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

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. Indeed, many biologically inspired systems that try to achieve agreements on some opinion must implement resilient dynamics, i.e. that are not strongly affected by noisy communications. In this work, we study the 3-MAJORITY dynamics, an opinion dynamics that has been shown 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. (2022). 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 reach in logarithmic time an almost-consensus metastable phase which lasts for a polynomial number of rounds with high probability. We characterize this phase by showing that there exists an attractive equilibrium value s eq is an element of [n] for the bias of the system, i.e. the difference between the majority community size and the minority one. Moreover, we show that t he agreement opinion is the initial majority one root if the bias towards it is of magnitude Omega(root nlog n) in the initial configuration. If, instead, p > 1/3, we show that 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 being allowed, the 3-MAJORITY dynamics surprisingly turns out to be less resilient to noise than the UNDECIDED-STATE dynamics, whose noise threshold value is p = 1/2.