Tree Representation, Growth Rate of Blockchain and Reward Allocation in Ethereum with Multiple Mining Pools

It is interesting but difficult and challenging to study Ethereum with multiple mining pools. One of the main difficulties comes from not only how to represent such a general tree with multiple block branches (or sub-chains) related to the multiple mining pools, but also how to analyze a multi-dimensional stochastic system due to the mining competition among the multiple mining pools. In this paper, we first set up a mathematical representation for the tree with multiple block branches. Then we provide a block classification of Ethereum: Regular blocks (in the main chain), orphan blocks, uncle blocks, stale blocks, and nephew blocks, and give some key probabilities of generating the different types of blocks by applying the law of large numbers. Based on this, we further discuss the growth rate of blockchain, and the reward allocation among the multiple mining pools through applying the renewal reward theorem. Finally, we use some simulation experiments to verify our theoretical results, and show that the approximate computation approaches developed, such as the key probabilities, the long-term growth rate of blockchain, and the long-term reward allocation (rate) among the multiple mining pools, can have a faster convergence. Therefore, we provide a powerful tool for observing and understanding the influence of the selfish mining attacks on the performance of Ethereum with multiple mining pools. We believe that the methodology and results developed in this paper will shed light on the study of Ethereum with multiple mining pools, such that a series of promising research can be inspired potentially.