On the Incentive Compatibility of Block Propagation in Bitcoin

Akira Sakurai; Fumichika Maeda; Kazuyuki Shudo; Taishi Nakai

arxiv: 2606.06860 · v1 · pith:CAV3LI5Knew · submitted 2026-06-05 · 💻 cs.CR

On the Incentive Compatibility of Block Propagation in Bitcoin

Fumichika Maeda , Akira Sakurai , Taishi Nakai , Kazuyuki Shudo This is my paper

Pith reviewed 2026-06-27 22:02 UTC · model grok-4.3

classification 💻 cs.CR

keywords block propagationincentive compatibilityBitcoin miningtie-breaking rulesforksmining rewardshashrate distribution

0 comments

The pith

Miners have no mining-reward incentive to relay blocks generated by other miners.

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

The paper derives closed-form expressions for each miner's expected reward under several tie-breaking rules, using a network model that incorporates how propagation delays produce forks and alter effective hashrate fairness. These expressions show that, for most rules, a miner gains nothing by forwarding a competitor's block and may even lose by doing so. Under the first-seen rule the expressions reverse: every miner below majority hashrate benefits from both receiving foreign blocks faster and sending its own blocks faster. The same formulas also quantify a direct trade-off in which stronger propagation incentives coincide with larger deviations from hashrate-proportional rewards.

Core claim

Using analytical reward expressions derived from a blockchain network model that captures the effect of forks on mining fairness, the paper shows that miners have no mining-reward incentive to relay blocks generated by other miners. By contrast, under the first-seen rule every non-majority miner is incentivized to receive other miners' blocks more quickly and to propagate its own blocks more quickly, while the first-seen rule also produces the largest deviation from hashrate-proportional rewards among the rules examined.

What carries the argument

Analytical reward expressions obtained from a blockchain network model that accounts for how propagation delays create forks and change effective mining fairness, expressed as functions of propagation delays, hashrate shares, and the chosen tie-breaking rule.

If this is right

Miners receive no additional mining reward for relaying blocks produced by competitors under standard tie-breaking rules.
Under the first-seen rule every non-majority miner gains reward by both reducing its own block propagation delay and by receiving competitors' blocks with lower delay.
The first-seen rule supplies the strongest incentive to reduce propagation delays of any rule considered.
Any rule that strengthens propagation incentives simultaneously increases the deviation of realized rewards from hashrate shares.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

A protocol change that adopts first-seen could be evaluated by measuring whether the resulting increase in propagation speed outweighs the measured increase in reward variance.
If propagation incentives prove more important than strict fairness, hybrid tie-breaking rules could be tested that blend first-seen behavior only among smaller miners.
Mechanisms outside pure mining rewards, such as explicit relay fees or reputation scores, might be needed if no tie-breaking rule simultaneously satisfies both goals.

Load-bearing premise

The blockchain network model that captures the effect of forks on mining fairness is accurate enough to derive the reward expressions for each tie-breaking rule.

What would settle it

A direct measurement or controlled simulation that shows whether actual mining rewards change exactly as the derived expressions predict when propagation delays between specific miners are altered under a given tie-breaking rule.

read the original abstract

Bitcoin is permissionless and does not rely on any central administrator, which gives it strong censorship resistance. At the same time, it is important to incentivize miners to behave in ways that align with the interests of the system as a whole. This paper asks whether miners are individually incentivized to propagate blocks, one of the most fundamental processes in Bitcoin. Miners collectively maintain the blockchain by generating blocks and disseminating them across the network. If miners have an incentive not to propagate some blocks, this would indicate a fundamental flaw in Bitcoin's incentive design. Although prior work has studied how propagation delays affect forks and mining rewards, it has not fully characterized miners' incentives to improve block propagation under different tie-breaking rules. To address this gap, we derive analytical reward expressions for each tie-breaking rule based on a blockchain network model that captures the effect of forks on mining fairness. These expressions explicitly characterize how block propagation delays, hashrate distribution, and tie-breaking rules jointly determine mining rewards. We then use them to analyze miners' incentives to improve block propagation. Our results show, for example, that miners have no mining-reward incentive to relay blocks generated by other miners. By contrast, under the first-seen rule, every non-majority miner is incentivized to receive other miners' blocks more quickly and to propagate its own blocks more quickly. Finally, we compare tie-breaking rules and identify a trade-off between propagation incentives and mining fairness. In particular, the first-seen rule provides the strongest incentives to reduce propagation delays, but it also worsens mining fairness the most.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper derives closed-form reward expressions under different tie-breaking rules and concludes miners lack incentive to relay blocks, but those signs depend on an unvalidated network model for fork rates.

read the letter

The main result is that the authors produce analytical expressions for mining rewards as functions of propagation delay, hashrate distribution, and tie-breaking rule. Differentiating those expressions yields the claim that miners have no reward incentive to forward other miners' blocks, while the first-seen rule gives smaller miners a clear incentive to both receive blocks faster and propagate their own faster, though at the cost of reduced fairness.

The work does a straightforward job of closing the gap the abstract identifies: earlier papers looked at delays and forks but did not map the full incentive picture across multiple tie rules. The trade-off they flag between propagation incentives and fairness is a useful observation to have written down.

The soft spot is the underlying network model that converts delays and hashrate into fork probabilities and fairness measures. All the incentive conclusions come from the partial derivatives of the reward formulas with respect to delay, so the sign of those derivatives is only as good as the model's fork mapping. The paper does not report simulation validation, sensitivity checks, or comparison against measured Bitcoin propagation traces, which leaves the quantitative incentive results resting on untested assumptions about delay independence, arrival processes, and connectivity.

This is for people who already work on Bitcoin incentive design or protocol security. A reader who knows the fork literature would get value from the explicit expressions, but would treat the incentive signs as provisional until the model is checked.

Send it to peer review. The question is worth referee time and the analytical framing is reasonable; the referees can require the model validation that is currently missing.

Referee Report

2 major / 1 minor

Summary. The paper derives closed-form analytical expressions for per-miner rewards under several tie-breaking rules for resolving forks in Bitcoin, using a network model that takes hashrate distribution and propagation delays as inputs and maps them to fork probabilities and mining fairness. These expressions are then differentiated with respect to propagation delays to determine individual miners' incentives to relay blocks or reduce their own delays. The central conclusions are that miners have no mining-reward incentive to relay blocks generated by others, that the first-seen rule creates strong propagation incentives for non-majority miners, and that first-seen improves propagation incentives at the cost of greater unfairness compared with other rules.

Significance. If the derivations and model are correct, the work supplies an analytical framework for quantifying how tie-breaking rules affect both fairness and propagation incentives, which could inform protocol-level choices in Bitcoin and similar systems. The explicit characterization of the joint dependence on delays, hashrate shares, and tie-breaking rules is a strength relative to purely simulation-based studies.

major comments (2)

[Abstract and model description] The reward expressions and all incentive conclusions rest on an unvalidated blockchain network model that converts propagation delays and hashrate shares into fork rates and fairness metrics. No simulation validation, sensitivity analysis, or comparison to real-network measurements is provided for the fork-probability mapping (e.g., assumed independence of delays or Poisson block arrivals), which directly determines the sign of the partial derivatives that produce the incentive results.
[Incentive analysis] The claim that miners have 'no mining-reward incentive to relay blocks generated by other miners' is obtained by showing that the partial derivative of a miner's reward with respect to another miner's propagation delay is zero or negative; this sign depends entirely on the functional form chosen for how delays affect fork probabilities in the network model, yet no robustness checks or alternative delay models are reported.

minor comments (1)

[Abstract] The abstract states that expressions 'explicitly characterize' the joint dependence on delays, hashrate, and rules, but does not display any of the expressions; including at least the key functional forms in the abstract or introduction would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below and outline planned revisions to strengthen the manuscript.

read point-by-point responses

Referee: [Abstract and model description] The reward expressions and all incentive conclusions rest on an unvalidated blockchain network model that converts propagation delays and hashrate shares into fork rates and fairness metrics. No simulation validation, sensitivity analysis, or comparison to real-network measurements is provided for the fork-probability mapping (e.g., assumed independence of delays or Poisson block arrivals), which directly determines the sign of the partial derivatives that produce the incentive results.

Authors: The analysis is intentionally analytical and derives closed-form reward expressions under standard assumptions (Poisson arrivals, independent delays) that are widely used in the blockchain literature to enable exact differentiation for incentive results. We acknowledge the absence of simulation validation or sensitivity checks in the submitted version. In revision we will add an explicit limitations subsection discussing these assumptions, their justification from prior work, and a sensitivity analysis varying the delay-to-fork mapping parameters to confirm that the sign of the key partial derivatives is robust within the modeled regime. Full empirical calibration against live-network traces lies outside the paper's analytical scope but can be noted as future work. revision: partial
Referee: [Incentive analysis] The claim that miners have 'no mining-reward incentive to relay blocks generated by other miners' is obtained by showing that the partial derivative of a miner's reward with respect to another miner's propagation delay is zero or negative; this sign depends entirely on the functional form chosen for how delays affect fork probabilities in the network model, yet no robustness checks or alternative delay models are reported.

Authors: The zero/negative sign for the cross-partial follows directly from the chosen functional form, which encodes that a miner's own reward is unaffected or harmed when another miner's block arrives faster (increasing the chance it wins the fork). This form is derived from the standard network model in the paper and is consistent with the literature on fork probabilities. We will add a short robustness paragraph examining an alternative linear delay-to-fork mapping and confirming that the qualitative incentive conclusion (no positive incentive to reduce others' delays) is preserved. The contribution remains the closed-form characterization under the stated model rather than a claim of universality across all possible mappings. revision: partial

Circularity Check

0 steps flagged

No circularity: analytical reward expressions derived from explicit model inputs without reduction to fits or self-citations

full rationale

The paper states a blockchain network model with explicit inputs (hashrate shares, propagation delays) and derives closed-form reward expressions for each tie-breaking rule from it. Incentive results follow by differentiating those expressions w.r.t. delays. No quoted step shows a reward quantity reducing to a fitted parameter by construction, no self-citation chain is load-bearing for the central claim, and the model is presented as an assumption rather than derived from the target result. This is a standard model-based derivation and receives the default non-finding.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

Central claim rests on an unelaborated blockchain network model whose parameters (hashrate distribution and propagation delays) enter the reward expressions; no invented entities are introduced.

free parameters (2)

hashrate distribution
Jointly determines mining rewards in the analytical expressions for each tie-breaking rule
propagation delays
Jointly determines mining rewards together with hashrate and tie-breaking rules

axioms (1)

domain assumption Blockchain network model captures the effect of forks on mining fairness
Explicitly invoked as the foundation for deriving analytical reward expressions

pith-pipeline@v0.9.1-grok · 5821 in / 1348 out tokens · 23460 ms · 2026-06-27T22:02:01.126128+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

32 extracted references · 8 canonical work pages

[1]

Bitcoin: A peer-to-peer electronic cash system,

S. Nakamoto, “Bitcoin: A peer-to-peer electronic cash system,” 2008

2008
[2]

On the security and performance of proof of work blockchains,

A. Gervais, G. O. Karame, K. W ¨ust, V . Glykantzis, H. Ritzdorf, and S. Capkun, “On the security and performance of proof of work blockchains,” inProceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security, ser. CCS ’16. New York, NY , USA: Association for Computing Machinery, 2016, p. 3–16. [Online]. Available: https://doi.org/...

work page doi:10.1145/2976749.2978341 2016
[3]

In: Proceedings of the 2020 ACM SIGSAC Conference on Computer and Communica- tions Security

A. Dembo, S. Kannan, E. N. Tas, D. Tse, P. Viswanath, X. Wang, and O. Zeitouni, “Everything is a race and nakamoto always wins,” inProceedings of the 2020 ACM SIGSAC Conference on Computer and Communications Security, ser. CCS ’20. New York, NY , USA: Association for Computing Machinery, 2020, p. 859–878. [Online]. Available: https://doi.org/10.1145/33722...

work page doi:10.1145/3372297.3417290 2020
[4]

Tight consistency bounds for bitcoin,

P. Ga ˇzi, A. Kiayias, and A. Russell, “Tight consistency bounds for bitcoin,” inProceedings of the 2020 ACM SIGSAC Conference on Computer and Communications Security, ser. CCS ’20. New York, NY , USA: Association for Computing Machinery, 2020, p. 819–838. [Online]. Available: https://doi.org/10.1145/3372297.3423365

work page doi:10.1145/3372297.3423365 2020
[5]

Information propagation in the bitcoin network,

C. Decker and R. Wattenhofer, “Information propagation in the bitcoin network,” inIEEE P2P 2013 Proceedings, 2013, pp. 1–10

2013
[6]

On scaling decentralized blockchains,

K. Croman, C. Decker, I. Eyal, A. E. Gencer, A. Juels, A. Kosba, A. Miller, P. Saxena, E. Shi, E. G ¨un Sirer, D. Song, and R. Wattenhofer, “On scaling decentralized blockchains,” inFinancial Cryptography and Data Security, J. Clark, S. Meiklejohn, P. Y . Ryan, D. Wallach, M. Brenner, and K. Rohloff, Eds. Berlin, Heidelberg: Springer Berlin Heidelberg, 20...

2016
[7]

Less is more: Understanding network bias in proof-of-work blockchains,

Y . Mao and S. B. Venkatakrishnan, “Less is more: Understanding network bias in proof-of-work blockchains,”Mathematics, vol. 11, no. 23, 2023. [Online]. Available: https://www.mdpi.com/2227- 7390/11/23/4741

2023
[8]

When is slower block propagation more profitable for large miners?

Z. Lu and R. Zhang, “When is slower block propagation more profitable for large miners?” inComputer Security – ESORICS 2023, G. Tsudik, M. Conti, K. Liang, and G. Smaragdakis, Eds. Cham: Springer Nature Switzerland, 2024, pp. 285–305

2023
[9]

Model-based calculation method of mining fairness in blockchain,

A. Sakurai and K. Shudo, “Model-based calculation method of mining fairness in blockchain,”IEEE Open Journal of the Computer Society, vol. 7, pp. 129–141, 2026

2026
[10]

Majority is not enough: Bitcoin mining is vul- nerable,

I. Eyal and E. G. Sirer, “Majority is not enough: Bitcoin mining is vul- nerable,” inFinancial Cryptography and Data Security, N. Christin and R. Safavi-Naini, Eds. Berlin, Heidelberg: Springer Berlin Heidelberg, 2014, pp. 436–454

2014
[11]

A fully local last-generated rule in a blockchain,

A. Sakurai and K. Shudo, “A fully local last-generated rule in a blockchain,” 2024, arXiv: 2411.08439. [Online]. Available: https://arxiv.org/abs/2411.08439

arXiv 2024
[12]

Tie-breaking rule based on partial proof of work in a blockchain,

A. Sakurai and K. Shudo, “Tie-breaking rule based on partial proof of work in a blockchain,”IEEE Access, vol. 12, pp. 197 999–198 014, 2024

2024
[13]

One weird trick to stop selfish miners: Fresh bitcoins, a solution for the honest miner (poster abstract),

E. Heilman, “One weird trick to stop selfish miners: Fresh bitcoins, a solution for the honest miner (poster abstract),” inFinancial Cryptogra- phy and Data Security, R. B ¨ohme, M. Brenner, T. Moore, and M. Smith, Eds. Berlin, Heidelberg: Springer Berlin Heidelberg, 2014, pp. 161– 162

2014
[14]

Analysis of bitcoin pooled mining reward systems,

M. Rosenfeld, “Analysis of bitcoin pooled mining reward systems,” 2011

2011
[15]

On subversive miner strategies and block withholding attack in bitcoin digital currency,

N. T. Courtois and L. Bahack, “On subversive miner strategies and block withholding attack in bitcoin digital currency,” 2014. [Online]. Available: https://arxiv.org/abs/1402.1718

Pith/arXiv arXiv 2014
[16]

Deeplog: Anomaly detection and diagnosis from system logs through deep learning,

Y . Kwon, D. Kim, Y . Son, E. Vasserman, and Y . Kim, “Be selfish and avoid dilemmas: Fork after withholding (faw) attacks on bitcoin,” inProceedings of the 2017 ACM SIGSAC Conference on Computer and Communications Security, ser. CCS ’17. New York, NY , USA: Association for Computing Machinery, 2017, p. 195–209. [Online]. Available: https://doi.org/10.114...

work page doi:10.1145/3133956.3134019 2017
[17]

Optimal selfish mining strategies in bitcoin,

A. Sapirshtein, Y . Sompolinsky, and A. Zohar, “Optimal selfish mining strategies in bitcoin,” inFinancial Cryptography and Data Security, J. Grossklags and B. Preneel, Eds. Berlin, Heidelberg: Springer Berlin Heidelberg, 2017, pp. 515–532

2017
[18]

Drammer: Deterministic rowhammer attacks on mobile platforms,

M. Carlsten, H. Kalodner, S. M. Weinberg, and A. Narayanan, “On the instability of bitcoin without the block reward,” in Proceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security, ser. CCS ’16. New York, NY , USA: Association for Computing Machinery, 2016, p. 154–167. [Online]. Available: https://doi.org/10.1145/2976749.2978408

work page doi:10.1145/2976749.2978408 2016
[19]

Blockchain mining games,

A. Kiayias, E. Koutsoupias, M. Kyropoulou, and Y . Tselekounis, “Blockchain mining games,” inProceedings of the 2016 ACM Conference on Economics and Computation, ser. EC ’16. New York, NY , USA: Association for Computing Machinery, 2016, p. 365–382. [Online]. Available: https://doi.org/10.1145/2940716.2940773

work page doi:10.1145/2940716.2940773 2016
[20]

Eclipse attacks on Bitcoin’s Peer-to-Peer network,

E. Heilman, A. Kendler, A. Zohar, and S. Goldberg, “Eclipse attacks on Bitcoin’s Peer-to-Peer network,” inProc. 24th USENIX Security Symposium (USENIX Security ’15). Washington, D.C.: USENIX Association, Aug. 2015, pp. 129–144. [Online]. Avail- able: https://www.usenix.org/conference/usenixsecurity15/technical- sessions/presentation/heilman

2015
[21]

Tendrilstaller: Block delay attack in bitcoin,

M. Walck, K. Wang, and H. S. Kim, “Tendrilstaller: Block delay attack in bitcoin,” in3rd IEEE International Conference on Blockchain, 2019, pp. 1–9

2019
[22]

Simblock: A blockchain network simulator,

Y . Aoki, K. Otsuki, T. Kaneko, R. Banno, and K. Shudo, “Simblock: A blockchain network simulator,” inProc. IEEE INFOCOM 2019 - IEEE Conference on Computer Communications Workshops (INFOCOM 2019 Workshops), 2019, pp. 325–329

2019
[23]

A concordance correlation coefficient to evaluate reproducibility,

L. I.-K. Lin, “A concordance correlation coefficient to evaluate reproducibility,”Biometrics, vol. 45, no. 1, pp. 255–268, 1989. [Online]. Available: http://www.jstor.org/stable/2532051

arXiv 1989
[24]

Compact block relay,

M. Corallo, “Compact block relay,” https://github.com/bitcoin/bips/blob/master/bip-0152.mediawiki, 2016, accessed: 2025-09-13

2016
[25]

Identifying impacts of protocol and internet development on the bitcoin network,

R. Nagayama, R. Banno, and K. Shudo, “Identifying impacts of protocol and internet development on the bitcoin network,” inProc. 25th IEEE Symposium on Computers and Communications (IEEE ISCC 2020), 2020, pp. 1–6

2020
[26]

Bitnodes,

“Bitnodes,” https://bitnodes.io/nodes/, accessed: 2026-04-26

2026
[27]

Mining pool stats,

“Mining pool stats,” Online, 2026, accessed: 2026-04-19. [Online]. Available: https://miningpoolstats.stream/

2026
[28]

On the peer degree distribution of the bitcoin p2p network,

M. Grundmann, M. Baumstark, and H. Hartenstein, “On the peer degree distribution of the bitcoin p2p network,” in2022 IEEE International Conference on Blockchain and Cryptocurrency (ICBC), 2022, pp. 1–5

2022
[29]

Discovering bitcoin’s public topology and influential nodes,

A. Miller, J. Litton, A. Pachulski, N. Gupta, D. Levin, N. Spring, and B. Bhattacharjee, “Discovering bitcoin’s public topology and influential nodes,” https://www.cs.umd.edu/projects/coinscope/coinscope.pdf, 2015, university of Maryland, Accessed: 2026-04-26

2015
[30]

Effects of a simple relay network on the bitcoin network,

K. Otsuki, Y . Aoki, R. Banno, and K. Shudo, “Effects of a simple relay network on the bitcoin network,” inProceedings of the Asian Internet Engineering Conference, ser. AINTEC ’19. New York, NY , USA: Association for Computing Machinery, 2019, pp. 41–46. [Online]. Available: https://doi.org/10.1145/3340422.3343640

work page doi:10.1145/3340422.3343640 2019
[31]

Calibrating the performance and security of blockchains via information propagation delays: revisiting an old approach with a new perspective,

J. Fechner, B. Chandrasekaran, and M. X. Makkes, “Calibrating the performance and security of blockchains via information propagation delays: revisiting an old approach with a new perspective,” in Proceedings of the 37th ACM/SIGAPP Symposium on Applied Computing, ser. SAC ’22. New York, NY , USA: Association for Computing Machinery, 2022, pp. 282–289. [On...

work page doi:10.1145/3477314.3507003 2022
[32]

Otherwise,T jk ≤T ik ≤T ij +T jk, and by Cond. A, pi,j,k = exp − Tik−Tjk T −exp − Tij T 1−exp − Tij T = 1− Tik−Tjk T − 1− Tij T Tij T +O(ε) = Tij +T jk −T ik Tij +O(ε).(74) Therefore, for alli, j, k∈V, Tij(1−p i,j,k) = (Tik −T jk)+ +O(T ε 2),(75) where(x) + := max{x,0}. Using (75), we obtain X j∈V αjTij(1−W ij) = X j∈V X k∈V αjαk(Tik −T jk)+ +O(T ε 2), (7...

[1] [1]

Bitcoin: A peer-to-peer electronic cash system,

S. Nakamoto, “Bitcoin: A peer-to-peer electronic cash system,” 2008

2008

[2] [2]

On the security and performance of proof of work blockchains,

A. Gervais, G. O. Karame, K. W ¨ust, V . Glykantzis, H. Ritzdorf, and S. Capkun, “On the security and performance of proof of work blockchains,” inProceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security, ser. CCS ’16. New York, NY , USA: Association for Computing Machinery, 2016, p. 3–16. [Online]. Available: https://doi.org/...

work page doi:10.1145/2976749.2978341 2016

[3] [3]

In: Proceedings of the 2020 ACM SIGSAC Conference on Computer and Communica- tions Security

A. Dembo, S. Kannan, E. N. Tas, D. Tse, P. Viswanath, X. Wang, and O. Zeitouni, “Everything is a race and nakamoto always wins,” inProceedings of the 2020 ACM SIGSAC Conference on Computer and Communications Security, ser. CCS ’20. New York, NY , USA: Association for Computing Machinery, 2020, p. 859–878. [Online]. Available: https://doi.org/10.1145/33722...

work page doi:10.1145/3372297.3417290 2020

[4] [4]

Tight consistency bounds for bitcoin,

P. Ga ˇzi, A. Kiayias, and A. Russell, “Tight consistency bounds for bitcoin,” inProceedings of the 2020 ACM SIGSAC Conference on Computer and Communications Security, ser. CCS ’20. New York, NY , USA: Association for Computing Machinery, 2020, p. 819–838. [Online]. Available: https://doi.org/10.1145/3372297.3423365

work page doi:10.1145/3372297.3423365 2020

[5] [5]

Information propagation in the bitcoin network,

C. Decker and R. Wattenhofer, “Information propagation in the bitcoin network,” inIEEE P2P 2013 Proceedings, 2013, pp. 1–10

2013

[6] [6]

On scaling decentralized blockchains,

K. Croman, C. Decker, I. Eyal, A. E. Gencer, A. Juels, A. Kosba, A. Miller, P. Saxena, E. Shi, E. G ¨un Sirer, D. Song, and R. Wattenhofer, “On scaling decentralized blockchains,” inFinancial Cryptography and Data Security, J. Clark, S. Meiklejohn, P. Y . Ryan, D. Wallach, M. Brenner, and K. Rohloff, Eds. Berlin, Heidelberg: Springer Berlin Heidelberg, 20...

2016

[7] [7]

Less is more: Understanding network bias in proof-of-work blockchains,

Y . Mao and S. B. Venkatakrishnan, “Less is more: Understanding network bias in proof-of-work blockchains,”Mathematics, vol. 11, no. 23, 2023. [Online]. Available: https://www.mdpi.com/2227- 7390/11/23/4741

2023

[8] [8]

When is slower block propagation more profitable for large miners?

Z. Lu and R. Zhang, “When is slower block propagation more profitable for large miners?” inComputer Security – ESORICS 2023, G. Tsudik, M. Conti, K. Liang, and G. Smaragdakis, Eds. Cham: Springer Nature Switzerland, 2024, pp. 285–305

2023

[9] [9]

Model-based calculation method of mining fairness in blockchain,

A. Sakurai and K. Shudo, “Model-based calculation method of mining fairness in blockchain,”IEEE Open Journal of the Computer Society, vol. 7, pp. 129–141, 2026

2026

[10] [10]

Majority is not enough: Bitcoin mining is vul- nerable,

I. Eyal and E. G. Sirer, “Majority is not enough: Bitcoin mining is vul- nerable,” inFinancial Cryptography and Data Security, N. Christin and R. Safavi-Naini, Eds. Berlin, Heidelberg: Springer Berlin Heidelberg, 2014, pp. 436–454

2014

[11] [11]

A fully local last-generated rule in a blockchain,

A. Sakurai and K. Shudo, “A fully local last-generated rule in a blockchain,” 2024, arXiv: 2411.08439. [Online]. Available: https://arxiv.org/abs/2411.08439

arXiv 2024

[12] [12]

Tie-breaking rule based on partial proof of work in a blockchain,

A. Sakurai and K. Shudo, “Tie-breaking rule based on partial proof of work in a blockchain,”IEEE Access, vol. 12, pp. 197 999–198 014, 2024

2024

[13] [13]

One weird trick to stop selfish miners: Fresh bitcoins, a solution for the honest miner (poster abstract),

E. Heilman, “One weird trick to stop selfish miners: Fresh bitcoins, a solution for the honest miner (poster abstract),” inFinancial Cryptogra- phy and Data Security, R. B ¨ohme, M. Brenner, T. Moore, and M. Smith, Eds. Berlin, Heidelberg: Springer Berlin Heidelberg, 2014, pp. 161– 162

2014

[14] [14]

Analysis of bitcoin pooled mining reward systems,

M. Rosenfeld, “Analysis of bitcoin pooled mining reward systems,” 2011

2011

[15] [15]

On subversive miner strategies and block withholding attack in bitcoin digital currency,

N. T. Courtois and L. Bahack, “On subversive miner strategies and block withholding attack in bitcoin digital currency,” 2014. [Online]. Available: https://arxiv.org/abs/1402.1718

Pith/arXiv arXiv 2014

[16] [16]

Deeplog: Anomaly detection and diagnosis from system logs through deep learning,

Y . Kwon, D. Kim, Y . Son, E. Vasserman, and Y . Kim, “Be selfish and avoid dilemmas: Fork after withholding (faw) attacks on bitcoin,” inProceedings of the 2017 ACM SIGSAC Conference on Computer and Communications Security, ser. CCS ’17. New York, NY , USA: Association for Computing Machinery, 2017, p. 195–209. [Online]. Available: https://doi.org/10.114...

work page doi:10.1145/3133956.3134019 2017

[17] [17]

Optimal selfish mining strategies in bitcoin,

A. Sapirshtein, Y . Sompolinsky, and A. Zohar, “Optimal selfish mining strategies in bitcoin,” inFinancial Cryptography and Data Security, J. Grossklags and B. Preneel, Eds. Berlin, Heidelberg: Springer Berlin Heidelberg, 2017, pp. 515–532

2017

[18] [18]

Drammer: Deterministic rowhammer attacks on mobile platforms,

M. Carlsten, H. Kalodner, S. M. Weinberg, and A. Narayanan, “On the instability of bitcoin without the block reward,” in Proceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security, ser. CCS ’16. New York, NY , USA: Association for Computing Machinery, 2016, p. 154–167. [Online]. Available: https://doi.org/10.1145/2976749.2978408

work page doi:10.1145/2976749.2978408 2016

[19] [19]

Blockchain mining games,

A. Kiayias, E. Koutsoupias, M. Kyropoulou, and Y . Tselekounis, “Blockchain mining games,” inProceedings of the 2016 ACM Conference on Economics and Computation, ser. EC ’16. New York, NY , USA: Association for Computing Machinery, 2016, p. 365–382. [Online]. Available: https://doi.org/10.1145/2940716.2940773

work page doi:10.1145/2940716.2940773 2016

[20] [20]

Eclipse attacks on Bitcoin’s Peer-to-Peer network,

E. Heilman, A. Kendler, A. Zohar, and S. Goldberg, “Eclipse attacks on Bitcoin’s Peer-to-Peer network,” inProc. 24th USENIX Security Symposium (USENIX Security ’15). Washington, D.C.: USENIX Association, Aug. 2015, pp. 129–144. [Online]. Avail- able: https://www.usenix.org/conference/usenixsecurity15/technical- sessions/presentation/heilman

2015

[21] [21]

Tendrilstaller: Block delay attack in bitcoin,

M. Walck, K. Wang, and H. S. Kim, “Tendrilstaller: Block delay attack in bitcoin,” in3rd IEEE International Conference on Blockchain, 2019, pp. 1–9

2019

[22] [22]

Simblock: A blockchain network simulator,

Y . Aoki, K. Otsuki, T. Kaneko, R. Banno, and K. Shudo, “Simblock: A blockchain network simulator,” inProc. IEEE INFOCOM 2019 - IEEE Conference on Computer Communications Workshops (INFOCOM 2019 Workshops), 2019, pp. 325–329

2019

[23] [23]

A concordance correlation coefficient to evaluate reproducibility,

L. I.-K. Lin, “A concordance correlation coefficient to evaluate reproducibility,”Biometrics, vol. 45, no. 1, pp. 255–268, 1989. [Online]. Available: http://www.jstor.org/stable/2532051

arXiv 1989

[24] [24]

Compact block relay,

M. Corallo, “Compact block relay,” https://github.com/bitcoin/bips/blob/master/bip-0152.mediawiki, 2016, accessed: 2025-09-13

2016

[25] [25]

Identifying impacts of protocol and internet development on the bitcoin network,

R. Nagayama, R. Banno, and K. Shudo, “Identifying impacts of protocol and internet development on the bitcoin network,” inProc. 25th IEEE Symposium on Computers and Communications (IEEE ISCC 2020), 2020, pp. 1–6

2020

[26] [26]

Bitnodes,

“Bitnodes,” https://bitnodes.io/nodes/, accessed: 2026-04-26

2026

[27] [27]

Mining pool stats,

“Mining pool stats,” Online, 2026, accessed: 2026-04-19. [Online]. Available: https://miningpoolstats.stream/

2026

[28] [28]

On the peer degree distribution of the bitcoin p2p network,

M. Grundmann, M. Baumstark, and H. Hartenstein, “On the peer degree distribution of the bitcoin p2p network,” in2022 IEEE International Conference on Blockchain and Cryptocurrency (ICBC), 2022, pp. 1–5

2022

[29] [29]

Discovering bitcoin’s public topology and influential nodes,

A. Miller, J. Litton, A. Pachulski, N. Gupta, D. Levin, N. Spring, and B. Bhattacharjee, “Discovering bitcoin’s public topology and influential nodes,” https://www.cs.umd.edu/projects/coinscope/coinscope.pdf, 2015, university of Maryland, Accessed: 2026-04-26

2015

[30] [30]

Effects of a simple relay network on the bitcoin network,

K. Otsuki, Y . Aoki, R. Banno, and K. Shudo, “Effects of a simple relay network on the bitcoin network,” inProceedings of the Asian Internet Engineering Conference, ser. AINTEC ’19. New York, NY , USA: Association for Computing Machinery, 2019, pp. 41–46. [Online]. Available: https://doi.org/10.1145/3340422.3343640

work page doi:10.1145/3340422.3343640 2019

[31] [31]

Calibrating the performance and security of blockchains via information propagation delays: revisiting an old approach with a new perspective,

J. Fechner, B. Chandrasekaran, and M. X. Makkes, “Calibrating the performance and security of blockchains via information propagation delays: revisiting an old approach with a new perspective,” in Proceedings of the 37th ACM/SIGAPP Symposium on Applied Computing, ser. SAC ’22. New York, NY , USA: Association for Computing Machinery, 2022, pp. 282–289. [On...

work page doi:10.1145/3477314.3507003 2022

[32] [32]

Otherwise,T jk ≤T ik ≤T ij +T jk, and by Cond. A, pi,j,k = exp − Tik−Tjk T −exp − Tij T 1−exp − Tij T = 1− Tik−Tjk T − 1− Tij T Tij T +O(ε) = Tij +T jk −T ik Tij +O(ε).(74) Therefore, for alli, j, k∈V, Tij(1−p i,j,k) = (Tik −T jk)+ +O(T ε 2),(75) where(x) + := max{x,0}. Using (75), we obtain X j∈V αjTij(1−W ij) = X j∈V X k∈V αjαk(Tik −T jk)+ +O(T ε 2), (7...