This paper studies a fundamental problem regarding the security of blockchain PoW consensus on how the existence of multiple misbehaving miners influences the profitability of selfish mining. Each selfish miner maintains a private chain and makes it public opportunistically for acquiring more rewards incommensurate to his Hash power. We first establish a general Markov chain model to characterize the state transition of public and private chains for Basic Selfish Mining (BSM), and derive the stationaryprofitable thresholdof Hash power in closed form. It reduces from 25% for a single attacker to below 21.48% for two symmetric attackers theoretically, and further reduces to around 10% with eight symmetric attackers experimentally. We next explore the profitable threshold when one of the attackers performs strategic mining based on Partially Observable Markov Decision Process (POMDP) that only half of the attributes pertinent to a mining state are observable to him. An online algorithm is presented to compute the nearly optimal policy efficiently despite the large state space and high dimensional belief space. The profitable threshold is much lower for the strategic attacker. Last, we formulate a simple model of absolute mining revenue that yields an interesting observation: selfish mining is never profitable at the first difficulty adjustment period, but relies on the reimbursement of stationary selfish mining gains in future periods. The delay till being profitable of an attacker increases with the decrease of his Hash power, making blockchain miners more cautious about performing selfish mining.

Ethereum is one of the most popular blockchain systems that support more than half a million transactions every day and foster miscellaneous decentralized applications with its Turing-complete smart contract machine. Whereas it remains mysterious what the transaction pattern of Ethereum is and how it evolves over time. In this article, we study the evolutionary behavior of Ethereum transactions from a temporal graph point of view. We first develop a data analytic platform to collect external transactions associated with users as well as internal transactions initiated by smart contracts. Three types of temporal graphs, user-to-user, contract-to-contract, and user-contract graphs, are constructed according to trading relationships and are segmented with an appropriate time window. We observe a strong correlation between the size of the user-to-user transaction graph and the average Ether price in a time window, while no evidence of such linkage is shown at the average degree, average edge weights, and average triplet closure duration. The macroscopic and microscopic burstiness of Ethereum transactions are validated. We analyze the Gini indexes of the transaction graphs and the user wealth in which Ethereum is found to be very unfair since the very beginning, in a sense, “the rich is already very rich.”