## Abstract

We review probabilistic models known as majority dynamics (also known as threshold voter models) and discuss their possible applications for achieving consensus in cryptocurrency systems. In particular, we show that using this approach in a straightforward way for practical consensus in a Byzantine setting can be problematic and requires extensive further research. We then discuss the Fast Probabilistic Consensus (FPC) protocol (Popov and Buchanan, J Parallel Distrib Comput 147:77–86, 2021), which circumvents the problems mentioned above by using external randomness.

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## Notes

From the side of theoretical computer science.

It is worth noting at this point that our discussion will sometimes switch from one particular model to another—after all, our intention is to consider a broad class of models and understand the related phenomena and challenges.

Basically, this is because these configurations are absorbing states due to the consistency property of the decision rule and the fact that the evolution of the system is not conservative.

In the following, for more clarity, we use the verb “to select” for the node selected to update its preference in the current round, and the verb “to choose” for the three randomly chosen peers whose current opinions it will use to make the decision.

That is, tracked at the time moments when nodes “do something”; those (random) time moments form an increasing sequence, which allows us to introduce that discrete-time process in a natural way.

One could try to argue that it would be possible to recover the nearest-neighbor property if, during the round, one updates the states of the nodes according to some pre-determined ordering; but that would destroy the Markov property.

Recall that, if \(\eta \sim \text {Binomial}(3,h)\), then \({\mathbb P}[\eta =0]=(1-h)^{3}\), \({\mathbb P}[\eta =1]=3(1-h)^{2}h\), \({\mathbb P}[\eta =2]=3(1-h)h^{2}\), \({\mathbb P}[\eta =3]=h^{3}\).

Formally, this also holds for

*m*∈{0,*n*}.Let us mention that this is only one possible definition of the potential; for example, in the classical papers on random walks in random environment, one would rather use the summation \({\sum }_{j=0}^{k-1}\) in Eq. 3. One can note, though, that these definitions lead essentially to similar objects that are used in similar ways.

We use the notation \(a\wedge b := \min \limits \{a,b\}\).

Even though it is possible to make a reasonable guess about how the terms in Eq. 10 would behave, writing it all rigorously is yet another story.

We had to introduce

*δ*_{n}for that reason; for following the subsequent calculations in an easier way, one can just assume that \(\sqrt {n}\) is an integer so*δ*_{n}= 0.That is, up to factors of a smaller order.

Up to a, typically bounded, multiplicative factor.

Here, the word “state” is used in a loose sense: it does not mean a particular point of the state space of the Markov chain, but rather a region close to a given local minimum of the potential.

For sure, complete omniscience is unrealistic; but, on the other hand, it is unclear how exactly to quantify the degree of knowledge that the adversary has about the network, and it is also unclear up to “which point” (whatever that means) it can gain that knowledge (now or in the future). However, it is clear that the adversary can obtain

*some*knowledge about the state of the network (by directly observing it and also maybe by doing some statistical analysis) and will try to do that in the case when the security depends on the network state’s obfuscation. Therefore, it is reasonable to assume omniscience to be on the safe side.“Try not! Do, or do not!” Ⓒ

It might be

*really*long, though.Also, using the technique of stochastic domination, it is even possible to show that the same holds for

*any*adversarial strategy, not only for “help the weakest.”Or, at least,

*with positive probability*the adversary is not able to predict it.Also known as a

*covert adversary*, cf. [6].Note that \(\frac {3-\sqrt {5}}{2}=\frac {1}{1+\phi }=\phi ^{-2}\approx 0.38\), where \(\phi =\frac {1+\sqrt {5}}{2}\) is the Golden Ratio.

This notation does not have anything to do with \({\hat p}_{m}\)s of Section 2.2.

One may think of Ψ as the first time the systems enters the “very steep” part of the potential, as explained in the previous section.

This gives rise to the second negative term in the right-hand side of Eq. 23; as before, it corresponds to the probability of Ψ not happening during

*m*_{0}consecutive rounds.As usual, with random fluctuations typically of order

*k*^{− 1/2}.In other words, it may make sense that different parts of the system are decentralized to a different degree.

## References

Abdullah MA, Draief M (2015) Global majority consensus by local majority polling on graphs of a given degree sequence. Discret Appl Math 180:1–10

Aguilera MK, Toueg S (2012) The correctness proof of Ben-Or’s randomized consensus algorithm. Distrib Comput 25(5):371–381

Aldous DJ (1982) Markov chains with almost exponential hitting times. Stoch Process Appl 13 (3):305–310

Aldous David J. , Brown M (1992) Inequalities for rare events in time-reversible Markov chains. I, volume 22 of Lecture Notes–Monograph Series, pp 1–16. Institute of Mathematical Statistics, Hayward, CA

Aldous DJ, Brown M (1993) Inequalities for rare events in time-reversible Markov chains II. Stoch Process Appl 44(1):15–25

Aumann Y, adversaries Yehuda Lindell. (2007) Security against covert adversaries: efficient protocols for realistic adversaries. In: Theory of cryptography conference. Springer, pp 137–156

Becchetti L, Clementi A, Natale E, Pasquale F, Trevisan L (2016) Stabilizing consensus with many opinions. In: Proceedings of the twenty-seventh annual ACM-SIAM symposium on Discrete algorithms. SIAM, pp 620–635

Belotti M, Božić N., Pujolle G (2019) S. Secci. A vademecum on blockchain technologies When, which, and how. IEEE Communications Surveys Tutorials 21(4):3796–3838

Ben-Or M (1983) Another advantage of free choice: completely asynchronous agreement protocols (extended abstract). In: Proceedings of the 2nd ACM Annual symposium on principles of distributed computing, Montreal, Quebec, pp 27–30

Benjamini I, Chan Siu-On, O’Donnell R, Tamuz O, Tan Li-Yang (2016) Convergence, unanimity and disagreement in majority dynamics on unimodular graphs and random graphs. Stoch Process Appl 126(9):2719–2733

Bovier A (2016) Metastability. A potentialtheoretic approach. Springer

Bovier A, Eckhoff M, Gayrard V, Klein M (2004) Metastability in reversible diffusion processes I Sharp asymptotics for capacities and exit times. J Eur Math Soc 6:399–424

Bracha G (1987) Asynchronous Byzantine agreement protocols. Inf Comput 75(2):130–143

Bramson M, Cox J, Le J-F, Gall T (2001) Super-Brownian limits of voter model clusters. Ann Probab 29(3):1001–1032

Capossele A, Mueller S, Penzkofer A (2019) Robustness and efficiency of leaderless probabilistic consensus protocols within Byzantine infrastructures. https://www.sciencedirect.com/science/article/pii/S2096720921000026

Cascudo I, David B (2017) Scrape: scalable randomness attested by public entities. In: International conference on applied cryptography and network security. Springer, pp 537–556

Cassandro M, Galves A, Olivieri E, Vares ME (June 1984) Metastable behavior of stochastic dynamics: a pathwise approach. J Stat Phys 35(5-6):603–634

Clifford P, Sudbury A (1973) A model for spatial conflict. Biometrika 60(3):581–588

Codd EF (1968) Cellular automata. Cambridge, Academic Press

Comets F, Popov S (2003) Limit law for transition probabilities and moderate deviations for Sinai’s random walk in random environment. Probab Theory Related Fields 126(4):571–609

Cooper C, Elsässer R., Radzik T (2014) The power of two choices in distributed voting. In: International colloquium on automata, languages, and programming. Springer, pp 435–446

Cooper C, Elsässer R., Radzik T, Rivera N, Shiraga T (2015) Fast consensus for voting on general expander graphs. In: International Symposium on Distributed Computing. Springer, pp 248–262

Theodore Cox J, Durrett R, Perkins Edwin A. (2000) Rescaled voter models converge to super-Brownian motion. Ann Probab., 28(1) 01:185–234

Theodore Cox J, Griffeath D (1983) Occupation time limit theorems for the voter model. Ann Probab 11(4):876–893

Cruise J, Ganesh A (2014) Probabilistic consensus via polling and majority rules. Queueing Syst Theory Appl 78(2):99–120

Dembo A, Guionnet A, Zeitouni O (2001) Aging properties of Sinai’s model of random walk in random environment. In: St. Flour Summer School 2001, Springer’s lecture notes in mathematics, vol 1837

Doerr B, Goldberg LA, Minder L, Sauerwald T, Scheideler C (2011) Stabilizing consensus with the power of two choices. In: Proceedings of the Twenty-Third Annual ACM Symposium on Parallelism in Algorithms and Architectures, SPAA ’11. Association for Computing Machinery, New York, pp 149–158

Durrett R (2010) Probability: theory and examples. Cambridge Series in Statistical and Probabilistic Mathematics, fourth. Cambridge University Press, Cambridge

Elsässer R., Friedetzky T, Kaaser D, Mallmann-Trenn F, Trinker H (2016) Rapid asynchronous plurality consensus. arXiv:1602.04667

Enriquez N, Sabot C, Zindy O (2009) Aging and quenched localization for one-dimensional random walks in random environment in the sub-ballistic regime. Bulletin de la Société Mathématique de France 137(3):423–452

Fanti G, Holden N, Peres Y, Ranade G (2020) Communication cost of consensus for nodes with limited memory. Proc Natl Acad Sci 117(11):5624–5630

Feldman P, Micali S (1989) An optimal probabilistic algorithm for synchronous Byzantine agreement. In: International Colloquium on Automata, Languages, and Programming. Springer, pp 341–378

Freidlin MI, Szücs J., Wentzell AD (2012) Random perturbations of dynamical systems grundlehren der mathematischen wissenschaften. Springer, Berlin

Friedman R, Mostefaoui A, Raynal M (2005) Simple and efficient oracle-based consensus protocols for asynchronous Byzantine systems. IEEE Trans Dependable Secure Comput 2(1):46– 56

Gács P., Kurdyumov GL, Levin LA (1978) One-dimensional uniform arrays that wash out finite islands. In: Problemy Peredachi Informatsii

Gärtner B., Zehmakan AN (2018) Majority model on random regular graphs. Lect Notes Comput Sci: 572–583

Gogolev A, Marchenko N, Marcenaro L, Bettstetter C (2015) Distributed binary consensus in networks with disturbances. ACM Trans Auton Adapt Syst 10(3):19:1–19:17

Gutierrez A, Müller S., Šebek S (2021) On asymptotic fairness in voting consensus protocols

Holley RA, Liggett TM (1975) Ergodic theorems for weakly interacting infinite systems and the voter model. The Annals of Probability 3(4):643–663

Jedrzejewski A, Sznajd-Weron K (2019) Statistical physics of opinion formation Is it a spoof? Comptes Rendus Physique 20(4):244–261

Kar S, Moura JMF (2007) Distributed average consensus in sensor networks with random link failures. In: 2007 IEEE International Conference on Acoustics, Speech and Signal Processing - ICASSP ’07, vol 2, pp II–1013–II–1016

Keilson J (1980) Markov chain models – rarity and exponentiality. Applied Mathematical Sciences 28. Berlin-Heidelberg-New York, Springer–Verlag 1979. ZAMM - J Appl Math Mech / Zeitschrift fü,r Angewandte Mathematik und Mechanik 60(5):272–272

Kurtz TG, Swanson J (2021) Finite Markov chains coupled to general Markov processes and an application to metastability I

Lenstra AK, Wesolowski B (2017) Trustworthy public randomness with sloth, unicorn, and trx. International Journal of Applied Cryptography 3(4):330–343

Thomas M (2012) Liggett. Interacting particle systems, vol 276. Springer Science & Business Media

Manzo F, Quattropani M, Scoppola E, Quattropani M, Scoppola E. (2021) A probabilistic proof of Cooper and Frieze’s. “First Visit Time Lemma”

Manzo F, Scoppola E (2019) Exact results on the first hitting via conditional strong quasi-stationary times and applications to metastability. J Stat Phys 174:1239–1262

Menshikov M, Popov S, Wade A (2017) Non-homogeneous random walks – Lyapunov function methods for near-critical stochastic systems. Cambridge University Press, Cambridge

Moreira AA, Mathur A, Diermeier D, Amaral LAN (2004) Efficient system-wide coordination in noisy environments. Proc Natl Acad Sci USA 101:12085–12090

Mossel E, Neeman J, Tamuz O (2013) Majority dynamics and aggregation of information in social networks. Auton Agent Multi-Agent Syst 28(3):408–429

Müller S, Penzkofer A, Camargo D, Saa O On fairness in voting consensus protocols. Computing Conference 21, to appear

Müller S., Penzkofer A, Kuśmierz B., Camargo D, weighted William J. Buchanan. Fast Probabilistic Consensus with weighted votes. FTC 2020. Advances in intelligent systems and computing, vol 1289, 2

Nakomoto S (2008) Bitcoin: A peer-to-peer electronic cash system. https://bitcoin.org/bitcoin.pdf

NKN L. (2018) NKN: a scalable self-evolving and self-incentivized decentralized network

Nyczka P, Cisło J., Sznajd-Weron K (2012) Opinion dynamics as a movement in a bistable potential. Physica A: Stat Mech Appl 391(1):317–327

Olivieri E, Vares ME (2005) Large deviations and metastability: Encyclopedia of mathematics and its applications. Cambridge University Press, Cambridge

Penrose O, Lebowitz JL (1971) Rigorous treatment of metastable states in the van der Waals-Maxwell theory. J Stat Phys 3(2):211–236

Popov S (2017) On a decentralized trustless pseudo-random number generation algorithm. J Math Crypt 11(1):37–43

Popov S (2021) Two-dimensional random walk – from path counting to random interlacements. Cambridge University Press, Cambridge

Popov S, Buchanan WJ (2021) FPC-BI Fast probabilistic consensus within Byzantine infrastructures. J Parallel Distrib Comput 147:77–86

Presutti E, Spohn H (1983) Hydrodynamics of the voter model. Ann Probab 11(4):867–875

Rabin MO (1983) Randomized Byzantine generals. In: 24th Annual symposium on foundations of computer science (sfcs 1983). IEEE, pp 403–409

Redner S (2019) Reality-inspired voter models: a mini-review. Comptes Rendus Physique 20 (4):275–292

Rocket T. (2018) Snowflake to avalanche: a novel metastable consensus protocol family for cryptocurrencies

Rocket T, Yin M, Sekniqi K, van Renesse R, Sirer EG (2019) Scalable and probabilistic leaderless BFT consensus through metastability

Ross SM (2009) Introduction to probability models, 10th edn. Academic Press, Cambridge

Schindler P, Judmayer A, Stifter N, Weippl E (2018) Hydrand: Practical continuous distributed randomness. Cryptology ePrint Archive, Report 2018/319. https://eprint.iacr.org/2018/319

Sinaj YG (1982) Limit behavior of the one-dimensional random walks in random environments. Teor Veroyatn Primen 27:247–258

Sowers RB (1999) Hydrodynamical limits and geometric measure theory: Mean curvature limits from a threshold voter model. J Funct Anal 169(2):421–455

Syta E, Jovanovic P, Kogias EK, Gailly N, Gasser L, Khoffi I, Fischer MJ, Ford B (2017) Scalable bias-resistant distributed randomness. In: 2017 IEEE Symposium on Security and Privacy (SP). IEEE, pp 444–460

Tanaka-Yamawaki M, Kitamikado S, Fukuda T (1996) Consensus formation and the cellular automata. Robot Auton Syst 19(1):15–22

Tran LV, Vu V (2020) Reaching a consensus on random networks: the power of few. In: Byrka J, Meka R (eds) Approximation, randomization, and combinatorial optimization. Algorithms and techniques, APPROX/RANDOM 2020, August 17-19, 2020, Virtual Conference, volume 176 of LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, pp 20:1–20:15

Wesolowski B (2020) Efficient verifiable delay functions. J Cryptol 33:2113–2147

Wolfram S (1994) Cellular automata and complexity: collected papers. Westview Press, Boulder

## Acknowledgements

The authors thank Olivia Saa for obtaining the solution of Eq. 18, Alexandre Reiffers-Masson and Yao-Hua Xu for suggesting several corrections to the arguments in Section 2.1, and David Phillips for helping us to improve the writing style. We also thank the anonymous referees for their useful comments and suggestions.

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Popov, S., Müller, S. Voting-based probabilistic consensuses and their applications in distributed ledgers.
*Ann. Telecommun.* **77**, 77–99 (2022). https://doi.org/10.1007/s12243-021-00875-7

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DOI: https://doi.org/10.1007/s12243-021-00875-7