How the interplay between power concentration, competition, and propagation affects the resource efficiency of distributed ledgers

Carlo Campajola; Jiahua Xu; Paolo Barucca

REVIEW 3 major objections 5 minor 22 references

How the interplay between power concentration, competition, and propagation affects the resource efficiency of distributed ledgers

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Bitcoin's natural fork rate reduces, to first order, to one formula in propagation time, total hash power, and mining concentration.

desk verdict A clean analytical fork-rate formula with a genuinely new HHI factorization, softened by an acknowledged synchronized-start assumption and a somewhat post-hoc empirical validation. read the letter →

arxiv 2411.10249 v1 pith:KFYWTPL7 submitted 2024-11-15 cs.DC cs.SIphysics.soc-ph

Paolo Barucca , Carlo Campajola , Jiahua Xu This is my paper · ORCID

classification cs.DCcs.SIphysics.soc-ph MSC 60G7062G30

keywords Bitcoinproof-of-worksoftforksblockpropagationtimehashrateconcentrationHerfindahl-Hirschmanindexorderstatisticsminingpools

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper sets out to explain how often a Proof-of-Work blockchain like Bitcoin naturally produces competing blocks — soft forks — and to do it analytically rather than by simulation. Its central claim is that the fork probability is, to first order, $C(\Delta_0) \approx \Delta_0 \Lambda (1 - \mathrm{HHI})$: the block propagation time $\Delta_0$ times the total hash rate $\Lambda$, scaled down by a factor that grows with how concentrated mining power is, measured by the Herfindahl-Hirschman index. Plugging in the time a block needs to reach half of the network, the model reproduces the empirically observed stale-block rate over a decade of Bitcoin data. If the paper is right, fork frequency — and the wasted energy it represents — can be forecast and partly controlled through propagation speed and the distribution of hash rates, and the efficiency-versus-concentration trade-off becomes explicit: fewer natural forks come at the cost of greater mining centralization.

What carries the argument

The load-bearing object is the joint distribution of the first two order statistics of $N$ independent exponential mining clocks. The paper computes the density of the gap $\Delta$ between the two fastest miners (Eq. 5); evaluating that density at $\Delta = 0$ gives $\Lambda(1-\mathrm{HHI})$, and the first-order Taylor expansion of the gap's cumulative distribution then yields $C(\Delta_0) \approx \Delta_0 \Lambda (1 - \mathrm{HHI})$ (Eq. 8). The Herfindahl-Hirschman index $\mathrm{HHI} = \sum_i (\lambda_i/\Lambda)^2$ is the single summary of hash-rate heterogeneity that enters the formula, and the empirical pipeline that feeds it — estimating $\lambda_i$ from mined-block counts, fitting exponential, log-normal, and truncated power-law null distributions by the method of moments, plus a semi-empirical Bayesian posterior for the hash rates — is what allows the comparison with observed fork rates.

What would settle it

Measure, over many blocks on the live network, the actual times at which each mining pool begins hashing the new tip (for example from stratum traffic or pool-published timestamps), and compare the empirical distribution of the gap between the two fastest solutions with the prediction of Eq. (5) built from the same $\{\lambda_i\}$; if the measured gap distribution is systematically wider when propagation is slow, the synchronized-start premise fails and the first-order fork formula with it. A cheaper, already available check: in periods where the paper's Fig. 6b yields negative implied HHI, the inversion of Eq. (8) is meaningless, and those episodes are the places a staggered-start correction would have to bite.

Watch

Extended reading notes

Core claim

The paper derives the probability that two miners solve a block within a propagation-time window. Writing each miner's mining time as an independent exponential clock with rate $\lambda_i$, the gap between the first two successful miners has a distribution whose density at zero equals $\Lambda(1-\mathrm{HHI})$, so for small propagation delays the fork rate is $C(\Delta_0) \approx \Delta_0 \Lambda (1 - \mathrm{HHI})$. The authors validate this against Bitcoin data from 2015 to 2024: with $\Delta_0$ taken as the time to reach 50% of the network, predicted fork rates are comparable with the measured stale-block rate. They further show that empirical hash rates over the decade follow a truncated power law, consistent with a rich-get-richer dynamic capped by total energy supply, that the number of active mining pools fell from about 100 to about 35 while concentration rose, and that these two trends partly offset each other, leaving falling propagation time as the dominant driver of the decline in fork rates. On the security side, the model's inversion — implied $\Delta_0$ and implied HHI — gives a diagnostic: periods where the implied values contradict the measured ones indicate miners with better-than-median connectivity, intra-pool coordination, or strategic behaviour such as selfish mining.

Load-bearing premise

The model hinges on every miner beginning to mine the next block at the same instant, so the first two solutions behave like the first two order statistics of independent exponential clocks; staggered starts in the real network would change the gap distribution and could break the clean $\Delta_0 \Lambda (1-\mathrm{HHI})$ dependence — an assumption the authors explicitly flag as 'relatively strong' in the Discussion.

Editorial extensions

If this is right

Fork rates in any Proof-of-Work ledger can be estimated directly from three measurable quantities — propagation time, total hash rate, and the HHI of the mining distribution — without simulating the consensus protocol.
The observed drop in Bitcoin's fork rate from roughly 1% in 2015 to about 0.1% today is attributable mainly to faster block propagation, because the opposing effects of fewer miners and greater concentration have largely cancelled out.
Higher hash-rate concentration suppresses natural forks, so a ledger designer faces an explicit trade-off: accept more waste and orphaned blocks for a more decentralized mining population, or accept centralization risk for smoother consensus.
Because the formula can be inverted, deviations between measured and implied fork rates become detection signals for non-competitive behaviour — selfish mining, block withholding, or a network core in which miners learn of blocks faster than the median node.
For faster blockchains where the ratio of propagation time to expected mining time exceeds about 0.4, the linear approximation degrades and the full order-statistics expression — including the hash-rate distribution's shape — must be used.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The identity $p(0) = \Lambda(1-\mathrm{HHI})$ is a statement about any race between exponential clocks, not only blockchains: wherever the gap between the first two finishers determines wasted work or a tie-break, such as sharded block production, DAG ordering, or replicated commit protocols, the same concentration discount should appear.
Because Eq. (8) needs only the fork rate, $\Delta_0$, and $\Lambda$, it could serve as a passive monitor of effective mining concentration in ledgers that do not publish pool attribution — a use the paper gestures at but does not develop.
A direct way to probe the model's limits is to relax the synchronized-start assumption in simulation by drawing each miner's start time from the observed propagation kernel; the deviation from $\Delta_0\Lambda(1-\mathrm{HHI})$ should grow with the spread of start times, and the episodes in the paper's Fig. 6b where implied HHI turns negative are a natural place to look for exactly that failure.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper is worth a serious look for one result: a closed-form approximation for natural fork rates in PoW ledgers with heterogeneous miners, C ≈ Δ0Λ(1−HHI). That is a real extension beyond the homogeneous-miner models of Decker-Wattenhofer and Fadda et al., and it gives a simple, interpretable diagnostic for anomaly detection. The derivation from exponential mining times to Eqs. (6)–(8) is clean, and the Monte Carlo in Fig. 10 matches the analytics across the tested distributions. The empirical work also shows genuine care: long time series, multiple distribution fits, and a sensible discussion of implied versus measured propagation times.

The soft spots are in the empirical validation, not the math. The fork-rate series is rescaled by a factor of 1.476 to pin one point to Gervais et al., and the headline comparison uses the 50% propagation threshold, selected after seeing the agreement. So Fig. 4b is suggestive, not independent confirmation. The stress-test concern about synchronized miner starts is legitimate: the HHI factorization in Eq. (8) inherits the common-start, memoryless exponential setup, and realistic propagation delays desynchronize starts. The authors acknowledge this in the Discussion, but they do not show how much the formula degrades under staggered starts. That is a gap, not a fatal flaw—the model is explicitly a baseline and the paper is honest about it.

One more issue: the abstract says the truncated power law is “justified by a rich-get-richer effect,” but the paper never derives that. It is a verbal justification, consistent with prior literature, not a result of this model. It should be softened or backed up.

Overall, the central contribution—the HHI dependence of the natural fork rate—is solid and useful. The paper deserves a serious referee. I would send it to peer review with the expectation that the empirical claims get robustness checks (alternative thresholds, no rescaling) and the TPL claim gets toned down or derived. If those are fixed, it is a solid contribution for a blockchain or distributed-systems venue.

Referee Report

3 major / 5 minor

Summary. The paper proposes an analytical model for the rate of soft forks in Proof-of-Work blockchains with heterogeneous miners. The central result is Eq. (8), C(Δ0) ≈ Δ0 Λ (1 − HHI), which expresses the fork probability as the product of the propagation-time threshold, the total hash rate, and a concentration term. The model is tested against Bitcoin data on mining-pool shares, block propagation times, and stale blocks, and is used to argue that both greater hash-rate concentration and lower propagation time reduce fork rates. The paper also fits exponential, log-normal, and truncated power-law distributions to empirical hash rates and uses the model to extrapolate fork rates under different miner counts and hash-rate dispersions.

Significance. The analytical result is clean and potentially useful: if valid, Eq. (8) gives a simple, falsifiable relationship between an observable fork rate and measurable aggregates (total hash rate, propagation time, and HHI). The derivation from exponential order statistics to Eqs. (6)–(8) is mathematically sound under the stated assumptions, and the simulations in Fig. 10 confirm the analytical formulas for several hash-rate distributions. The empirical contribution is more mixed: the hash-rate distribution analysis and the historical concentration trend are informative, but the central validation claim that model-estimated fork rates are 'comparable with the empirical stale blocks rate' is weakened by the rescaling of the fork-rate data and by the posterior choice of the 50% propagation-time percentile. The stress-test concern about synchronized miner starts lands: it identifies a potentially load-bearing assumption that is acknowledged but not quantified.

major comments (3)

[§2, Eqs. (1)–(8); §3 Discussion] The central approximation C(Δ0) ≈ Δ0 Λ (1 − HHI) is derived from N exponential clocks that all start at t = 0. In a real network, miner i can begin working on a new block only after receiving it, so its completion time is better described as T_i = S_i + E_i, where S_i is the arrival time of the previous block at miner i and E_i ~ Exp(λ_i). Once the S_i are non-degenerate, the joint law of the two smallest T_i is not the common-start order-statistic law in Eqs. (1)–(6), and the HHI factorization in Eq. (7) relies on the memoryless property of synchronized exponentials. The paper acknowledges this in the Discussion ('the relatively strong assumption that miners start mining at the same time'), but it does not quantify the error. Please add a sensitivity analysis, for example simulating T_i = S_i + E_i with S_i drawn from an empirically plausible propagation-time distribution or a two-miner calculation with fixed start-time offsets, and report how C(Δ0) and its HHI dependence change. Without such a check, the claim that Eq. (8) is validated in the regime of interest is not established.
[§4.1.2, Fig. 4] The empirical validation is not independent as presented. The fork-rate series is rescaled by a constant factor 1.476 so that its value in February 2016 matches the 0.41% value from Gervais et al. [18], and the comparison in Fig. 4b uses Δ0 = Δ0^(50), a choice made after inspecting the agreement with the model. Therefore the match in Fig. 4b is not a free prediction: one point is imposed by construction, and the time-series agreement is not out-of-sample. Please show the raw, unrescaled fork-rate series, and either preselect Δ0 on a training period and evaluate on the remainder or otherwise provide an out-of-sample test. The current Fig. 4b supports only the weaker statement that the model is compatible with the rescaled data for one particular propagation threshold.
[Fig. 4a, §2, Fig. 6b] The choice of Δ0 as the time to reach 50% of nodes is not derived from the model; in the model, Δ0 should be the time within which a second miner must solve the puzzle for a fork to occur, i.e., the time by which the first block reaches the second successful miner, not a network percentile. The paper shows in Fig. 4a that using 50%, 90%, or 99% propagation times changes the predicted fork rate by up to an order of magnitude, and Fig. 6b shows that the implied HHI is sometimes negative, which the authors correctly attribute to Δ0 being too large. The authors should clarify the interpretation of Δ0 in Eq. (8) in terms of the peer-to-peer propagation process, and report how the conclusions change if a different propagation-time percentile or a fitted Δ0 is used. This is necessary to substantiate the 'comparable with empirical stale blocks rate' claim.

minor comments (5)

[Eq. (5) and the preceding joint density] There is a sign error or typo in the exponent of the joint density before Eq. (5): as printed, e^{λ_i(t−t′)+Σ_k λ_k t′} does not integrate to Eqs. (5)–(6). The correct exponent should be e^{λ_i(t′−t)−Σ_k λ_k t′} (or an equivalent decaying form). Please fix this to make the derivation self-contained for readers.
[Eq. (12), §4.1.5] The method-of-moments estimators for the truncated power-law parameters are stated as α = 1 − (m/s)^2 and β = m/s^2; the expression for α should be 1 − (m/s)^2 only if the variance is such that s > m, which holds here, but the notation is likely a typo for 1 − (m/s)^2. Please clarify the formula and its domain of validity.
[§4.1.2] The fork rate is defined as the number of block heights with at least one stale or orphan block divided by the total number of blocks, but the term 'fork rate' is later used interchangeably with the probability that a block becomes a stale block. These are not the same quantity, and the distinction should be stated explicitly.
[Fig. 4b, Fig. 5] The 90% confidence band around the historical fork-rate series is mentioned but its construction is not described. Please specify the source of uncertainty (e.g., Poisson sampling, resampling, or measurement noise) used to produce the shaded area.
[§C.3.3 and Fig. 8/9] The 'truncated power law' in Eq. (24) is a Gamma-like density with an exponential cutoff, not a truncated power law in the sense of Burroughs and Tebbens [22]; please justify the naming or use a standard term such as 'power law with exponential cutoff.' Also, the text in §C.3.4 refers to Fig. 9 when describing the scenarios of zero miners, but the fork-rate results are in Fig. 8; please correct the cross-references.

Circularity Check

1 steps flagged · score 2.0 of 10

The fork-rate formula Eq. (8) is derived from independent inputs; the empirical validation is partly calibrated and self-confirming, and a minor non-load-bearing self-citation appears in the distributional rationale.

other [Methods §4.1.2 (Forks) and Results §2 (Fig. 4b)]
"The bitcoin-data/stale-blocks is likely to under-report stale blocks due to its crowd-sourced nature: to reconcile it with other data sources, we adjust the computed fork rate to match the value of 0.41% reported by Gervais et al. [18] at the end of February 2016. This results in a rescaling of the fork rates by a factor of 1.476. ... We show in Fig. 4b that the model-predicted fork rate is largely consistent with the empirically estimated fork rate when Δ0 = Δ(50)0."

The empirical fork-rate series used as the benchmark is rescaled to hit a single external target (0.41% at Feb 2016), so it is not a purely observed validation series. The model input Δ0 is then selected among the 50/90/99 percentiles after seeing which yields agreement (Fig. 4b uses Δ0(50)). Thus the claim of empirical consistency is partly self-confirming: the comparison scale is anchored to a literature value and the propagation threshold is chosen post hoc, rather than being an out-of-sample test of Eq. (8). This does not make the derivation of Eq. (8) circular, as the hash-rate distribution and Δ0 are independent inputs, but it weakens the empirical validation.

full rationale

The central fork-rate formula, Eq. (8) ≈ Δ0·Λ·(1−HHI), is derived analytically from the joint distribution of the first two order statistics of N independent exponential mining times (Eqs. 1–6). The inputs N, {λi}, and Δ0 are measured independently from blockchain and network data; no parameter is fitted to fork rates. The derivation is therefore self-contained and not circular: the HHI dependence emerges from the exponential order statistics, not from a fit. The empirical comparison is partly softened by the rescaling of the fork-rate series to match a Gervais et al. value and by the post-hoc choice of Δ0(50) among three percentiles, which introduces an element of self-confirmation in the validation step. Additionally, the paper's Discussion acknowledges the strong synchronized-start assumption; this is a modeling limitation rather than circularity. Finally, the cited support for the log-normal distribution includes two self-citations ([16] and [21]), but these are not load-bearing because the main results hold across all considered distributions. Overall, the theoretical result is not reduced to its inputs, so the circularity score is low.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard Poisson/exponential assumptions about mining, the synchronized-start simplification, and the assumption that mining pools act as single miners with zero internal delay. The free parameters are the fitted null-distribution parameters (per period) and two calibration choices: the fork-rate rescaling factor and the propagation-time percentile used for the headline comparison.

free parameters (5)

Exponential rate r (per period) = e.g., 49,565 (2015-06); 19,159 (2023-12)
Fitted to empirical hash rates via method of moments (Eq. 12); r = 1/m where m is the empirical mean hash rate.
Log-normal parameters μ, σ2 (per period) = e.g., μ = -11.96, σ = 1.52 (2015-06)
Fitted to empirical hash rates via method of moments (Eq. 12).
Truncated power-law parameters α, β (per period) = e.g., α = 0.89, β = 5,518 (2015-06)
Fitted to empirical hash rates via method of moments (Eq. 12).
Fork-rate rescaling factor = 1.476
Chosen so the computed fork rate matches the 0.41% value reported by Gervais et al. at end of February 2016 (Methods 4.1.2). This is a calibration of the empirical fork-rate series.
Propagation-time percentile used for headline comparison = 50% (median)
The model is compared with empirical fork rates using Δ0^(50), Δ0^(90), Δ0^(99); the 50% value produces the closest match (Fig. 4b, Fig. 6). The choice is post hoc, as the implied Δ0 is smaller than the median broadcast time.

assumptions (6)

domain assumption Mining times are exponentially distributed (Poisson arrivals)
Invoked in Eq. (1); standard for PoW mining, but not empirically tested in the paper.
domain assumption All miners start mining the next block at the same time
Used to set up the order-statistics derivation (Fig. 1, Eqs. 2-6); acknowledged as a limitation in Discussion.
domain assumption Zero propagation delay within mining pools
Stated in Methods 4.1.3: 'we assume that there is no propagation delay among the members within the same mining pool.'
domain assumption Hash rates are independent across miners
Required to derive Eq. (10) from Eq. (9) by factorizing the integral over {λ_i}.
domain assumption Hash rates are constant within each 20,000-block period
Dividing data into stationary periods (Methods 4.1); acknowledged as a limitation in Discussion.
standard math Order statistics of independent exponential random variables
Background result used to derive the distribution of the two fastest mining times (Eqs. 2-6).

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Pith. "Pith review of How the interplay between power concentration, competition, and propagation affects the resource efficiency of distributed ledgers." pith.science (2026). https://pith.science/paper/KFYWTPL7

@misc{pith2026241110249,
  author       = {Pith},
  title        = {Pith review of: How the interplay between power concentration, competition, and propagation affects the resource efficiency of distributed ledgers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KFYWTPL7}},
  note         = {Machine review of arXiv:2411.10249}
}

read the original abstract

Forks in the Bitcoin network result from the natural competition in the blockchain's Proof-of-Work consensus protocol. Their frequency is a critical indicator for the efficiency of a distributed ledger as they can contribute to resource waste and network insecurity. We introduce a model for the estimation of natural fork rates in a network of heterogeneous miners as a function of their number, the distribution of hash rates and the block propagation time over the peer-to-peer infrastructure. Despite relatively simplistic assumptions, such as zero propagation delay within mining pools, the model predicts fork rates which are comparable with the empirical stale blocks rate. In the past decade, we observe a reduction in the number of mining pools approximately by a factor 3, and quantify its consequences for the fork rate, whilst showing the emergence of a truncated power-law distribution in hash rates, justified by a rich-get-richer effect constrained by global energy supply limits. We demonstrate, both empirically and with the aid of our quantitative model, that the ratio between the block propagation time and the mining time is a sufficiently accurate estimator of the fork rate, but also quantify its dependence on the heterogeneity of miner activities. We provide empirical and theoretical evidence that both hash rate concentration and lower block propagation time reduce fork rates in distributed ledgers. Our work introduces a robust mathematical setting for investigating power concentration and competition on a distributed network, for interpreting discrepancies in fork rates -- for example caused by selfish mining practices and asymmetric propagation times -- thus providing an effective tool for designing future and alternative scenarios for existing and new blockchain distributed mining systems.

Figures

Figures reproduced from arXiv: 2411.10249 by the authors.

**Figure 1.** To simplify the dynamics, we assume that all miners start mining at t1. At t2 Miner 1 solves the PoW puzzle for block x and starts broadcasting it to the network. By t4, the message has spread to most of the nodes. However, Miner 2 successfully mines the block at t3, which is later than t2 and earlier than t4. Therefore, based on the definition above, the propagation time is t4 − t2, and the fork is present between … view at source ↗

**Figure 2.** Share of blocks mined among miners in selected observation periods. Unknown miners - for [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Complementary cumulative distribution function (ccdf) of hash rates’ empirical distribution [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: Time series of historically measured fork rates (dotted line) compared with model-estimated [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: Fork rate estimation at different block propagation times, comparing the one obtained numer [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: Time series of implied ∆0 and implied HHI versus their empirical values. We show in Fig. 4b that the model-predicted fork rate is largely consistent with the empirically estimated fork rate when ∆0 = ∆(50) 0 . We also find that there is close to no difference between t…

**Figure 7.** Figure 7: Extrapolation of fork rates under the log-normal or truncated power law hash rate distributions. [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 8.** Figure 8: Fork rates by number of zero miners and block propagation delay at various given hash rate [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]

**Figure 9.** Figure 9: Complementary cumulative distribution function (ccdf) of hash rates when different numbers [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]

**Figure 10.** Figure 10: Fork rates computed from the simulated environment (thick, transparent curves) compared [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]

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