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Voting-based probabilistic consensuses and their applications in distributed ledgers

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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

  1. From the side of theoretical computer science.

  2. 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.

  3. 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.

  4. 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.

  5. 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.

  6. 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.

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

  8. Formally, this also holds for m ∈{0,n}.

  9. 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.

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

  11. 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.

  12. We had to introduce δn for that reason; for following the subsequent calculations in an easier way, one can just assume that \(\sqrt {n}\) is an integer so δn = 0.

  13. That is, up to factors of a smaller order.

  14. Up to a, typically bounded, multiplicative factor.

  15. 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.

  16. 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.

  17. “Try not! Do, or do not!” Ⓒ

  18. It might be really long, though.

  19. 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.”

  20. Or, at least, with positive probability the adversary is not able to predict it.

  21. Also known as a covert adversary, cf. [6].

  22. 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.

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

  24. 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.

  25. 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 m0 consecutive rounds.

  26. As usual, with random fluctuations typically of order k− 1/2.

  27. In other words, it may make sense that different parts of the system are decentralized to a different degree.

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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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Authors and Affiliations

  1. Centro de Matemática, University of Porto, Porto, Portugal

    Serguei Popov

  2. IOTA Foundation, 10405, Berlin, Germany

    Serguei Popov & Sebastian Müller

  3. Aix Marseille Université, CNRS, Centrale Marseille, I2M - UMR 7373, 13453, Marseille, France

    Sebastian Müller

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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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