Explaining Temporal Fluctuations of Broadcast Communications between Validator Nodes in a Proof-of-stake Blockchain

Abstract

In traditional blockchains under the Proof-of-Work consensus algorithm, such as Bitcoin and the initial Ethereum, temporal fluctuations of block generation time follow an exponential distribution.Since September 15, 2022, Ethereum has changed its consensus algorithm to Proof of Stake to become Ethereum 2, which qualitatively changes the nature of the fluctuations.Careful analysis reveals that block creation time of Ethereum 2 also slightly fluctuates around 12 seconds, but the origin of this fluctuations is not well understood theoretically.This research contributes to the understanding of this fluctuations, especially focusing on Ethereum 2, by considering a mathematical model on broadcast communications between validator nodes theoretically.As a result, we find that this fluctuation directly comes from the time that it takes for the broadcast communications between validator nodes and it follows a Gumbel distribution.We also show some results of real data analysis which supports our theoretical findings.As an application of the theoretical findings, furthermore, we formulat a blockchain trilemma inequality to explain trade-offs between transaction processing performance, security, and decentralization.