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Phase transition of the 3-majority opinion dynamics with noisy interactions

Francesco d'Amore, Isabella Ziccardi

TL;DR

This work analyzes the phase transition behavior of the 3-Majority opinion dynamics under uniform communication noise on the complete graph with binary opinions. By modeling the system through the bias $s_t$ and deriving the mean-field update, the authors identify a sharp threshold at $p=1/3$ that separates rapid almost-consensus formation from rapid loss of consensus, and they characterize a metastable equilibrium $s_{\mathrm{eq}}$ that governs the dynamics for $p<1/3$. They establish precise time scales: logarithmic-time convergence to the metastable state and polynomial-time persistence within a stable interval, as well as symmetry-breaking results when the initial bias is small. The results are complemented by experiments on Erdős–Rényi and random-regular graphs, demonstrating robustness of the phase transition and highlighting the impact of topology on noise resilience. Overall, the paper advances understanding of non-linear, noise-tolerant consensus in bio-inspired and distributed systems, and contrasts the 3-Majority dynamics with the Undecided-State dynamics in terms of noise thresholds and resilience.

Abstract

Communication noise is a common feature in several real-world scenarios where systems of agents need to communicate in order to pursue some collective task. In particular, many biologically inspired systems that try to achieve agreements on some opinion must implement resilient dynamics that are not strongly affected by noisy communications. In this work, we study the popular 3-Majority dynamics, an opinion dynamics which has been proved to be an efficient protocol for the majority consensus problem, in which we introduce a simple feature of uniform communication noise, following (d'Amore et al. 2020). We prove that in the fully connected communication network of n agents and in the binary opinion case, the process induced by the 3-Majority dynamics exhibits a phase transition. For a noise probability $p < 1/3$, the dynamics reaches in logarithmic time an almost-consensus metastable phase which lasts for a polynomial number of rounds with high probability. Furthermore, departing from previous analyses, we further characterize this phase by showing that there exists an attractive equilibrium value $s_{\text{eq}} \in [n]$ for the bias of the system, i.e. the difference between the majority community size and the minority one. Moreover, the agreement opinion turns out to be the initial majority one if the bias towards it is of magnitude $Ω(\sqrt{n\log n})$ in the initial configuration. If, instead, $p > 1/3$, no form of consensus is possible, and any information regarding the initial majority opinion is lost in logarithmic time with high probability. Despite more communications per-round are allowed, the 3-Majority dynamics surprisingly turns out to be less resilient to noise than the Undecided-State dynamics (d'Amore et al. 2020), whose noise threshold value is $p = 1/2$.

Phase transition of the 3-majority opinion dynamics with noisy interactions

TL;DR

This work analyzes the phase transition behavior of the 3-Majority opinion dynamics under uniform communication noise on the complete graph with binary opinions. By modeling the system through the bias and deriving the mean-field update, the authors identify a sharp threshold at that separates rapid almost-consensus formation from rapid loss of consensus, and they characterize a metastable equilibrium that governs the dynamics for . They establish precise time scales: logarithmic-time convergence to the metastable state and polynomial-time persistence within a stable interval, as well as symmetry-breaking results when the initial bias is small. The results are complemented by experiments on Erdős–Rényi and random-regular graphs, demonstrating robustness of the phase transition and highlighting the impact of topology on noise resilience. Overall, the paper advances understanding of non-linear, noise-tolerant consensus in bio-inspired and distributed systems, and contrasts the 3-Majority dynamics with the Undecided-State dynamics in terms of noise thresholds and resilience.

Abstract

Communication noise is a common feature in several real-world scenarios where systems of agents need to communicate in order to pursue some collective task. In particular, many biologically inspired systems that try to achieve agreements on some opinion must implement resilient dynamics that are not strongly affected by noisy communications. In this work, we study the popular 3-Majority dynamics, an opinion dynamics which has been proved to be an efficient protocol for the majority consensus problem, in which we introduce a simple feature of uniform communication noise, following (d'Amore et al. 2020). We prove that in the fully connected communication network of n agents and in the binary opinion case, the process induced by the 3-Majority dynamics exhibits a phase transition. For a noise probability , the dynamics reaches in logarithmic time an almost-consensus metastable phase which lasts for a polynomial number of rounds with high probability. Furthermore, departing from previous analyses, we further characterize this phase by showing that there exists an attractive equilibrium value for the bias of the system, i.e. the difference between the majority community size and the minority one. Moreover, the agreement opinion turns out to be the initial majority one if the bias towards it is of magnitude in the initial configuration. If, instead, , no form of consensus is possible, and any information regarding the initial majority opinion is lost in logarithmic time with high probability. Despite more communications per-round are allowed, the 3-Majority dynamics surprisingly turns out to be less resilient to noise than the Undecided-State dynamics (d'Amore et al. 2020), whose noise threshold value is .
Paper Structure (14 sections, 20 theorems, 58 equations, 3 figures)

This paper contains 14 sections, 20 theorems, 58 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Our result
  3. Related Works
  4. Consensus problem.
  5. General consensus algorithms in the GOSSIP model.
  6. Consensus problem in the presence of noise or stubborn agents.
  7. Preliminaries
  8. Communication noise.
  9. Further notation.
  10. Victory of the majority
  11. Symmetry breaking
  12. Victory of noise
  13. Experiments
  14. Tools

Key Result

Theorem 1

Let $\{s_t\}_{t \geq 0}$ be the bias of the process induced by the 3-Majority dynamics with uniform noise probability $p$, and let $s_0$ the initial value of the bias. The following statements hold.

Figures (3)

  • Figure 1: Average convergence times to almost-consensus in s and Erdos-Rényi graphs. The average is computed over 1000 different runs, and the random underlying graph is sampled at each run. The shaded areas represent the sample standard deviations.
  • Figure 2: Average convergence times to almost-consensus in random regular graphs with degrees 5 and 3. The average is computed over 1000 different runs, and the random underlying graph is sampled at each run. The shaded areas represent the sample standard deviations.
  • Figure 3: Behaviors of the bias ratio $\left\lvert \frac{\text{bias}}{\text{size}} \right\rvert$ in different topologies and for different noise values over a single run of the dynamics. The noise parameter $p$ is chosen from set $\{1/8,1/6,1/5,1/4,5/12,1/2\}$. The dotted colored lines represent the bias ratio's equilibria in cliques, depending on the corresponding noise value. All graphs have $2^{16}$ nodes. For the sake of readability, we report here the values of the noise parameters for each colored line: red = 1/8, blue = 1/6, green = 1/5, purple = 1/4, gold = 5/12, brown = 1/2.

Theorems & Definitions (34)

  • Definition 1: Opinion dynamics -- informal
  • Theorem
  • Lemma 1
  • proof
  • Theorem 2: Victory of the majority
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • ...and 24 more