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

TL;DR

This paper analyzes the phase transition behavior of the 3-Majority opinion dynamics under uniform communication noise, revealing a critical noise threshold at p=1/3 that determines whether consensus is achievable or lost.

Contribution

It introduces a detailed phase transition analysis of the noisy 3-Majority dynamics, including the characterization of metastable states and the impact of initial bias.

Findings

For p < 1/3, the system reaches an almost-consensus metastable phase in logarithmic time.

The system exhibits an attractive equilibrium bias value depending on initial conditions.

For p > 1/3, no consensus is possible, and initial information is lost rapidly.

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 $\Omega(\sqrt{n\log n})$ 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 $p = 1/2$.