- Open Access
The bow tie structure of the Bitcoin users graph
Applied Network Science volume 4, Article number: 56 (2019)
The availability of the entire Bitcoin transaction history, stored in its public blockchain, offers interesting opportunities for analysing the transaction graph to obtain insight on users behaviour. This paper presents an analysis of the Bitcoin users graph, obtained by clustering the transaction graph, to highlight its connectivity structure and the economical meaning of the different obtained components. In fact, the bow tie structure, already observed for the graph of the web, is augmented, in the Bitocoin users graph, with the economical information about the entities involved. We study the connectivity components of the users graph individually, to infer their macroscopic contribution to the whole economy. We define and evaluate a set of measures of nodes inside each component to characterize and quantify such a contribution. We also perform a temporal analysis of the evolution of the resulting bow tie structure. Our findings confirm our hypothesis on the components semantic, defined in terms of their economical role in the flow of value inside the graph.
This paper presents an analysis of the Bitcoin users graph, obtained by heuristic clustering of the Bitcoin transaction graph. In the users graph nodes represent Bitcoin users and edges model the flow of value between them. This graph contains information which may be used to conduct rich analyses. Indeed, the nodes are augmented with the users balance and the edges are weighted according to the Bitcoin value exchanged. Moreover, the information contained in the Blockchain reports also the creation dates of each edge, and this can be exploited to perform a set of temporal analysis.
The analysis takes inspiration from the seminal paper (Broder et al. 2000), introducing the concept of a bow tie structure for the graph representing the Web (subsequently refined in Meusel et al. (2014); Donato et al. (2008)). In this graph, each node corresponds to a web page and two nodes are connected by a direct arc whether there is an hyperlink from one to the other. Differently from the graph collected by Broder et al. (2000), we have a richer set of information which allows us to link the structure of the graph with the economical activities of the users.
The macroscopic representation of the graph as a bow tie derives from the partitioning of the graph in separate components according to the connectivity of its nodes, i.e. each node is assigned to a given component according to its reachable nodes set. The nodes in the biggest strongly connected component are called SCC. The remaining nodes reaching (resp. reached by) the ones in the SCC are called IN (resp. OUT). The other nodes in the biggest weakly connected component are called TUBE, TENDRIL, or FRINGE (see “Formal definitions and method” section for the formal definition), and the remaining nodes of the graph are called DISCONNECTED.
In the first part of the paper, we support our conjecture that each component gives a different contribution to the graph from an economical point of view. In this sense, the macroscopic bow tie structure of the graph reflects the flow of value between the different components in the Bitcoin economy. In such light, we might think of the SCC component as the dynamic core of the economic community, the component where value exchanges take place. Following the same model, the IN component would contain the nodes moving value towards the SCC and OUT would represent the set of nodes where value is credited from the SCC. We verify our conjecture on actual data, proving that the purely topological structure reflects on the different measures we consider to monitor the economical activity of the nodes.
In the second part of the paper, we perform a temporal analysis, studying how the different components change over time. Since by our hypothesis the topology is linked to the economical activity, our observations give also insights on how said economical activity changes over time from a macroscopic point of view in the Bitcoin economy.
We have presented a preliminary evaluation of the Bitcoin User graph connectivity structure in Di Francesco Maesa et al. (2018b). Beside a general revision and improvement, this paper extends our previous work in the following directions:
we include a “Data acquisition” section, to better explain how the dataset has been acquired and built;
we provide the high level definition of the algorithm used to assign roles to the nodes in the graph (Formal definitions and method);
we double the set of measures performed on the connectivity components, by including a set of measures regarding cluster lifespans. We define such new measures, comment their relevance regarding the components and present a new graph to outline the most interesting result (The bow tie structure of the Bitcoin users graph);
we extend the temporal analysis of the graph components not only by including and commenting a new graph to corroborate our previous observations, but also by discussing more deeply node activity and disconnected component behaviour (Temporal analysis). A new set of experimental results is provided to support our considerations.
The paper is structured as follows. In “Background and related work” section we present the relevant background and related work on Bitcoin users graph analysis. In “Data acquisition” section we show our data acquisition method, and in “Formal definitions and method” section we present our formal definitions of role components and method used to compute them. In “The bow tie structure of the Bitcoin users graph” section we present our main findings, further refined by a temporal analysis presented in “Temporal analysis” section. Finally, we present our conclusions and future work in “Conclusions” section.
Background and related work
Bitcoin (Nakamoto 2008) is a cryptocurrency relying on blockchain technology. This means that the entire history of the system is saved inside a secure and decentralised ledger (called blockchain) in an append only fashion. The blockchain defines the state of the system and each new block added to it contains an ordered set of transactions expressing a state update. From an high level point of view it can be modelled as a mapping of values to addresses, where transactions are the only tool available to change such mapping by transferring value between different addresses. This abstraction suffices for the scope of this paper, for more precise explanation of the Bitcoin protocol see Bonneau et al. (2015). Each transaction is many to many, i.e. it can have multiple addresses providing value to be spent as inputs (named multi-input transaction) and more than one address receiving part of such value as output. The couple (address, value) receiving a payment through a transaction is called transaction output in the rest of this paper. Furthermore there exists a special transaction type, called coinbase, to reward a fixed value and some collected fees to some miner addresses for each block. A miner is a voluntary validator node, willing to dedicate some computational power to take part into the distributed consensus algorithm behind the Bitcoin blockchain security guarantees. Since validating new blocks, i.e. mining, is a computationally intensive task, randomized rewards are proportionally assigned to miners. The rationality behind such rewards and the mining process is beyond the scope of this paper, for further reading see Bonneau et al. (2015). It suffices to say that newly generated value is constantly entered into the system through special transactions with no inputs (i.e. coinbase transactions).
Users take part in the system through addresses, that are just representations of a public key owned by the user. Users anonymity is only protected through addresses pseudonymity, i.e. the fact that addresses owned by the same user do not share any information between themselves and do not carry on any real world information about that user. This has lead to the development of deanonymization attacks, aimed at breaking the addresses pseudonymity property. Usually this is achieved through heuristic clustering, i.e. the grouping together of different addresses belonging to the same user in a single cluster. For example, the most used heuristic rule, called common inputs heuristic states that all input addresses of a transaction belong to the same user (Nakamoto 2008; Fergal and Harrigan 2013). By parsing all transactions in the blockchain it is possible to build a transactions graph (representing value exchanges between addresses), that can be then refined into an users graph (representing payments between approximated users) by applying the heuristic clustering.
Several analysis of the Bitcoin graph have been presented. Some of them only consider the transactions graph (Kondor et al. 2014; Popuri and Gunes 2016), but this may lead to less meaningful results due to the high number of different addresses that are controlled by the same user. Other analysis have been performed on the users graph (Ron and Shamir 2013; Meiklejohn et al. 2013; Androulaki et al. 2013; Lischke and Fabian 2016; Di Francesco Maesa D et al. 2017; Di Francesco Maesa et al. 2016a), as for our own study first presented in Di Francesco Maesa D et al. (2016b) and later expanded in Di Francesco Maesa et al. (2018a). Relying on the users graph instead of the transactions graph often results in more interesting insight, but the accuracy of the clustering step needs to be taken in account not to skew the analysis (Harrigan and Fretter 2016). Despite some efforts have been made trying to represent the Bitcoin users graph structure, mainly in the area of graph visualization (McGinn et al. 2016), at the best of our knowledge no previous work is available on its macroscopic bow tie structure analysis, as performed by this work.
The analysis presented in this paper are performed on a set of snapshots of the Bitcoin users graph obtained from the dataset presented in Di Francesco Maesa et al. (2018a). For a detailed description of the dataset and tools used to retrieve it, the interested reader can refer to Di Francesco Maesa et al. (2018a), in this section we only highlight its characteristics more relevant to this work. We also remark how this work is based on the Bitcoin main chain, no forks of the Bitcoin official history are taken into account (such as the BitcoinCash one).
The Bitcoin users graph is obtained by applying the common inputs heuristic (see “Background and related work” section) and is built through the algorithm presented in Di Francesco Maesa et al. (2018a) whose complexity is linear in the number of addresses and transaction outputs. The clustering algorithm works by building an auxiliary graph were each node represents an address. For any multi-input transaction, it then adds a new edge between the first input address and each one of the other input addresses. When all transactions have been so processed, the connected components of the obtained graph are computed, and each connected component is mapped to a cluster, containing the addresses of that component. It is proven in Di Francesco Maesa et al. (2018a) how this methods is the same as computing the common input heuristic. The heuristic is applied to all transactions contained in the first 398 800 blocks of the official Bitcoin blockchain, i.e. taking into account all Bitcoin transactions from 2009-01-03 18:15:05 GMT to 2015-12-23 09:40:52 GMT. Among these transactions only the outputs containing Pay to PubKey Hash (p2pkh), Pay to PubKey (p2pk) and Pay to Script Hash (p2sh) standard script types are interpreted, that alone represent 99.5% of all transaction outputs (see Di Francesco Maesa et al. (2018a)). We obtained from (Blockchain Info Tags 2019) a set of identity tags associated to addresses, we used such tagged addresses to identify the clusters containing them.
The temporal analysis is performed by considering a set of temporal snapshots of the users graph. For each temporal snapshot, the analysis is applied to the induced graph, which is the graph containing only transactions with a timestamp less than the considered cut-off timestamp, pruned of its periphery, i.e. by cutting-off nodes that only have one incoming arc. The main aim of this pruning is to penalize the nodes corresponding to users that just received a payment and might have had no time to spend it due to the artificial cut introduced by the time snapshot cut-off. Do note that the pruned nodes have no influence on the connectivity of the graph (because they only have a single incoming edge).
Finally, since each node of the graph corresponds to a cluster of addresses, in the following we will use the term node and cluster interchangeably. We also remark how we use the term node according to the graph theory notation, i.e. a node belongs to the users graph and, in turn, it represents a set of addresses, not to be confused with a node of the Bitcoin peer-to-peer communication network.
Formal definitions and method
Given the users graph G(V,E) obtained by the clustering heuristic of the Bitcoin transactions graph (see “Background and related work” section), we assign roles to the nodes in V following the notation of Broder et al. (2000), and summarized in the well-known bow tie structure scheme shown in Fig. 1. The role of a node x is based on the set of nodes x can reach and that can reach x, as formalized next. In the following we refer to G∗ as the undirected graph obtained symmetrizing G, i.e. obtained by G without considering the direction of the edges. Moreover, we call G−1 as the inverse graph of G obtained by reverting the direction of all the edges of G.
Given a graph G(V,E), the role of the nodes in V is defined as follows.
disconnected: nodes not connected to the giant connected component of G∗.
scc: nodes in the giant strongly connected component of G.
in: nodes not in scc and able to reach the nodes in scc.
out: nodes not in scc and reachable by the nodes in scc.
tendril: nodes not in the previous categories, that either can reach at least a node in out (tendrilToOut) or can be reached by at least a node in in (tendrilFromIn), but not both.
tube: nodes not in scc, that can reach at least a node in out, and can be reached by at least one node in in.
fringe: nodes not in any of the previous categories.
The nodes having role FRINGE include the ones which are connected to TENDRIL (eventually using an undirected path). For instance, this is the case of a node y reaching x∈TENDRIL, where x can be reached by IN (i.e. x∈TENDRILFROMIN).
For the sake of completeness, we report here the linear algorithm we have used to assign to each node of G its corresponding role. It should be remarked that, due to the size of the network, the reachability tests for sets of nodes have to be linear. We have addressed this task using a BFS (Breadth First Search) multi-source, which, given a set of nodes X, computes the set of nodes reachable from at least one node in X. Namely, a BFS multi-source is a modified BFS whose starting queue is filled with all the nodes in X.
The resulting algorithm is shown in Algorithm 1. It firstly starts computing G−1,G∗ and the connected components of G∗ to assign the label DISCONNECTED. After this, the algorithm computes the strongly connected components of G to assign the role SCC. It then computes the nodes reachable from the nodes in SCC (assigning to them the role OUT). This is done using a BFS multi-source which marks the visited nodes. Similarly, but using the symmetric graph G−1, the algorithm computes the nodes able to reach the nodes in SCC (assigning to them the role IN). At Lines 11 and 12, the algorithm computes the nodes reachable from IN in G, assigning to them the role TENDRIL. Finally, the nodes reaching OUT are computed: the ones who were reached also by IN are labelled as TUBE, the other ones are labeled as TENDRIL. All the remaining nodes are labeled as FRINGE.
As a result, the following holds.
Given a graph G(V,E), Algorithm 1 computes the roles of all the nodes in V according to Definition 1.
The bow tie structure of the Bitcoin users graph
By classifying the nodes of the Bitcoin users graph according to Definition 1, we obtained a bow tie structure for the Bitcoin users graph, showed in Fig. 1. In this section, we first present some statistics on the shape and size of the structure and then we introduce and support our economical interpretation. In particular, by dividing the graph in different components according to the bow tie model, we show how they exhibit a different behaviour considering non topologically dependent measures. Moreover a semantic explanation of the role of each component can be inferred from the way the Bitcoin protocol works.
Bow Tie components size For the sake of completeness, we report in Fig. 1 the relative sizes of the different components of the bow tie we found. We can note how these sizes are very different from those presented for the web graph analyzed in Broder et al. (2000), whose nodes are web pages and whose edges represent hyperlinks between themFootnote 1. The results about the web graph presented in Broder et al. (2000) show the IN, OUT, and TENDRIL components all with almost same dimensions (21.29%, 21.29% and 21.52% respectively), while the SCC component is slightly bigger (27.74%) and DISCONNECTED about one third the size of them (8.24%). In comparison in Fig. 1 the SCC component is dominant over all the others (containing 86.36% of all the nodes in the graph), the next component by size is DISCONNECTED (9.08%) followed by OUT (2.48%) and IN (2.06%), with the size of all other components almost irrelevant. We should remark that the component sizes distribution reliability has been questioned as a measure in the literature, since it is thought to depend on the web crawler adopted. For example in Meusel et al. (2014), the authors state that “While it is always possible to compute the components of the bow tie ofBroder et al. (2000), the proportion of the components is not intrinsic”. In the same paper the computed component sizes differ greatly from Broder et al. (2000) (SCC 51.28%, IN 31.96%, OUT 6.05%, TENDRIL 4.61% and DISCONNECTED 5.84%), probably both because of the different crawler used and because of the different time of data collection.
Economical interpretation of the graph components In the remaining part of this section, we link the bow tie structure to the economical activity of the nodes involved in the different components. We aim to show that SCC represents the center of the economical activity, where IN nodes move value towards the SCC and OUT nodes correspond to nodes with value credited from the SCC.
In this scenario OUT would contain the yet unspent outputs from the SCC, either because the owner did not have time to spend them before the data acquisition time cut-off or because they were deposited for cold storage. The IN nodes instead should represent mainly miners obtaining newly minted value in the form of mining rewards (see “Background and related work” section). Such value is then injected (i.e. spent) in the main economy of Bitcoin, represented by the SCC. In fact a new node is created in the graph as soon as its corresponding cluster in the blockchain receives a payment, so, inside the giant weakly connected component, value flows by design from nodes with no incoming arcs (that can only be part of IN) through multiple intermediate nodes until they reach nodes with no outgoing arcs (that can only be members of OUT). Of course this is not in general the only case, since the same node can receive both new value as mining rewards as well as payments (and so arcs) from other nodes. In this scheme TENDRILFROMIN (TENDRILTOOUT) are anomalies that send value to (receive value from) nodes not part of the main economy (i.e. outside the SCC). Similarly TUBE nodes transmit value from IN to OUT bypassing the SCC completely.
To get a first insight supporting such hypothesis we used a dataset of 12 413 deanonymized nodes (with identities obtained from (Blockchain Info Tags 2019)), i.e. clusters containing an address associated to a known real world entity, to map them to the relative component in the graph. Deanonymized nodes were only found inside the four main components (SCC,IN,OUT and DISCONNECTED). In particular IN only contained three nodes representing known entities. By manual inspection we found these entities to belong to two minor minersFootnote 2 part of pools and a mining poolFootnote 3. Conversely most of the famous services and hubs of the Bitcoin economy, such as all of the named nodes observed in Di Francesco Maesa et al. (2018a) to be the most central nodes in the graph, are part of the SCC. E.g. consider Table 4 in Di Francesco Maesa et al. (2018a) showing the ten most central nodes according to Harmonic, Eigenvector and Page-Rank node centrality.
Evaluation of component metrics
To better understand the type of nodes with different roles we evaluated the following metrics with respect to each role. We do remark how the introduced measures, with the exception of CLUSTNUM, are not classical graph measures, and, as such, are independent from the structure of the graph considered.
ClustNum: represents the number of clusters for each role component, since clusters correspond to graph nodes, it is the same as number of nodes for each component.
Balance: measures the current (at data collection time) balance of a cluster (i.e. sum of balances of all its addresses).
ValueRec: expresses the total value received during a cluster lifespan, or up to the time of data collection if the cluster is still active. It is defined as the sum of all payments received. This measure helps to point out clusters with small current balance (for example zero), that have owned a lot of value in the past.
TransIn: represents the number of payments received by a cluster (coinbase rewards included). This measure is useful in estimating the economical importance of a cluster in the way of collecting payments.
TransOut: measures the number of payments done by a cluster. It is computed as the number of transactions originated from a cluster. This measure represents the economical importance of a cluster as its weight in issuing payments. Note that both TransIn and TransOut measures the in and out-degrees of the clusters, including self-loops and multi-edges. These kind of edges happen whenever two addresses of the same cluster exchange money or there is a repeated payment between two clusters.
We remark how this measure is not the same as the number of arcs incoming and outgoing, respectively, in the graph. In fact, due to the addresses clustering step, some transactions may become self loops or repeated arcs. Furthermore, repeated arcs may naturally be present between addresses, not consequence of our clustering. Since both self loops and repeated arcs do not influence the graph connectivity they are ignored during our connectivity analysis. This is why the two measures can vary greatly from nodes indegree and outdegree respectively.
AddrNum: counts the number of addresses in a cluster. This measure can be used to indirectly estimate the cluster activity.
CLifespan: measures the lifespan (i.e. activity time), in milliseconds, of a cluster performing at least one payment. We name this measure closed lifespan because the cluster has both a creation date, i.e. the date that it received its first payment (when one of its addresses first appeared in a transaction output in the blockchain), and a last activity date, i.e. the date of the last payment issued by the cluster. The average of this value is computed only among the closed lifespan clusters in each role component. Do note that by timestamp of a transaction we mean the timestamp of the block such transaction is part of in the blockchain.
CLifePerc: represents the fraction of clusters in a given role component that have a closed lifespan, i.e. the number of clusters making at least one payment divided by the number of clusters in that component.
OLifespan: measures the lifespan, in milliseconds, of a cluster that never performs a payment. We name this measure open lifespan because the cluster only has a creation date. Since there is no last activity date for such clusters we measure their lifespan as time passed from their creation date to the data collection cut-off. Same as for CLifespan, the average of this value is computed among open lifespan clusters only in each role component.
OLifePerc: represents the fraction of clusters in a given role component that have an open lifespan. As a consequence of their definitions it holds that OLifePerc+CLifePerc=1 for each role component.
The cumulative (i.e. the sum of the values of all nodes with the same role) and average (i.e. the cumulative value divided by the number of nodes with a given role) values of the introduced measures for each component are presented in Table 1. To better understand the contribution of each role component to each measure (excluding lifespan ones) we also depict the percentages of the cumulative values with respect to the cumulative value for the entire graph (ignoring role components) in Fig. 2.
As a side remark, since Table 1 reports the number of addresses in each component, by considering the ratio between the cumulative value of each measure in each component and the number of addresses in that component, one can characterize the average behaviour of the addresses independently from the clustering phase. This allows to obtain, for instance, the average balance and the average value of the incoming transactions of the addresses present in each component.
Interestingly, we will show how the above measures, not dependent from the topology of the graph, are consistent with the topological partition found.
Clusters density From Fig. 2 we can observe how ADDRNUM follows a distribution similar to the size of components, with an advantage for the SCC. In fact, inspecting the average values for such measure in Table 1, we can note how, among the four main components (SCC, IN, OUT and DISCONNECTED), SCC is the only one to exhibit a value sensibly greater than one. The DISCONNECTED component in particular exhibits a value very close to one, meaning that most of its clusters are actually addresses singletons. The fact that nodes in the SCC have the greatest average value for such measure supports our hypothesis that it contains the really active clusters of the economy.
First of all we remind that the lifespan values are computed from the block timestamps. The timestamp of each block is chosen by the miner actually creating such block. Since there is no concept of universal clock, the timestamps can be inconsistent between adjacent blocks, i.e. a given block can have a timestamp lower than a preceding block. As such, in few cases, the lifespan of a node could be negative. To avoid such problem we have introduced a lower bound of zero for the lifespans, i.e. nodes with negative lifespan are corrected to zero. We do remark how a lifespan of zero is, in general, a valid one, since it might represent a cluster receiving its first transaction and creating all its outgoing payments inside a single block.
From the values of OLIFEPERC and CLIFEPERC shown in Table 1 we can derive interesting considerations on clusters activity depending on the component they belong to. Obviously, 100% of clusters in SCC, IN, TENDRILTOOUT and TUBE have a closed lifespan (i.e. they performed at least one payment). This is a direct consequence of how such components are defined. Conversely more than 99% of clusters in DISCONNECTED have an open lifespan, meaning that they have never spent the value received. This is consisted with our assumption that they represent isolated users with little to no economical interaction between themselves. Similarly about 75% of clusters in OUT never spend the value received (do note that nodes in OUT can only transfer their value to other nodes in OUT), supporting our hypothesis on the role of clusters in such component.
Figure 3 depicts the average values of CLIFESPAN, i.e. the average time that clusters remain active, from Table 1, expressed in days instead of seconds. We can see from the figure how IN and SCC contain the most short lived clusters (on average), while clusters in OUT with a closed lifespan remain active for about three times longer. This is consistent with our assumption that IN clusters pay out the value received quickly to SCC, while OUT clusters contain temporary storage of value not immediately spent (and not as fast as it happens in the SCC). Moreover, by looking at the average OLIFESPAN of clusters in OUT we can measure the average time (i.e. age) of value in OUT still waiting to be spent. Clusters that never have spent their value in OUT have been waiting about 526 days (i.e. about one and a half years) on average. Even if this value encompasses also the value that has simply been credited too soon to be spent (i.e. too close to the data collection cut-off)Footnote 4, it can still be considered as lower value for average age of funds kept in cold storage or forgotten. Computing the same value for the DISCONNECTED component from Table 1, we obtain an average age of approximately 1960 days, i.e. about five years and an half of funds inactivity (out of the approximately seven years span of the data), consistent with the assumed semantic of the component.
Balance analysis Our assumption on OUT, i.e. that it contains unspent outputs from the SCC, while the real value exchanges take part inside the SCC, is supported by the presented results. We can observe in Fig. 2 the marked discrepancy between the current balance and total value received by clusters. In fact even if the SCC dominates all measures in the figure, including the cumulative value received by its clusters, it also shows a surprisingly relatively low cumulative current balance. The proportional high value received indicates how value is mostly exchanged inside the SCC, while the low current balance indicates how a big part of such value is in the end credited to the clusters in the OUT component, behaviour explained by our conjecture.
Coinbase transactions To better understand the flow of value in the graph we also performed a study on the presence of coinbase transactions (see “Background and related work” section) in clusters depending on their role. In particular we measured NUMCOINB, as the number of transaction outputs (i.e. payments) received by clusters from coinbase transactions, UNIQUECOINB, as the number of clusters that received at least one payment from a coinbase transaction and BALANCECOINB as the total value received by a cluster from coinbase transactions, i.e. the sum of all transaction outputs amounts received from coinbase transactions by the same cluster.
The fractions of the cumulative values by role component over the entire graph for these three measures are shown in Fig. 4, supporting our assumption on the IN component role. In fact we can note how the IN component contains most of the clusters that have received at least one coinbase transaction. Furthermore the evident difference between the first two columns in the same figure, i.e. the number of coinbase payments received by nodes (NUMCOINB) and number of nodes that receive at least a coinbase transaction (UNIQUECOINB) tells us that SCC actually contains less miners than IN even if it receives more rewards. By looking at the third column, i.e. the cumulative value obtained from coinbase transactions (BALANCECOINB) we can also notice how the SCC miners receive less value than the ones in IN despite receiving more individual rewards. This means that the miners in SCC not only are fewer but also weaker. In fact if we compute the average coinbase payment (i.e. the cumulative value of the received rewards divided by the number of rewards received) received by the nodes in each component we find out the following values: 4.13 BTC for SCC, 0.35 BTC for OUT, 37.94 BTC for IN and 20.96 BTC for DISCONNECTED nodes. This shows how each nodes receiving a coinbase reward in IN receives on average a reward nine times higher than the ones in SCC.
Final remarks We can conclude by saying that the introduced measures indeed support our hypothesis about the semantic of the bow tie components. In particular, the assumption that the SCC represents the real economically active component is supported by the previous observations on clusters lifespan, density and balance. Especially so, by our remarks on the marked discrepancy between the current value hold and cumulative value received during time by the SCC clusters. At the same time, the balance and lifespan analysis corroborate our assumptions on the OUT and IN components, while the provided coinbase transactions analysis clearly supports our hypothesis on the IN and DISCONNECTED components.
A clear advantage of building a graph from historical data contained in a blockchain is that, since the entire history of the system is available, it is possible to perform a temporal analysis of the graph structure to study the evolution of its components. This was not performed for the web in Broder et al. (2000) or Meusel et al. (2014), because of the difficulty of obtaining a sequence of temporal snapshots.
Such greater insight is always provided for free by design in current cryptocurrencies, because a node must have the possibility of recomputing and checking the entire history of transactions, starting from the very beginning, i.e. the genesis block.
To better understand the evolution of the components analysed in “The bow tie structure of the Bitcoin users graph” section we performed a temporal analysis of the connectivity of the graph. We choose to use the same timestamp cut-offs asDi Francesco Maesa et al. (2018a), i.e. to divide the timespan of our dataset in twenty temporal snapshots all equal in duration except the first one. The first snapshot was chosen to be longer (four years unlike the approximately fifty five days of all other snapshots) to allow the graph to better represent a mature users usage, avoiding the anomalies and uncertainties of bootstrapping and early adoption years.
Measures We computed the same measures presented in “Evaluation of component metrics” section for each snapshot, obtaining their temporal behaviour. In the following we only show the evolution of the two we deem more interesting: CLUSTNUM (i.e. component sizes, Fig. 5), and BALANCE (Fig. 6).
Component sizes Regarding the component sizes, we studied how nodes change their role over time. Concerning this, we remark that any older graph is a subgraph of a newer one, implying that nodes have a forced way of changing roles, ultimately favouring the SCC component. In fact nodes with role IN or OUT can only change role by becoming of role SCC, since they already reach or are reachable, respectively from the SCC, and so any new edge that would cause them to change role can only result in them being able to both reach and being reachable by the SCC thus becoming nodes of role SCC themselves. Analogously, nodes belonging to a TENDRIL can only become IN (from TENDRILFROMIN) or OUT (from TENDRILTOOUT), TUBE or SCC. Finally a node with role TUBE can change role to either IN, OUT or SCC. But once a node has role SCC it can no longer change role. Of course, new nodes can get any role.
In order to measure how nodes change their role, we analysed the temporal stability of the components. By “analysing the temporal stability of a set of nodes” we mean counting the number of nodes that remain in the set between a temporal snapshot and the next as well as counting the number of nodes that leave or join the set. We show here the results for the IN and OUT components only, which are the two most relevant components by number of nodes in the biggest connected component (excluding the SCC).
Concerning the size of the different components over time, we observe the growth in proportional size of the SCC component, mostly at the expenses of the IN component, as shown in Fig. 5a. This behaviour is more evident in Fig. 5b where the continue growth in terms of number of nodes of both DISCONNECTED and OUT components rapidly overtakes the initially larger IN component that, conversely, remains pretty much stable in size over timeFootnote 5. The same consideration is even clearer by comparing Fig. 7a and b. In fact the IN component, except for a moderate growth around the end of 2014, is pretty much temporally stable, i.e. it only varies minimally during time the set of nodes it contains. Moreover relatively few nodes leave such component during time. The OUT component instead follows an opposite evolution. Indeed, it continuously grows over time thanks to a number of new nodes that join the component higher than the non negligible number of nodes that leaves it over time.
In fact, the IN component contains a lot of old rewards that have been used to supply value to the SCC, as well as some negligible value to the TENDRILFROMIN. This is observable by the stability of such component, shown in Fig. 7a as well as by the fact that in Fig. 5a the relative size of the component keeps shrinking, which indicates that it grows slower than the other components as the graph grows. It also explains why the current balance of the component proportionally diminishes over time (since any value it holds is mostly transferred to the SCC), see Fig. 6a.
Components cumulative balance The results obtained regarding the cumulative current balance values per component support our conjecture about the different economical roles of the different components. In fact, we can observe from Fig. 6b that the current balance of the SCC remains somewhat stable, while the cumulative current balance of OUT increases over time. The same can be seen more clearly in Fig. 6a where the percentage of value held by the OUT component increases over time at the expenses of the other components (mainly the SCC). This is because more and more value actually passes and is used by the nodes in the SCC, but it is temporary (potentially for a long time in case of cold storage clusters) stored in the OUT component in the form of currently unspent outputs.
As a comparison we show in Fig. 8 the plot about the total value received over time for the four main components considered in Fig. 6b. The behaviour of SCC compared to OUT is very different between Fig. 8. In fact, considering the cumulative received value over time SCC and OUT show a similar behaviour, as both approximately double their respective starting values (do note how the value received axis is in logarithmic scale). Of course, SCC still moves value two orders of magnitude greater than OUT, despite the final cumulative balance similarity. Such behaviour is also consistent with the relatively high instability of the OUT component observable in Fig. 7b. In fact a relatively high dynamicity in the set is expected as the value stored in its nodes is spent (making them become a part of the SCC and leaving such value to circulate in the SCC) while, at the same time, new unspent outputs are created from the SCC.
Temporal node activity Having a concept of temporal evolution, it becomes possible to evaluate nodes activity in each role component over time. Unfortunately, giving a definition of active node is not trivial, since it depends on what we consider activity. The easiest approach would be to consider active only nodes performing at least one payment (i.e. creating at least one transaction) during a certain time frame. But this would consider as inactive, nodes that only receive payments. Arguably a node receiving payments might be considered an active part of an economy, so both receiving or issuing payments might be considered a sign of activity. Consequently we decided to evaluate both definitions, naming a node active during time t if the relative cluster performs at least one transaction during time t, and passively active during time t if it is active during time t or if the relative cluster is the beneficiary of at least one transaction output during time t. We do remark that the second definition is a generalization of the first, i.e. if a node is active during a given time t it is also passively active. Furthermore, we remark that node activity is measured at cluster level and not node level, so clusters with transactions between addresses inside themselves would still be considered active, even if, from a graph point of view, the corresponding arcs would result in ignored self loops. In Fig. 9 are shown the results for both definitions.
We can observe from Fig. 9, how the SCC is alone responsible for most of the activity of nodes in both graphs, in line with our interpretation of such component. We also note how the only perturbation in a pretty uniform behaviour in Fig. 9a, is due to an increased activity of the IN component at the end of 2014, consistent with what already observed in Fig. 7a. From the comparison between Fig. 9a and b, we can notice the unsurprisingly increased importance of the OUT component if incoming payments are also considered as node activity.
Finally we can formulate some considerations about the DISCONNECTED component. In fact we can see from Fig. 5b how the component size increases over time, while Fig. 6b tells us that the value it holds does not increase, it actually decreases a little over time. This happens as few nodes get connected over time to the giant connected component by spending their value, and so changing component altogether. In fact the actual value hold by the component is only the one it has received from coinbase transactions. This is observable by comparing the second column of Fig. 2 with the third column of Fig. 4: the current cumulative balance of the component matches with the amount received by coinbase transactions alone.
The increase in the size of the DISCONNECTED component is instead explainable by looking at the nodes it contains. In fact such component contains mostly singletons with no edges, representing clusters with one single address (as already observed in “Evaluation of component metrics” section). In fact 99.994% of all nodes in the DISCONNECTED component are isolated (i.e. they have no incoming or outgoing arcs) and have zero balance (i.e. they receive no value from coinbase transactions, and of course neither from regular ones since they have no incoming arcs). Moreover the DISCONNECTED component is proportionally relevant in Fig. 9b, but it is not present in Fig. 9a. This means that such isolated nodes appear in a transaction output, but they are not connected to other nodes (from nodes of components other than DISCONNECTED) and have zero balance. A possible explanation, that would satisfy all such anomalies, is that such nodes are created as recipients of special transaction outputs, carrying no value. Such special outputs are often used in the Bitcoin protocol to encapsulate arbitrary data inside the blockchain instead of transferring value (see Bartoletti and Pompianu (2017)). Since the script used in such transactions is none of the ones we decoded (see “Data acquisition” section) the resulting edge is not created, and the involved recipient address results in an isolated node.
In this paper we have presented a study of the connectivity defined components of the Bitcoin users graph following the bow tie structure presented in Broder et al. (2000). After formally defining the components and presenting the methods and dataset employed, we developed a semantic hypothesis on the economical role of each component. To verify our hypothesis, first we have defined a set of measures aimed at modelling node activity (focused on node balance, connections and lifespan). The aggregated values for all nodes part of each given component allowed us to derive macroscopic considerations. We have then performed a temporal analysis of the components evolution over time, by considering the evolution of the graph in a sequence of temporal snapshots.
Our analysis has shown that most of the economical exchanges are performed by the clusters belonging to the SCC component of the graph, while the current balance is mostly contained in the OUT component. Furthermore, we have discovered that most miners are contained in the IN component and that these miners receive higher rewards with respect to those belonging to the SCC component. These conjectures are further corroborated by the temporal analysis we performed.
We plan to extend our work in two different directions. First, we aim to better study the smallest components of the bow tie (i.e. TENDRIL and TUBE) to aid possible deanonymization attempts. In fact, despite their small size, it is worth of investigation trying to understand why the nodes involved avoid the main economical community represented by SCC. Second, it would be interesting to analyze other graph methodologies for decomposing a network, studying the (eventual) corresponding economical meaning, extracting outlier users, and comparing them with the decomposition we have used, i.e. the bow tie in this paper. Finally, we are currently performing the same analysis for the graph obtained from other cryptocurrencies (e.g. Ethereum), with the goal of comparing the economies of the two cryptocurrencies.
Availability of data and materials
The terminology used in Broder et al. (2000) is the same as the one we use in this paper.
Even if we try to mitigate such bias by pruning the periphery, as explained in “Data acquisition” section.
We choose not to show the scc in Fig. 5b because its predominant size in the complete graph would not allow to appreciate finer changes in the smaller components
- The Abbreviations introduced are explained in Definition 1 and regard the different connectivity components according to the bow tie structure of the Bitcoin users graph. In particular:
- ∙disconnected :
nodes not connected to the giant connected component of the undirected graph
- ∙scc :
nodes in the giant strongly connected component.
- ∙in :
nodes not in scc and able to reach the nodes in scc.
- ∙out :
nodes not in scc and reachable by the nodes in scc.
- ∙tendril :
nodes not in the previous categories, that either can reach at least a node in out (tendrilToOut) or can be reached by at least a node in in (tendrilFromIn), but not both.
- ∙tube :
nodes not in scc, that can reach at least a node in out, and can be reached by at least one node in in.
- ∙fringe :
nodes belonging to the graph and not in any of the previous categories.
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Maesa, D.D., Marino, A. & Ricci, L. The bow tie structure of the Bitcoin users graph. Appl Netw Sci 4, 56 (2019). https://doi.org/10.1007/s41109-019-0163-y
- Graph analysis
- Bow tie
- Complex networks