Francesco Maria De Collibus, Alberto Partida, Matija Piškorec
We analyse the transaction networks of four representative ERC-20 tokens that run on top of the public blockchain Ethereum and can be used as collateral in DeFi: Ampleforth (AMP), Basic Attention Token (BAT), Dai (DAI) and Uniswap (UNI). We use complex network analysis to characterize structural properties of their transaction networks. We compute their preferential attachment and we investigate how critical code-controlled nodes ( smart contracts , SC) executed on the blockchain are in comparison to human-owned nodes ( externally owned accounts , EOA), which are be controlled by end users with public and private keys or by off-blockchain code. Our findings contribute to characterise these new financial networks. We use three network dismantling strategies on the transaction networks to analyze the criticality of smart contract and known exchanges nodes as opposed to EOA nodes. We conclude that smart contract and known exchanges nodes play a structural role in holding up these networks, theoretically designed to be distributed but in reality tending towards centralisation around hubs. This sheds new light on the structural role that smart contracts and exchanges play in Ethereum and, more specifically, in Decentralized Finance (DeFi) networks and casts a shadow on how much decentralised these networks really are. From the information security viewpoint, our findings highlight the need to protect the availability and integrity of these hubs.
This paper is concerned with the stability of shares in a cryptocurrency where the new coins are issued according to the Proof of Stake protocol. We identify large, medium and small investors under various rewarding schemes, and show that the limiting behaviors of these investors are different -- for large investors their shares are stable, while for medium to small investors their shares may be volatile or even shrink to zero. For instance, with a geometric reward there is chaotic centralization, where all the shares will eventually concentrate on one investor in a random manner. This leads to the phase transition phenomenon, and the thresholds for stability are characterized. In response to the increasing activities in blockchain networks, we also propose and analyze a dynamical population model for the PoS protocol, which allows the number of investors to grow over the time. Numerical experiments are provided to corroborate our theory.
In this article, we present a Social Network Analysis–based approach to investigate user behaviour during a cryptocurrency speculative bubble in order to extract knowledge patterns about it. Our approach is general and can be applied to any past, present and future cryptocurrency speculative bubble. To verify its potential, we apply it to investigate the Ethereum speculative bubble happened in the years 2017 and 2018. We also describe several interesting knowledge patterns about the behaviour of specific categories of users that we obtained from this investigation. Furthermore, we describe how our approach can support the construction of an identikit of the speculators who maneuvered behind the Ethereum bubble analysed. Finally, we show that this capability of supporting the hunting for speculators is intrinsic of our approach and can cover past, present and future bubbles.
Anwar Said, Muhammad Umar Janjua, Saeed‐Ul Hassan, Zeeshan Muzammal · 8 authors
Ethereum, the second-largest cryptocurrency after Bitcoin, has attracted wide attention in the last few years and accumulated significant transaction records. However, the underlying Ethereum network structure is still relatively unexplored. Also, very few attempts have been made to perform link predictability on the Ethereum transactions network. This paper presents a Detailed Analysis of the Ethereum Network on Transaction Behavior, Community Structure, and Link Prediction (DANET) framework to investigate various valuable aspects of the Ethereum network. Specifically, we explore the change in wealth distribution and accumulation on Ethereum Featured Transactional Network (EFTN) and further study its community structure. We further hunt for a suitable link predictability model on EFTN by employing state-of-the-art Variational Graph Auto-Encoders. The link prediction experimental results demonstrate the superiority of outstanding prediction accuracy on Ethereum networks. Moreover, the statistic usages of the Ethereum network are visualized and summarized through the experiments allowing us to formulate conjectures on the current use of this technology and future development.
We show that knowledge of wallet addresses from the current time state of a blockchain network, such as Bitcoin, increases the performance of illicit activity detection. Based on this finding we introduce two new methods for the sampling of classifier training data so that precedence is given to transaction information from the recent past and the current time state. This sampling enables streaming classification in which a decision on the class of a transaction needs to be made based on data seen to date. Our new approach provides insight into how the dynamics of the blockchain network plays a central role in the detection of illicit transactions, and is independent of the classifier choice. Our proposed sampling methods enable graph convolution network (GCN) and random forest (RF) classifiers to better adapt to changes in the network due to significant events, such as the closure of a large ‘Darknet’ marketplace. We introduce Graphlet spectral correlation analysis for exposing the effect of such network re-organisation due to major events. Finally, based on our analysis, we propose a new two-stage random forest classifier that feeds back intermediate predictions of neighbours to improve the classification decision. Our methodology enables practical streaming classification, even in the scenario of very limited information on the feature space of each transaction.
Deepesh Chaudhari, Rachit Agarwal, Sandeep K. Shukla
The temporal aspect of blockchain transactions enables us to study the address's behavior and detect if it is involved in any illicit activity. However, due to the concept of change addresses (used to thwart replay attacks), temporal aspects are not directly applicable in the Bitcoin blockchain. Several pre-processing steps should be performed before such temporal aspects are utilized. We are motivated to study the Bitcoin transaction network and use the temporal features such as burst, attractiveness, and inter-event time along with several graph-based properties such as the degree of node and clustering coefficient to validate the applicability of already existing approaches known for other cryptocurrency blockchains on the Bitcoin blockchain. We generate the temporal and non-temporal feature set and train the Machine Learning (ML) algorithm over different temporal granularities to validate the state-of-the-art methods. We study the behavior of the addresses over different time granularities of the dataset. We identify that after applying change-address clustering, in Bitcoin, existing temporal features can be extracted and ML approaches can be applied. A comparative analysis of results show that the behavior of addresses in Ethereum and Bitcoin is similar with respect to in-degree, out-degree and inter-event time. Further, we identify 3 suspects that showed malicious behavior across different temporal granularities. These suspects are not marked as malicious in Bitcoin.
Jianhong Lin, Emiliano Marchese, Claudio J. Tessone, Tiziano Squartini
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 - each weight representing the total capacity of the channels the two involved nodes have established on a given temporal snapshot. As the study of the BLN weighted structural properties reveals, it is becoming increasingly ‘centralized’ 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’.
Surfing Online Social Media (OSM) websites have become a daily activity for a large number of people worldwide. People use OSMs to satisfy their innate need to socialise, but also as a source of information or to share personal facts. Thanks to the massive success of cryptocurrencies, the blockchain technology gained popularity among researchers, giving birth to a new generation of social media. Steemit is the most well-known blockchain-based social media, and it is based on the public blockchain Steem. Steemit employs Steem as data storage, and to implement a rewarding mechanism that grants cryptocurrency to pieces of content that are considered relevant by the users. Steem represents the first experiment that integrates OSMs and an economic rewarding system on the same platform, and in this paper, we inspect the interactions among the users from a community perspective. We apply two community detection algorithms on five graphs that model just as many facets of the Steem blockchain and test the detected structure against three measures for community structure evaluation. Findings show that communities tend to be very large, index of how much users are encouraged to interact as much as possible, and in particular, in the monetary graph, we detect a large number of the block producers of Steem.
Reputation plays a fundamental role in the blockchain ecosystem, as it provides means to assign trust levels to participating nodes. Reputation is usually gained in time: the higher the fair participation of a node to the network, the higher its reputation. IOTA’s Coordicide proposes a reputation system based on an asset named Mana. This reputation system is the base of the Coordicide’s primary security protection. At present, IOTA assigns to nodes an amount of Mana equivalent of the amount of transferred funds only. However, this may lead to privacy leakage on nodes’ status and may not be sufficient to fully characterize the behavior of a node in the network. Furthermore, the Mana reputation system is currently susceptible to monopoly, eclipse, and spamming attacks. An efficient and secure reputation system shall comprehend a larger number of factors to provide security and reliability.In this paper, we propose Enhanced Mana (eMana), a theoretical model for the IOTA reputation system accounting for nodes’ good or bad behaviour based on their assigned tasks. We model each of the Coordicide modules having direct or indirect connections with the reputation system. In eMana the behavior of a peer node can be transparently analyzed by the other peers; thus, it is infeasible to modify the reputation value. Theoretically, eMana is flexible enough to include possible new modules’ behaviour and provides a strict hard-to-gain reputation. To the best of our knowledge, eMana is the first mathematical model for Coordicide’s enhanced reputation based on nodes’ multi-dimensional behaviour. Additionally, eMana is generic and flexible, and may also be adaptable to other Distributed Ledger Technology (DLT) systems.
Bishenghui Tao, Hong‐Ning Dai, Jiajing Wu, Ivan Wang‐Hei Ho · 6 authors
In this brief, we conduct a complex-network analysis of the Bitcoin transaction network. In particular, we design a new sampling method, namely random walk with flying-back (RWFB), to conduct effective data sampling. We then conduct a comprehensive analysis of the Bitcoin network in terms of the degree distribution, clustering coefficient, the shortest-path length, connected component, centrality, assortativity, and the rich-club coefficient. We obtain several important observations including the small-world phenomenon, multi-center status, preferential attachment, and non-rich-club effect of the current network. This work brings up an in-depth understanding of the current Bitcoin blockchain network and offers implications for future directions in malicious activity and fraud detection in cryptocurrency blockchain networks.
Ana Todorovska, Eva Spirovska, Gorast Angelovski, Hristijan Peshov · 9 authors
In a world where no country, market, or economy is an island, interconnectivity is becoming a fundamental feature of almost all social and economic systems. In the case of digital assets like cryptocurrencies, the impact of interconnectivity on their performance and price trajectory is amplified. Studying these phenomena is essential for understanding the processes driving the crypto-markets. In this paper, we propose seven different approaches to create a network of eighteen most important cryptocurrencies. The first three approaches discover correlations between cryptocurrencies based on their daily prices, daily returns, and sentiment extracted from Reddit data. The following two approaches offer insights from the frequency of joint appearance of cryptocurrencies in Google news and Reddit data. The remaining two approaches determine each cryptocurrency’s impact over the others when forecasting prices and returns. Furthermore, we explore the networks’ interdependencies to explore the similarities of the cryptocurrency networks generated by different approaches. The proposed methodology allows us to understand the dynamics in the cryptocurrency markets and the different processes that influence their performance.
Summary In the Bitcoin system, transactions and their collections (i.e., blocks) are distributed over a peer‐to‐peer (P2P) network (i.e., Bitcoin network) constructed by participating nodes. Each node maintains a distributed ledger (i.e., blockchain) consisting of retrieved blocks. Therefore, speedy block distribution over the Bitcoin network is essential for all nodes to reach a global consensus on the blockchain. On the other hand, Bitcoin clients are developed as open source software, and thus they can be modified by malicious users. Existing work has pointed out that an attacker can delay the block propagation between neighboring nodes by exploiting the regular timeout mechanism for unexpected slow block transfer caused by temporal network trouble. In this paper, we focus on block diffusion delay attacks, where multiple attackers collude with a specific miner (i.e., a special node that creates new blocks and broadcasts these blocks to the other miners) to disturb the propagation of blocks generated by competing miners. Through simulation experiments, we first reveal that about 30% of honest nodes cannot normally retrieve a block when there are only 1% of the nodes in the system are high‐degree adversary nodes in the system. This indicates that the malicious miner colluding with the attackers can intentionally delay the diffusion of the block mined by the competing miner, so as to win the competitive block diffusion even if it loses at the competitive block mining. To alleviate the block diffusion delay attack, we propose two kinds of countermeasures: a proactive approach that is a speedy recovery method from the interruption by adjusting the timeout value and a reactive approach that is a block retrieval node selection method based on the past download rate from each neighbor. Through simulation experiments, we show the countermeasures can effectively alleviate the risk.
Stefan Kitzler, Friedhelm Victor, Pietro Saggese, Bernhard Haslhofer
We present a measurement study on compositions of Decentralized Finance (DeFi) protocols, which aim to disrupt traditional finance and offer services on top of distributed ledgers, such as Ethereum. Understanding DeFi compositions is of great importance, as they may impact the development of ecosystem interoperability, are increasingly integrated with web technologies, and may introduce risks through complexity. Starting from a dataset of 23 labeled DeFi protocols and 10,663,881 associated Ethereum accounts, we study the interactions of protocols and associated smart contracts. From a network perspective, we find that decentralized exchange (DEX) and lending protocol account nodes have high degree and centrality values, that interactions among protocol nodes primarily occur in a strongly connected component, and that known community detection methods cannot disentangle DeFi protocols. Therefore, we propose an algorithm to decompose a protocol call into a nested set of building blocks that may be part of other DeFi protocols. This allows us to untangle and study protocol compositions. With a ground truth dataset that we have collected, we can demonstrate the algorithm’s capability by finding that swaps are the most frequently used building blocks. As building blocks can be nested, that is, contained in each other, we provide visualizations of composition trees for deeper inspections. We also present a broad picture of DeFi compositions by extracting and flattening the entire nested building block structure across multiple DeFi protocols. Finally, to demonstrate the practicality of our approach, we present a case study that is inspired by the recent collapse of the UST stablecoin in the Terra ecosystem. Under the hypothetical assumption that the stablecoin USD Tether would experience a similar fate, we study which building blocks — and, thereby, DeFi protocols — would be affected. Overall, our results and methods contribute to a better understanding of a new family of financial products.
Francesco Maria De Collibus, Alberto Partida, Matija Piškorec, Claudio J. Tessone
In this study, we analyse the aggregated transaction networks of Ether (the native cryptocurrency in Ethereum) and the three most market-capitalised ERC-20 tokens in this platform at the time of writing: Binance, USDT, and Chainlink. We analyse a comprehensive dataset from 2015 to 2020 (encompassing 87,780,546 nodes and 856,207,725 transactions) to understand the mechanism that drives their growth. In a seminal analysis, Kondor et al. (PLoS ONE, 2014, 9: e86197) showed that during its first year, the aggregated Bitcoin transaction network grew following linear preferential attachment. For the Ethereum-based cryptoassets, we find that they present in general super-linear preferential attachment, i.e., the probability for a node to receive a new incoming link is proportional to k α , where k is the node’s degree. Specifically, we find an exponent α = 1.2 for Binance and Chainlink, for Ether α = 1.1, and for USDT α = 1.05. These results reveal that few nodes become hubs rapidly. We then analyse wealth and degree correlation between tokens since many nodes are active simultaneously in different networks. We conclude that, similarly to what happens in Bitcoin, “the rich indeed get richer” in Ethereum and related tokens as well, with wealth much more concentrated than in-degree and out-degree.
Cryptocurrency, without exaggeration, can be called one of the most popular and demanded digital assets. The use of electronic means guarantees anonymity for users. Blockchain technology allows you to trace transactions, but does not provide an opportunity to determine the owner of the wallet. However, the issue of exchanging digital assets for traditional money remains open for many users.\n\nWhat do you need to work with digital currency?\nTo buy cryptocurrency and start working with digital assets, you need a virtual wallet and a Visa or MasterCard bank card. There are the following types of wallets:\n\nLocal. Also called desktop. A local wallet is a program that is installed on a computer.\n\n\nMobile. Application for installation on a mobile device. Programs should be downloaded from the official stores AppStore and Play Market.\n\n\nOnline wallet. It is available from any device, but it is not the safest option, so it is not recommended to use it for storing large amounts. An online wallet is perfect for those who are just getting started with digital assets.\n\n\nHardware. It is a USB device. Suitable for storing large amounts.\n\n\nWhen choosing a wallet, you need to focus on your needs. The most convenient way to pay is from a mobile phone. If you plan to make frequent financial transactions, it is better to use an online wallet. The registration procedure will differ depending on the platform you choose.\n\nOnline exchangers\nExchange services offer the fastest and easiest way to exchange cryptocurrencies. Such sites allow you to withdraw money by various methods, including to a bank card. As a rule, such services operate according to a similar algorithm.\n\nTo exchange funds, you must:\n\n - choose the currency of interest;\n - indicate the method of receiving funds;\n - enter the required information;\n - pass verification.\nThen the exchange service will send the number of the virtual account to which the transaction is made in the specified amount. After the money is transferred, you need to confirm the action. As a rule, the exchange takes no more than 15 minutes, but there are times when the application is processed for more than 2 hours. This is usually due to DDoS attacks on the blockchain platform or digital asset renewal. In this case, it is better to contact the technical support of the service. If the operation does not go through because of your mistake, most likely, the money will be lost.\n\nDepending on the service, the commission can range from 1-2% to 10% and higher. It is necessary to choose a site taking into account the needs and parameters of interest. For this, you can use special aggregators. This will allow you to quickly find the option you are interested in. It is better to choose from services that have been operating for at least three years.\n\nCompany Alligator offers to exchange Bitcoins and other cryptocurrencies on favorable terms. The service has been offering its services since 2015 and is one of the most reliable in Russia, Ukraine and the CIS countries. Millions of users work with Alligator every month. The latest software and servers ensure maximum security for user assets. All user data is securely protected.\n\nThe service offers favorable conditions, a good rate and a small commission. Thanks to qualified technical support, any issues are resolved before they become problems. Transactions usually take several minutes to complete. All transactions are performed quickly, and in total, the exchanger supports more than 150 digital currencies.\n\nP2P sites and their features\nYou can withdraw funds from a cryptocurrency wallet to an electronic or bank account through the P2P platform. There are a lot of such systems, so you can choose the option that will meet your exact requirements. On such platforms, digital assets are exchanged between two users. The P2P platform acts as a guarantor. The commission is set by the service, and in some cases it may be absent altogether.\nThe P2P platform acts as a guarantor of the transaction. The seller is guaranteed to receive money before his digital currency reaches the stranger's account. However, there is one point here: there is no guarantee that the user who buys your digital currency will not use a stolen bank card to transfer funds. To minimize risks, it is recommended to conclude deals with users who have a good reputation. On P2P sites, as a rule, there is a user rating, which determines the percentage of success.\n\nHow to exchange money on the exchange\nMany trading platforms provide for the possibility of withdrawing money to a bank card. This is a convenient and safe way. But many exchanges charge a high commission for direct withdrawals, as trading platforms usually cooperate with counterparties. When transferring funds, the platform first transfers the currency to a third-party service. The exchange partner transfers money to the client, after which the operation can be considered completed.\n\nIn this case, the costs are quite high, but this option is the most transparent and popular. It should be borne in mind that not all exchanges are equally safe. The vulnerability of such services is one of the main problems. Trading platforms attract the attention of hackers, and in the event of a hacked wallet, the probability of a refund is minimal.\n\nIf you decide to make money on digital assets, we recommend that you study several cryptocurrency reviews in order to understand which one is the most promising and which one should work with. It is most profitable to make money on the course races, but for this you need to regularly monitor any fluctuations.
Simone Casale-Brunet, Paolo Ribeca, Patrick Charles Doyle, Marco Mattavelli
Non-fungible tokens (NFTs) as a decentralized proof of ownership represent one of the main reasons why Ethereum is a disruptive technology. This paper presents the first systematic study of the interactions occurring in a number of NFT ecosystems. We illustrate how to retrieve transaction data available on the blockchain and structure it as a graph-based model. Thanks to this methodology, we are able to study for the first time the topological structure of NFT networks and show that their properties (degree distribution and others) are similar to those of interaction graphs in social networks. Time-dependent analysis metrics, useful to characterize market influencers and interactions between different wallets, are also introduced. Based on those, we identify across a number of NFT networks the widespread presence of both investors accumulating NFTs and individuals who make large profits.
With the aim of bitcoin mining recognition, a method based on community detection is built with small sample of Sichuan daily bitcoin mining electricity consumption data. The method precisely categorizes Sichuan bitcoin mining entities according to their electricity consumption patterns. It is found that Sichuan bitcoin mining activity is strongly featured with seasonality and highly correlated with local hydropower generation. Such a discovery is a key not only to explaining the characteristics of the electricity consumption curves, but also to extending the method summarized from Sichuan case to other provinces. In practice, from Xinjiang power-intensive consumers, suspicious bitcoin miners are detected with the method and their mining activity is found to be correlated with local wind power generation. It is inferred that difficulty in green power utilization during the abundant period results in low electricity price which attracts bitcoin mining activity. By comparing the electricity consumption curves of Sichuan and Xinjiang (suspicious) bitcoin mining entities, a character of peak-valley complementation in temporal distribution implies a nomad electricity consumption of the miners, which could be influential to green power utilization as well as power transmission and distribution.
Lijun Wang, Pinghai Xu, Wei Su, Yang Li · 5 authors
Aiming at the problem of low voting enthusiasm of nodes in the Delegated Proof of Stake (DPOS) consensus mechanism in the blockchain, an incentive mechanism based on a clustering algorithm is proposed. By establishing a node positivity model, the nodes are classified by a clustering algorithm, and rewards are distributed to different nodes according to the classification results, so as to improve the enthusiasm of nodes in voting. The traditional K-means clustering algorithm has problems such as the need to predict the number of clusters K and the inability to deal with outliers, so it is not suitable for the incentive mechanism of node classification. Aiming at the problem of K-means clustering algorithm, an Hegselmann-Krause (HK) opinion dynamics clustering algorithm is introduced, which can meet the needs of the reward distribution mechanism in this paper. Through simulation and comparison with the existing schemes, the effectiveness of the improved algorithm is verified.
Social bots can cause social, political, and economical disruptions by spreading rumours. The state-of-the-art methods to prevent social bots from spreading rumours are centralised and such solutions may not be accepted by users who may not trust a centralised solution being biased. In this paper, we developed a decentralised method to prevent social bots. In this solution, the users of a social network create a secure and privacy-preserving decentralised social network and may accept social media content if it is sent by its neighbour in the decentralised social network. As users only choose their trustworthy neighbours from the social network to be part of its neighbourhood in the decentralised social network, it prevents the social bots to influence a user to accept and share a rumour. We prove that the proposed solution can significantly reduce the number of users who are share rumour.
Kevin E. Martin, Mohamed Rahouti, Moussa Ayyash, Izzat Alsmadi
Abstract The vast majority of digital currency transactions rely on a blockchain framework to ensure quick and accurate execution. As such, understanding how a blockchain works is vital to understanding the dynamics of cryptocurrency operations. One of the key benefits of this type of system is the exhaustive records captured in a given marketplace. The interwoven movement between agents can effectively be expressed as a graph via the extraction of historical data from the blockchain. By looking at a specific blockchain as an interaction of its agents, network representation learning can be leveraged to examine these relationships. Furthermore, the analysis of a graph structure can be enhanced through the application of modern and sophisticated machine learning techniques. Leveraging the automated nature of these methods can create meaningful observations of the input network. In this paper, we utilize several machine learning models to detect anomalous transactions in various digital currency markets. We find that supervised learning techniques yield encouraging results, whereas unsupervised learning techniques struggle more with the classification.
Ningyu He, Weihang Su, Zhou Yu, Xinyu Liu · 10 authors
The continuing expansion of the blockchain ecosystems has attracted much attention from the research community. However, although a large number of research studies have been proposed to understand the diverse characteristics of individual blockchain systems (e.g., Bitcoin or Ethereum), little is known at a comprehensive level on the evolution of blockchain ecosystems at scale, longitudinally, and across multiple blockchains. We argue that understanding the dynamics of blockchain ecosystems could provide unique insights that cannot be achieved through studying a single static snapshot or a single blockchain network alone. Based on billions of transaction records collected from three representative and popular blockchain systems (Bitcoin, Ethereum and EOSIO) over 10 years, we conduct the first study on the evolution of multiple blockchain ecosystems from different perspectives. Our exploration suggests that, although the overall blockchain ecosystem shows promising growth over the last decade, a number of worrying outliers exist that have disrupted its evolution.
In the Bitcoin mining network, miners contribute computation power to solve crypto-puzzles in exchange for financial rewards. Due to the randomness and the competitiveness of mining, individual miners tend to join mining pools for low risks and steady incomes. Usually, a pool is managed by its central operator, who charges fees for providing risk-sharing services. This paper presents a hierarchical distributed computation paradigm where miners can distribute their power among multiple pools. By adding virtual pools, we separate miners’ dual roles of being the operator as well as being the member when solo mining. We formulate a multi-leader multi-follower Stackelberg game to study the joint utility maximization of pool operators and miners, thereby addressing a computation power allocation problem. We investigate two practical pool operation modes, a uniform-share-difficulty mode and a nonuniform-share-difficulty mode. We derive analytical results for the Stackelberg equilibrium of the game under both modes, based on which optimal strategies are designed for all operators and miners. Numerical evaluations are presented to verify the proposed model.