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DéjàVu: A Minimalistic Mechanism for Distributed Plurality Consensus

Francesco d'Amore, Niccolò D'Archivio, George Giakkoupis, Frédéric Giroire, Emanuele Natale

Abstract

We study the plurality consensus problem in distributed systems where a population of extremely simple agents, each initially holding one of k opinions, aims to agree on the initially most frequent one. In this setting, h-majority is arguably the simplest and most studied protocol, in which each agent samples the opinion of h neighbors uniformly at random and updates its opinion to the most frequent value in the sample. We propose a new, extremely simple mechanism called DéjàVu: an agent queries neighbors until it encounters an opinion for the second time, at which point it updates its own opinion to the duplicate value. This rule does not require agents to maintain counters or estimate frequencies, nor to choose any parameter (such as a sample size h); it relies solely on the primitive ability to detect repetition. We provide a rigorous analysis of DéjàVu that relies on several technical ideas of independent interest and demonstrates that it is competitive with h-majority and, in some regimes, substantially more communication-efficient, thus yielding a powerful primitive for plurality consensus.

DéjàVu: A Minimalistic Mechanism for Distributed Plurality Consensus

Abstract

We study the plurality consensus problem in distributed systems where a population of extremely simple agents, each initially holding one of k opinions, aims to agree on the initially most frequent one. In this setting, h-majority is arguably the simplest and most studied protocol, in which each agent samples the opinion of h neighbors uniformly at random and updates its opinion to the most frequent value in the sample. We propose a new, extremely simple mechanism called DéjàVu: an agent queries neighbors until it encounters an opinion for the second time, at which point it updates its own opinion to the duplicate value. This rule does not require agents to maintain counters or estimate frequencies, nor to choose any parameter (such as a sample size h); it relies solely on the primitive ability to detect repetition. We provide a rigorous analysis of DéjàVu that relies on several technical ideas of independent interest and demonstrates that it is competitive with h-majority and, in some regimes, substantially more communication-efficient, thus yielding a powerful primitive for plurality consensus.

Paper Structure

This paper contains 30 sections, 48 theorems, 288 equations.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Roadmap.
  4. Main Technical Ideas
  5. Poisson race in the $\mathcal{PULL}^*$ model for bias amplification.
  6. From $\mathcal{PULL}^*$ to $\mathcal{PULL}$($h$).
  7. Expected amplification of the bias in $\mathcal{PULL}$($h$).
  8. Amplification of the bias in concentration.
  9. Lower bound for $h$-Majority.
  10. Number of samples.
  11. Related Work
  12. Comparison of DéjàVu with h-majority
  13. Comparison of DéjàVu with 2-Choices
  14. Preliminaries
  15. Amplification bias Dejavu in PULL*
  16. ...and 15 more sections

Key Result

Theorem 1

Let $h\geq 2$ and $C=(C_1,\ldots,C_k)$ be an initial system configuration where each agent supports an opinion in $\{1,\ldots,k\}$, with $C_1\geq C_2 \geq \cdots \geq C_k$. Assume that $C_1 = \omega\left( \log n \right)$ and that, for a large enough constant $\lambda > 0$, DéjàVu converges to consensus on the first opinion w.h.p. in $O\left( \qty(\frac{n}{(h^2 C_1)}+1) \log n \right)$ rounds. $\

Theorems & Definitions (100)

  • Theorem 1
  • Theorem 2: Generalization of Theorem 4.12 in becchettiSimpleDynamicsPlurality2017
  • Theorem 3
  • Lemma 4
  • Theorem 5
  • proof
  • proof : Proof of \ref{['lemma:lower_upper_ratio_probability']}
  • Lemma 6
  • Claim 7
  • proof
  • ...and 90 more