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When analyzing Bitcoin users' balance distribution, we observed that it follows a log-normal pattern. Drawing parallels from the successful application of Gibrat's law of proportional growth in explaining city size and word frequency distributions, we tested whether the same principle could account for the log-normal distribution in Bitcoin balances. However, our calculations revealed that the exponent parameters in both the drift and variance terms deviate slightly from one. This suggests that Gibrat's proportional growth rule alone does not fully explain the log-normal distribution observed in Bitcoin users' balances. During our exploration, we discovered an intriguing phenomenon: Bitcoin users tend to fall into two distinct categories based on their behavior, which we refer to as ``poor" and ``wealthy" users. Poor users, who initially purchase only a small amount of Bitcoin, tend to buy more bitcoins first and then sell out all their holdings gradually over time. The certainty of selling all their coins is higher and higher with time. In contrast, wealthy users, who acquire a large amount of Bitcoin from the start, tend to sell off their holdings over time. The speed at which they sell their bitcoins is lower and lower over time and they will hold at least a small part of their initial holdings at last. Interestingly, the wealthier the user, the larger the proportion of their balance and the higher the certainty they tend to sell. This research provided an interesting perspective to explore bitcoin users' behaviors which may apply to other finance markets.

Cyclic arbitrage chances exist abundantly among decentralized exchanges (DEXs), like Uniswap V2. For an arbitrage cycle (loop), researchers or practitioners usually choose a specific token, such as Ether as input, and optimize their input amount to get the net maximal amount of the specific token as arbitrage profit. By considering the tokens' prices from CEXs in this paper, the new arbitrage profit, called monetized arbitrage profit, will be quantified as the product of the net number of a specific token we got from the arbitrage loop and its corresponding price in CEXs. Based on this concept, we put forward three different strategies to maximize the monetized arbitrage profit for each arbitrage loop. The first strategy is called the MaxPrice strategy. Under this strategy, arbitrageurs start arbitrage only from the token with the highest CEX price. The second strategy is called the MaxMax strategy. Under this strategy, we calculate the monetized arbitrage profit for each token as input in turn in the arbitrage loop. Then, we pick up the most maximal monetized arbitrage profit among them as the monetized arbitrage profit of the MaxMax strategy. The third one is called the Convex Optimization strategy. By mapping the MaxMax strategy to a convex optimization problem, we proved that the Convex Optimization strategy could get more profit in theory than the MaxMax strategy, which is proved again in a given example. We also proved that if no arbitrage profit exists according to the MaxMax strategy, then the Convex Optimization strategy can not detect any arbitrage profit, either. However, the empirical data analysis denotes that the profitability of the Convex Optimization strategy is almost equal to that of the MaxMax strategy, and the MaxPrice strategy is not reliable in getting the maximal monetized arbitrage profit compared to the MaxMax strategy.

The measurement of the velocity of money is still a significant topic. In this paper, we proposed a method to calculate the velocity of money by combining the holding-time distribution and lifespan distribution. By derivation, the velocity of money equals the holding-time distribution's value at zero. When we have much holding-time data, this problem can be converted to a regression problem. After a numeric simulation, we find that the calculating accuracy is high even if we used only a small part of the holding time data, which implies a potential application in measuring the velocity of money in reality, such as digital money. We also tested the methods on Cardano and found that the method can also provide a reasonable estimation of velocity in some cases.

In decentralized exchanges (DEXs), the arbitrage paths exist abundantly in the form of both arbitrage loops (e.g. the arbitrage path starts from token A and back to token A again in the end, A, B,..., A) and non-loops (e.g. the arbitrage path starts from token A and stops at a different token N, A, B,..., N). The Moore-Bellman-Ford algorithm, often coupled with the ``walk to the root" technique, is commonly employed for detecting arbitrage loops in the token graph of decentralized exchanges (DEXs) such as Uniswap. However, a limitation of this algorithm is its ability to recognize only a limited number of arbitrage loops in each run. Additionally, it cannot specify the starting token of the detected arbitrage loops, further constraining its effectiveness in certain scenarios. Another limitation of this algorithm is its incapacity to detect non-loop arbitrage paths between any specified pairs of tokens. In this paper, we develop a new method to solve these problems by combining the line graph and a modified Moore-Bellman-Ford algorithm (MMBF). This method can help to find more arbitrage loops by detecting at least one arbitrage loop starting from any specified tokens in the DEXs and can detect the non-loop arbitrage paths between any pair of tokens. Then, we applied our algorithm to Uniswap V2 and found more arbitrage loops and non-loops indeed compared with applying the Moore-Bellman-Ford (MBF) combined algorithm. The found arbitrage profit by our method in some arbitrage paths can be even as high as one million dollars, far larger than that found by the MBF combined algorithm. Finally, we statistically compare the distribution of arbitrage path lengths and the arbitrage profit detected by both our method and the MBF combined algorithm, and depict how potential arbitrage opportunities change with time by our method.

This study quantifies the potential non-atomic MEV on Layer-2 (L2) blockchains by measuring the arbitrage opportunities between cross-rollup and DEX-CEX. Over recent years, we observe a shift in trading activities from Ethereum to rollups, with swaps on rollups occurring 2-3 times more frequently, albeit with lower trade volumes. By analyzing the costs of swap on L2s and price discrepancies cross-rollup and DEX-CEX, we identify more than 500 000 unexplored arbitrage opportunities. In particular, we find that these opportunities persist, on average, for 10 to 20 blocks, necessitating the modification of the Loss Versus Rebalancing (LVR) metric to prevent double-counting. Our findings indicate that the arbitrage opportunities in Arbitrum, Base, and Optimism range between 0.03% and 0.05% of the trading volume, while in the ZKsync it fluctuates around 0.25%.

In this paper we describe a prototype of a blockchain-in-a-box system which allows users to easily bootstrap the whole Ethereum Proof-of-Work (PoW) network running on multiple Raspberry Pi nodes - an inexpensive modular computers. Users are able to orchestrate the whole blockchain network using a single web based interface, for example they are able to set the topology of the peer-to-peer (P2P) connections and control the initialization parameters. Each Raspberry Pi has a screen attached which visualizes current state of local blockchain, allowing users to easily visualize the consensus of the network in real time. We show how this platform can be used to perform experiments on consensus quality while using different P2P topologies. Similar experiments can be used for demonstration purposes in a workshop or other educational settings.

Decentralized Finance (DeFi) refers to financial services that are not necessarily related to crypto-currencies. By employing blockchain for security and integrity, DeFi creates new possibilities that attract retail and institution users, including central banks. Given its novel applications and sophisticated designs, the distinction between DeFi services and understanding the risk involved is often complex. This work systematically presents the major categories of DeFi protocols that cover over 90\% of total value locked (TVL) in DeFi. It establishes a structured methodology to differentiate between DeFi protocols based on their design and architecture. Every DeFi protocol is classified into one of three groups: liquidity pools, pegged and synthetic tokens, and aggregator protocols, followed by risk analysis. In particular, we classify stablecoins, liquid staking tokens, and bridged (wrapped) assets as pegged tokens resembling similar risks. The full risk exposure of DeFi users is derived not only from the DeFi protocol design but also from how it is used and with which tokens.

ChainScience 2024, the second edition of the interdisciplinary conference, brought together academics, practitioners, and industry experts to explore novel developments in the realm of distributed ledger technologies. The conference aimed to bridge diverse fields such as informatics, business, economics, finance, regulation, law, mathematics, physics, and complexity science. The papers presented in these conference proceedings address emerging topics such as AI/ML applications to blockchain, DLTs interoperability, decentralized financial services, and tokenomics, alongside ethical, societal, and governance aspects of blockchain and DLTs. With a focus on promoting high-quality research and interdisciplinary collaboration, ChainScience24 aimed to unlock the collective potential of its diverse participants, embodying the ethos that the whole is greater than the sum of its parts.

Liquid Staking Tokens (LSTs) function as tokenized representations of staked native assets while also accruing staking rewards. They emerged as a preferred method of staking within Proof of Stake (PoS) blockchains, owing to their ease of use and tradability. In this Systematization of Knowledge (SoK), we establish a general framework describing the design choices and protocols underlying liquid staking. We then employ the framework to systematically compare the top LST implementations, examining their node operator selection, validator operations, and staking rewards distribution models. We further discuss security concerns associated with liquid staking, its implications for PoS blockchain security, and Distributed Validator technology (DVT) as a potential solution. Finally, we empirically analyze LSTs' performance and find that the design choices and market events affect peg stability; particularly, LSTs with centralized governance and operations are more efficient in tracking staking rewards.

Arbitrage can arise from the simultaneous purchase and sale of the same asset in different markets in order to profit from a difference in its price. This work systematically reviews arbitrage opportunities between Automated Market Makers (AMMs) on Ethereum ZK rollups, and Centralised Exchanges (CEXs). First, we propose a theoretical framework to measure such arbitrage opportunities and derive a formula for the related Maximal Arbitrage Value (MAV) that accounts for both price divergences and liquidity available in the trading venues. Then, we empirically measure the historical MAV available between SyncSwap, an AMM on zkSync Era, and Binance, and investigate how quickly misalignments in price are corrected against explicit and implicit market costs. Overall, the cumulative MAV from July to September 2023 on the USDC-ETH SyncSwap pool amounts to $104.96k (0.24% of trading volume)., Comment: In proceedings of Marble 2024

This paper studies liquid staking tokens (LSTs) on automated market makers (AMMs), both theoretically and empirically. LSTs are tokenized representations of staked assets on proof-of-stake blockchains. First, we model LST-liquidity on AMMs theoretically, categorizing suitable AMM types for LST liquidity and deriving formulas for the necessary returns from trading fees to adequately compensate liquidity providers under the particular price trajectories of LSTs. For the latter, two relevant metrics are considered: (1) losses compared to holding the liquidity outside the AMM (loss-versus-holding, or "impermanent loss"), and (2) the relative profitability compared to fully staking the capital (loss-versus-staking) which is specifically tailored to the case of LST-liquidity. Next, we empirically measure these metrics for Ethereum LSTs across the most relevant AMM pools. We find that, while trading fees often compensate for impermanent loss, fully staking is more profitable for many pools, raising questions about the sustainability of the current LST liquidity allocation to AMMs.

Ethereum 2.0, as the preeminent smart contract blockchain platform, guarantees the precise execution of applications without third-party intervention. At its core, this system leverages the Proof-of-Stake (PoS) consensus mechanism, which utilizes a stochastic process to select validators for block proposal and validation, consequently rewarding them for their contributions. However, the implementation of blockchain technology often diverges from its central tenet of decentralized consensus, presenting significant analytical challenges. Our study collects consensus reward data from the Ethereum Beacon chain and conducts a comprehensive analysis of reward distribution and evolution, categorizing them into attestation, proposer and sync committee rewards. To evaluate the degree of decentralization in PoS Ethereum, we apply several inequality indices, including the Shannon entropy, the Gini Index, the Nakamoto Coefficient, and the Herfindahl-Hirschman Index (HHI). Our comprehensive dataset is publicly available on Harvard Dataverse, and our analytical methodologies are accessible via GitHub, promoting open-access research. Additionally, we provide insights on utilizing our data for future investigations focused on assessing, augmenting, and refining the decentralization, security, and efficiency of blockchain systems.

Liquid staking has become the largest category of decentralized finance protocols in terms of total value locked. However, few studies exist on its implementation designs or underlying risks. The liquid staking protocols allow for earning staking rewards without the disadvantage of locking the capital at the validators. Yet, they are seen by some as a threat to the Proof-of-Stake blockchain security. This paper is the first work that classifies liquid staking implementations. It analyzes the historical performance of major liquid staking tokens in comparison to the traditional staking for the largest Proof-of-Stake blockchains. Furthermore, the research investigates the impact of centralization, maximum extractable value and the migration of Ethereum from Proof-of-Work to Proof-of-Stake on the tokens' performance. Examining the tracking error of the liquid stacking providers to the staking rewards shows that they are persistent and cannot be explained by macro-variables of the currency, such as the variance or return.

When considering a specific event, news that accurately reflects the ground truth is deemed as real information, while news that deviates from the ground truth is classified as false information. False information often spreads fast due to its novel and attention-grabbing content, which poses a threat to our society. By extending the Susceptible-Infected (SI) model, our research offers analytical decision boundaries that enable effective interventions to get desirable results, even when intermediate functions cannot be analytically solved. These analytical results may provide valuable insights for policymakers in false information control. When assessing intervention costs using the model, the results indicate that the sooner we intervene, the lower the overall intervention cost tends to be.

IOTA is a distributed ledger technology that relies on a peer-to-peer (P2P) network for communications. Recently an auto-peering algorithm was proposed to build connections among IOTA peers according to their "Mana" endowment, which is an IOTA internal reputation system. This paper's goal is to detect potential vulnerabilities and evaluate the resilience of the P2P network generated using IOTA auto-peering algorithm against eclipse attacks. In order to do so, we interpret IOTA's auto-peering algorithm as a random network formation model and employ different network metrics to identify cost-efficient partitions of the network. As a result, we present a potential strategy that an attacker can use to eclipse a significant part of the network, providing estimates of costs and potential damage caused by the attack. On the side, we provide an analysis of the properties of IOTA auto-peering network ensemble, as an interesting class of homophile random networks in between 1D lattices and regular Poisson graphs.

This paper proposes a novel multi-layer blockchain architecture for the cross-border trading of CBDCs. The permissioned layer-2, by relying on the public consensus of the underlying network, assures the security and integrity of the transactions and ensures interoperability with domestic CBDCs implementations. Multiple Layer-3s operate various Automated Market Makers (AMMs) and compete with each other for the lowest costs. To provide insights into the practical implications of the system, simulations of trading costs are conducted based on historical FX rates, with Project Mariana as a benchmark. The study shows that, even with liquidity fragmentation, a multi-layer and multi-AMM setup is more cost-efficient than a single AMM., Comment: This paper was presented at the Crypto Finance Conference (CfC) Academic Track 2024 in St. Moritz, Switzerland

An important virtue of distributed ledger technologies is their acclaimed higher level of decentralisation compared to traditional financial systems. Empirical literature, however, suggests that many systems tend towards centralisation as well. This study expands the current literature by offering a first-time, data-driven analysis of the degree of decentralisation of the platform Hedera Hashgraph, a public permissioned distributed ledger technology, employing data directly fetched from a network node. The results show a considerably higher amount of released supply compared to the release schedule and a growing number of daily active accounts. Also, Hedera Hashgraph exhibits a high centralisation of wealth and a shrinking core that acts as an intermediary in transactions for the rest of the network. However, the Nakamoto index and Theil index point to recent progress towards a more decentralised network.

The open availability of the entire history of the Bitcoin transactions opens up the possibility to study this system at an unprecedented level of detail. This contribution is devoted to the analysis of the mesoscale structural properties of the Bitcoin User Network (BUN), across its entire history (i.e. from 2009 to 2017). What emerges from our analysis is that the BUN is characterized by a core-periphery structure a deeper analysis of which reveals a certain degree of bow-tieness (i.e. the presence of a Strongly-Connected Component, an IN- and an OUT-component together with some tendrils attached to the IN-component). Interestingly, the evolution of the BUN structural organization experiences fluctuations that seem to be correlated with the presence of bubbles, i.e. periods of price surge and decline observed throughout the entire Bitcoin history: our results, thus, further confirm the interplay between structural quantities and price movements observed in previous analyses., Comment: 17 pages, 11 figures

This study explores the intricacies of waiting games, a novel dynamic that emerged with Ethereum's transition to a Proof-of-Stake (PoS)-based block proposer selection protocol. Within this PoS framework, validators acquire a distinct monopoly position during their assigned slots, given that block proposal rights are set deterministically, contrasting with Proof-of-Work (PoW) protocols. Consequently, validators have the power to delay block proposals, stepping outside the honest validator specs, optimizing potential returns through MEV payments. Nonetheless, this strategic behaviour introduces the risk of orphaning if attestors fail to observe and vote on the block timely. Our quantitative analysis of this waiting phenomenon and its associated risks reveals an opportunity for enhanced MEV extraction, exceeding standard protocol rewards, and providing sufficient incentives for validators to play the game. Notably, our findings indicate that delayed proposals do not always result in orphaning and orphaned blocks are not consistently proposed later than non-orphaned ones. To further examine consensus stability under varying network conditions, we adopt an agent-based simulation model tailored for PoS-Ethereum, illustrating that consensus disruption will not be observed unless significant delay strategies are adopted. Ultimately, this research offers valuable insights into the advent of waiting games on Ethereum, providing a comprehensive understanding of trade-offs and potential profits for validators within the blockchain ecosystem., Comment: 23 pages, 13 figures, 3 appendices

The proceedings of ChainScience 2023 epitomize the integration of various scientific disciplines with the dynamic world of blockchain and AI. This collection, encapsulating both full and short papers, as well as posters, delves into areas such as cryptoeconomics, machine learning, and analysis of blockchain networks, under the guiding principle of this year's conference: 'Applying insights and methods from the physical, mathematical, behavioral, and computational sciences to blockchain and the future of AI'. This diverse compilation offers a critical examination of theoretical constructs and innovative practical applications to empower the future of the blockchain industry., Comment: Held 15-17 June 2023 in Boston, Massachusetts, USA

This paper presents a study of the Poof-of-Stake (PoW) Ethereum consensus protocol, following the recent switch from Proof-of-Work (PoS) to Proof-of-Stake within Merge upgrade. The new protocol has resulted in reduced energy consumption and a shift in economic incentives, but it has also introduced new threat sources such as chain reorganizations and balancing attacks. Using a simple and flexible agent-based model, this study employs a time-continuous simulation algorithm to analyze the evolution of the blocktree and assess the impact of initial conditions on consensus quality. The model simulates validator node behavior and the information propagation throughout the peer-to-peer network of validators to analyze the resulting blockchain structure. Key variables in the model include the topology of the peer-to-peer network and average block and attestation latencies. Metrics to evaluate consensus quality are established, and means to observe the model's responsiveness to changes in parameters are provided. The simulations reveal a phase transition in which the system switches from a consensus state to a non-consensus state, with a theoretical justification presented for this observation., Comment: 8 pages, 4 figures

The behaviour of Bitcoin owners is reflected in the structure and the number of bitcoin transactions encoded in the Blockchain. Likewise, the behaviour of Bitcoin traders is reflected in the formation of bullish and bearish trends in the crypto market. In light of these observations, we wonder if human behaviour underlies some relationship between the Blockchain and the crypto market. To address this question, we map the Blockchain to a spin-lattice problem, whose configurations form ordered and disordered patterns, representing the behaviour of Bitcoin owners. This novel approach allows us to obtain time series suitable to detect a causal relationship between the dynamics of the Blockchain and market trends of the Bitcoin and to find that disordered patterns in the Blockchain precede Bitcoin panic selling. Our results suggest that human behaviour underlying Blockchain evolution and the crypto market brings out a fascinating connection between disorder and panic in Bitcoin dynamics., Comment: 6 pages, 3 figures

The core of many cryptocurrencies is the decentralised validation network operating on proof-of-work technology. In these systems, validation is done by so-called miners who can digitally sign blocks once they solve a computationally-hard problem. Conventional wisdom generally considers this protocol as secure and stable as miners are incentivised to follow the behaviour of the majority. However, whether some strategic mining behaviours occur in practice is still a major concern. In this paper we target this question by focusing on a security threat: a selfish mining attack in which malicious miners deviate from protocol by not immediately revealing their newly mined blocks. We propose a statistical test to analyse each miner's behaviour in five popular cryptocurrencies: Bitcoin, Litecoin, Monacoin, Ethereum and Bitcoin Cash. Our method is based on the realisation that selfish mining behaviour will cause identifiable anomalies in the statistics of miner's successive blocks discovery. Secondly, we apply heuristics-based address clustering to improve the detectability of this kind of behaviour. We find a marked presence of abnormal miners in Monacoin and Bitcoin Cash, and, to a lesser extent, in Ethereum. Finally, we extend our method to detect coordinated selfish mining attacks, finding mining cartels in Monacoin where miners might secretly share information about newly mined blocks in advance. Our analysis contributes to the research on security in cryptocurrency systems by providing the first empirical evidence that the aforementioned strategic mining behaviours do take place in practice.

Ranking the spreading influence of nodes is of great importance in practice and research. The key to ranking a node's spreading ability is to evaluate the fraction of susceptible nodes been infected by the target node during the outbreak, i.e., the outbreak size. In this paper, we present a dynamic Markov process (DMP) method by integrating the Markov chain and the spreading process to evaluate the outbreak size of the initial spreader. Following the idea of the Markov process, this method solves the problem of nonlinear coupling by adjusting the state transition matrix and evaluating the probability of the susceptible node being infected by its infected neighbours. We have employed the susceptible-infected-recovered (SIR) and susceptible-infected-susceptible (SIS) models to test this method on real-world static and temporal networks. Our results indicate that the DMP method could evaluate the nodes' outbreak sizes more accurately than previous methods for both single and multi-spreaders. Besides, it can also be employed to rank the influence of nodes accurately during the spreading process.

More than ten years ago the blockchain was acclaimed as the solution to overcome centralised trusted third parties for online payments. Through the years the crypto-movement changed and evolved, although decentralisation remained the core ideology and the necessary feature every new crypto-project should provide. In this paper we study the concept of centralisation in cryptocurrencies using a wide array of methodologies from the complex systems literature, on a comparative collection of blockchains, in order to define the many different levels a blockchain system may display (de-)centralisation and to question whether the present state of cryptocurrencies is, in a technological and economical sense, actually decentralised.

We propose a novel framework to analyse the velocity of money in terms of the contribution (MicroVelocity) of each individual agent, and to uncover the distributional determinants of aggregate velocity. Leveraging on complete publicly available transactions data stored in blockchains from four cryptocurrencies, we empirically find that MicroVelocity i) is very heterogeneously distributed and ii) strongly correlates with agents' wealth. We further document the emergence of high-velocity intermediaries, thereby challenging the idea that these systems are fully decentralised. Further, our framework and results provide policy insights for the development and analysis of digital currencies.

The Bitcoin Lightning Network (BLN) was launched in 2018 to scale up the number of transactions between Bitcoin owners. Although several contributions concerning the analysis of the BLN binary structure have recently appeared in the literature, the properties of its weighted counterpart are still largely unknown. The present contribution aims at filling this gap, by considering the Bitcoin Lightning Network over a period of 18 months, ranging from 12th January 2018 to 17th July 2019, and focusing on its weighted, undirected, daily snapshot representation. As the study of the BLN weighted structural properties reveals, it is becoming increasingly 'centralised' at different levels, just as its binary counterpart: 1) the Nakamoto coefficient shows that the percentage of nodes whose degrees/strengths 'enclose' the 51% of the total number of links/total weight is rapidly decreasing; 2) the Gini coefficient confirms that several weighted centrality measures are becoming increasingly unevenly distributed; 3)the weighted BLN topology is becoming increasingly compatible with a core-periphery structure, with the largest nodes 'by strength' constituting the core of such a network, whose size keeps shrinking as the BLN evolves. Further inspection of the resilience of the weighted BLN shows that removing such hubs leads to the network fragmentation into many components, an evidence indicating potential security threats - as the ones represented by the so called 'split attacks'., Comment: 18 pages, 12 figures

Reaching consensus -- a macroscopic state where the system constituents display the same microscopic state -- is a necessity in multiple complex socio-technical and techno-economic systems: their correct functioning ultimately depends on it. In many distributed systems -- of which blockchain-based applications are a paradigmatic example -- the process of consensus formation is crucial not only for the emergence of a leading majority but for the very functioning of the system. We build a minimalistic network model of consensus formation on blockchain systems for quantifying how central nodes -- with respect to their average distance to others -- can leverage on their position to obtain competitive advantage in the consensus process. We show that in a wide range of network topologies, the probability of forming a majority can significantly increase depending on the centrality of nodes that initiate the spreading. Further, we study the role that network topology plays on the consensus process: we show that central nodes in scale-free networks can win consensus in the network even if they broadcast states significantly later than peripheral ones., Comment: 7 pages, 6 figures

Blockchain and general purpose distributed ledgers are foundational technologies which bring significant innovation in the infrastructures and other underpinnings of our socio-economic systems. These P2P technologies are able to securely diffuse information within and across networks, without need for trustees or central authorities to enforce consensus. In this contribution, we propose a minimalistic stochastic model to understand the dynamics of blockchain-based consensus. By leveraging on random-walk theory, we model block propagation delay on different network topologies and provide a classification of blockchain systems in terms of two emergent properties. Firstly, we identify two performing regimes: a functional regime corresponding to an optimal system function; and a non-functional regime characterised by a congested or branched state of sub-optimal blockchains. Secondly, we discover a phase transition during the emergence of consensus and numerically investigate the corresponding critical point. Our results provide important insights into the consensus mechanism and sub-optimal states in decentralised systems.

Bitcoin is built on a blockchain, an immutable decentralised ledger that allows entities (users) to exchange Bitcoins in a pseudonymous manner. Bitcoins are associated with alpha-numeric addresses and are transferred via transactions. Each transaction is composed of a set of input addresses (associated with unspent outputs received from previous transactions) and a set of output addresses (to which Bitcoins are transferred). Despite Bitcoin was designed with anonymity in mind, different heuristic approaches exist to detect which addresses in a specific transaction belong to the same entity. By applying these heuristics, we build an Address Correspondence Network: in this representation, addresses are nodes are connected with edges if at least one heuristic detects them as belonging to the same entity. %addresses are nodes and edges are drawn between addresses detected as belonging to the same entity by at least one heuristic. %nodes represent addresses and edges model the likelihood that two nodes belong to the same entity %In this network, connected components represent sets of addresses controlled by the same entity. In this paper, we analyse for the first time the Address Correspondence Network and show it is characterised by a complex topology, signalled by a broad, skewed degree distribution and a power-law component size distribution. Using a large-scale dataset of addresses for which the controlling entities are known, we show that a combination of external data coupled with standard community detection algorithms can reliably identify entities. The complex nature of the Address Correspondence Network reveals that usage patterns of individual entities create statistical regularities; and that these regularities can be leveraged to more accurately identify entities and gain a deeper understanding of the Bitcoin economy as a whole., Comment: 21 pages, 13 figures