-likeness' emerge.See also [18].
users; still, the understanding of the mechanisms shaping the joint evolution of the three quantities above remains far from being complete.
This paper aims at providing an overview of the most recent results on the topic achieved in the last years.One of the main messages concerns the possibility to retrieve signals of exogenous events by analysing the blockchain-induced transaction networks: the best example is provided by the failure of Mt.Gox in 2014, an event deeply affecting the Bitcoin Address Network and the Bitcoin User Network structure (i.e.irrespective from the chosen representation).From this point of view, out-degrees have been found to be particularly informative properties: higher moments of the out-degrees distribution (as the standard deviation, the skewness and the kurtosis) diverge as the network size grows larger than the value observed in correspondence of the Mt.Gox failure; besides, the out-degrees heterogeneity rises during periods of price decline and vice-versa.Such a result is further refined by a Granger causality analysis, revealing that during the triennium 2010-2012 an increase of the out-degrees standard deviation causes a price decline [16].These results, in turn, suggest a sort of behavioural explanation for the price dynamics displayed during the early stages of Bitcoin: during periods in which the price continuously increases, always larger numbers of traders are, in turn, attracted to the system; the former ones, likely performing only few transactions, link to the network hubs (usually exchange markets) that gain a large number of connections over the course of the weeks and cause the price to rise even more.
Interestingly, the analysis of the Bitcoin Lightning Network reveals the same trends observed for the BAN and the BUN appear, as the emergence of a statistically-significant core-periphery structure, of an uneven distribution of the nodes' centrality and wealth, etc.: these results suggest tendency of the Bitcoin 'Layer 2' network to become less distributed.This process has the undesirable consequence of making this off-chain payment network less resilient to failures, malicious attacks.
All the results reviewed in this article ultimately -and consistently -point out a tendency to centralisation that has been observed in the Bitcoin structure at different levels [18,29], an evidence that deserves to be investigated in greater detail.with core-nodes drawn in red and periphery-nodes drawn in green.See also [18].
which is nothing else that the multinomial version of the surprise, originally proposed to carry out a community detection exercise [30].The presence of three different binomial coefficients allows three different 'species' of links to be accounted for: the binomial coefficient V• i enumerates the number of ways i links can redistributed within the first module (e.g. the core portion), the binomial coefficient V• j enumerates the number of ways j links can redistributed within the second module (e.g. the periphery portion) and the binomial coefficient enumerates the number of ways the remaining L − (i + j) links can be redistributed between the first and the second module, i.e. over the remaining V − (V • + V • ) node pairs; the values i and j are bounded by the values V • and V • , although the sum i + j can range between l * • + l * • and the minimum between L and V • + V • .From a technical point of view, S is the p-value of a multivariate hypergeometric distribution, describing the probability of i + j successes in L draws (without replacement), from a finite population of size V that contains exactly V • objects with a first specific feature and V • objects with a second specific feature.

APPENDIX B -CENTRALITY AND CENTRALIZATION MEASURES
Indices measuring the centrality of a node aim at quantifying the 'importance' of a node in a network, according to some specific topological property.Among the measures proposed so far, of particular relevance are the degree centrality, the closeness centrality, the betweenness centrality and the eigenvector centrality: • the degree centrality of node i coincides with the degree of node i, i.e. the number of its neighbours, normalised by the maximum attainable value, i.e.N − 1: From the definition above, it follows that the most central node, according to the degree variant, is the one connected to all the other nodes; • the closeness centrality of node i is defined as where d ij is the topological distance between nodes i and j, i.e. the length of the shortest path(s) connecting them.From the definition above, it follows that the most central node, according to the closeness variant, is the one lying at distance 1 from each other node; • the betweenness centrality of node i is given by where σ st is the total number of shortest paths between node s and t and σ st (i) is the number of shortest paths between nodes s and t that pass through node i.From the definition above, it follows that the most central node, according to the betweenness variant, is the one lying 'between' any two other nodes; • the eigenvector centrality of node i, e c i , is defined as the i-th element of the eigenvector corresponding to the largest eigenvalue of the binary adjacency matrix (whose existence is ensured by the Perron-Frobenius theorem).According to the definition above, a node with large eigenvector centrality is connected to other 'well connected' nodes [21].
The centrality indices defined above provide a rank of the nodes of a network.Sometimes, however, it is useful to compactly describe a certain network structure in its entirety.To this aim, a family of indices, known as centralisation indices, has been defined.In mathematical terms, any centralisation index reads can be employed.As for the UBCM case, the solutions to the system of equations defining the DBCM in the sparse case, i.e. x L , ∀ i, can be chosen to initialise the recipe above.The computation of the system above can be further sped up, by assigning to the nodes with the same (pair of) out-and in-degrees (k out i , k in i ) the same pair of values (x, y) [31].

FIG. 1 .
FIG.1.Evolution of basic statistics, i.e. the number of nodes (left panels) and the link density (right panels) for two Bitcoin network representations, i.e. the BAN (bottom panels) and the BUN (top panels) at the weekly time scale, from July 2010 to 18th December 2017 (i.e. for networks with at least 200 nodes).Although the network size increases, it becomes sparser (irrespectively from the considered representation).Similar results are observed for the BAN and the BUN at the daily time scale.See also[16].

FIG. 2 .
FIG.2.Correlation between the Bitcoin price in USD, the basic statistics (number of nodes and link density -top panels) and the moments of the out-degrees distribution (bottom panels) for the BUN at the weekly time scale.Additionally, each dot representing an observation is coloured according to the value of the Ratio between the current Price and its Moving Average (RPMA) indicator.The vertical, dashed line coincides with the bankruptcy of Mt.Gox.Purely structural quantities are correlated with exogenous quantities as the Bitcoin price; see, for example, the evolution of the out-degrees standard deviation whose larger values (observable after the Mt.Gox failure) correspond to price drops.See also[16].

FIG. 3 .
FIG. 3. Evolution of the percentage of nodes belonging to the core portion of the BUN at the weekly time scale.During the biennium 2012-2013 the core portion of the BUN steadily rises until it reaches 30% of the network; afterwards, during the biennium 2014-2015, it remains quite constant; then, during the last two years covered by our data set (i.e.2016-2018), the core portion of the BUN shrinks and the percentage of nodes belonging to it goes back to the pre-2012 values.The vertical, dashed line coincides with the bankruptcy of Mt.Gox.

FIG. 5 .
FIG.5.Evolution of the total number of nodes N , total number of links L and link density d = 2LN (N −1) for the BLN daily-block snapshot representation.As for the BAN and the BUN, the position d ∼ N −1 well describes the link density dependence on N , at least for the snapshots for which N ≤ 10 3 .See also[18].

FIG. 6 .
FIG.6.Top panels: comparison between the observed Gini index for the degree, closeness, betweenness and eigenvector centrality (x-axis) and their expected value, computed under the UBCM (y-axis) for the BLN daily-block snapshot representation.Bottom panels: comparison between the observed degree-, closeness-, betweenness-and eigenvector-centralisation measures and their expected value computed under the UBCM.Once the information contained into the degree sequence is properly accounted for, a (residual) tendency to centralisation is still visible, letting the picture of a network characterised by some kind of more-than-expected 'star-likeness' emerge.See also[18].

Figure 16 : 21 FIG. 7 .
Figure 16: (colour online) Comparison between the largest connected component of the BLN (daily-block snapshot representation) on day 16 (left -95 nodes and 155 links are present) and on day 34 (right -359 nodes and 707 links are present).A visual inspection of the network evolution suggests the presence of a core-periphery structure since its early stages.