[Paper Review] Block arrivals in the Bitcoin blockchain

Rhys Bowden, Paul Keeler+2 more|arXiv (Cornell University)|Jan 23, 2018

Blockchain Technology Applications and Security5 references57 citations

TL;DR

The paper shows that Bitcoin block arrivals are not a homogeneous Poisson process and develops refined point process models that incorporate difficulty adjustments and hash-rate dynamics to describe block generation over multiple timescales.

ABSTRACT

Bitcoin is a electronic payment system where payment transactions are verified and stored in a data structure called the blockchain. Bitcoin miners work individually to solve a computationally intensive problem, and with each solution a Bitcoin block is generated, resulting in a new arrival to the blockchain. The difficulty of the computational problem is updated every 2,016 blocks in order to control the rate at which blocks are generated. In the original Bitcoin paper, it was suggested that the blockchain arrivals occur according to a homogeneous Poisson process. Based on blockchain block arrival data and stochastic analysis of the block arrival process, we demonstrate that this is not the case. We present a refined mathematical model for block arrivals, focusing on both the block arrivals during a period of constant difficulty and how the difficulty level evolves over time.

Motivation & Objective

• Motivate and test the assumption that Bitcoin block arrivals follow a homogeneous Poisson process.
• Develop refined point process models for block arrivals that account for constant-difficulty segments and dynamic hash rate.
• Estimate the global hash rate from available blockchain data and study its time variation.
• Incorporate difficulty adjustment and block propagation effects into the arrival process analysis.
• Provide data-cleaning methods and practical estimators for analyzing the blockchain over long horizons.

Proposed method

• Review blockchain data and acknowledge limitations of timestamps and arrival definitions.
• Estimate hash rate H(t) from difficulty via equations D_{i+1}=1209600 D_i / (X_{2016i}-X_{2016(i-1)}) and Ĥ_i = (2^32 D_i)/(X_{2016i}-X_{2016(i-1)}).
• Propose a sliding-window hash-rate estimator H^W_{k,i} with window length k blocks.
• Propose a kernel-based estimator H^K_h(t) using a weighted sum of D_{⌈i/2016⌉} with bandwidth h.
• Model H(t) as piecewise exponential H(t) ≈ e^{a t + b} over intervals where log(H) is approximately linear, and derive a relation a = (1/T_{2016}) log(1209600/ T_{2016}).
• Discuss data cleaning to handle timestamp errors and out-of-order blocks, including a resampling scheme based on longest increasing subsequences.

Experimental results

Research questions

• RQ1 Do Bitcoin block arrivals behave as a homogeneous Poisson process across timescales?
• RQ2 How does the difficulty adjustment mechanism and varying hash rate affect the block arrival process?
• RQ3 How can we reliably estimate the global hash rate from observed block data and difficulty?
• RQ4 What is the impact of block propagation delays on modeling the blockchain arrival process?
• RQ5 Can we develop tractable point-process models that respect both constant-difficulty segments and dynamic hash-rate evolution?

Key findings

• Block arrivals are not well described by a homogeneous Poisson process at certain timescales.
• A suite of refined point-process models is proposed to capture block arrivals over multiple timescales, incorporating difficulty adjustments and hash-rate dynamics.
• The global hash rate can be estimated from difficulty and inter-arrival times using Ĥ_i = (2^32 D_i)/(X_{2016i}-X_{2016(i-1)}), and sliding-window and kernel-based estimators provide smoother views.
• Hash rate grows roughly exponentially on several intervals, enabling H(t) ≈ e^{a t + b} within sections, with a related to the segment-time T_{2016} by a = (1/T_{2016}) log(1209600/ T_{2016}).
• There is empirical support that the long-term average inter-arrival time and the slope a align with the theoretical relationship, as demonstrated by simulations and observed data.