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Statistical Confidence in Mining Power Estimates for PoW Blockchains

Mary Milad, Christina Ovezik, Dimitris Karakostas, Daniel W. Woods

TL;DR

The paper addresses the challenge of statistical uncertainty in decentralization estimates for PoW blockchains by formulating a binomial-hypothesis framework around the Nakamoto coefficient (NC). It models mining power as a multinomial distribution and tests the top-$C$ mining entities, yielding p-values and a range of plausible NC values at a chosen significance level $\alpha$. Through empirical analysis of five ledgers using sliding-window blocks, it shows that daily granularity often fails statistical tests, while a 7-day window substantially improves confidence and reduces overconfidence in centralization assessments. The authors provide open-source Python code to compute NC ranges, advocate reporting NC as a range rather than a single value, and highlight the practical importance for assessing 51\%-attack risk in real networks.

Abstract

The security of blockchain systems depends on the distribution of mining power across participants. If sufficient mining power is controlled by one entity, they can force their own version of events. This may allow them to double spend coins, for example. For Proof of Work (PoW) blockchains, however, the distribution of mining power cannot be read directly from the blockchain and must instead be inferred from the number of blocks mined in a specific sample window. We introduce a framework to quantify this statistical uncertainty for the Nakamoto coefficient, which is a commonly-used measure of blockchain decentralization. We show that aggregating blocks over a day can lead to considerable uncertainty, with Bitcoin failing more than half the hypothesis tests (α = 0.05) when using a daily granularity. For these reasons, we recommend that blocks are aggregated over a sample window of at least 7 days. Instead of reporting a single value, our approach produces a range of possible Nakamoto coefficient values that have statistical support at a particular significance level α.

Statistical Confidence in Mining Power Estimates for PoW Blockchains

TL;DR

The paper addresses the challenge of statistical uncertainty in decentralization estimates for PoW blockchains by formulating a binomial-hypothesis framework around the Nakamoto coefficient (NC). It models mining power as a multinomial distribution and tests the top- mining entities, yielding p-values and a range of plausible NC values at a chosen significance level . Through empirical analysis of five ledgers using sliding-window blocks, it shows that daily granularity often fails statistical tests, while a 7-day window substantially improves confidence and reduces overconfidence in centralization assessments. The authors provide open-source Python code to compute NC ranges, advocate reporting NC as a range rather than a single value, and highlight the practical importance for assessing 51\%-attack risk in real networks.

Abstract

The security of blockchain systems depends on the distribution of mining power across participants. If sufficient mining power is controlled by one entity, they can force their own version of events. This may allow them to double spend coins, for example. For Proof of Work (PoW) blockchains, however, the distribution of mining power cannot be read directly from the blockchain and must instead be inferred from the number of blocks mined in a specific sample window. We introduce a framework to quantify this statistical uncertainty for the Nakamoto coefficient, which is a commonly-used measure of blockchain decentralization. We show that aggregating blocks over a day can lead to considerable uncertainty, with Bitcoin failing more than half the hypothesis tests (α = 0.05) when using a daily granularity. For these reasons, we recommend that blocks are aggregated over a sample window of at least 7 days. Instead of reporting a single value, our approach produces a range of possible Nakamoto coefficient values that have statistical support at a particular significance level α.
Paper Structure (13 sections, 20 equations, 4 figures, 3 tables)

This paper contains 13 sections, 20 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Approach
  3. Framework
  4. Data Collection
  5. Results
  6. Related Work
  7. Discussion
  8. Conclusion
  9. Other Metrics
  10. Herfindahl–Hirschman Index (HHI)
  11. Shannon entropy
  12. Gini Coefficient
  13. Code

Figures (4)

  • Figure 1: A sliding window with a 3-day granularity.
  • Figure 2: How often a Nakamoto coefficient passes a hypothesis test at $\alpha=0.05$ with differing levels of granularity (based on our full dataset from 2018-2023).
  • Figure 3: With a smaller confidence level ($\alpha=0.01$), a smaller proportion of hypothesis tests are passed (based on our full dataset from 2018-2023).
  • Figure 4: The range of Nakamoto coefficient estimates (upper and lower) for Bitcoin with $\alpha=0.05$.