Effects of free will and massive opinion in majority rule model

TL;DR

This paper investigates how free will and mass opinion influence the dynamics of majority rule models in multi-agent systems, revealing phase transition shifts and thresholds for consensus.

Contribution

It introduces two models with different opinion-updating dynamics, analyzing how free will and mass opinion affect phase transitions and consensus in majority rule systems.

Findings

Phase transition points can shift depending on opinion dynamics.

A finite population has a threshold for free will below which consensus is achieved.

Analytical results agree with simulations.

Abstract

We study the effects of free will and massive opinion of multi-agents in a majority rule model wherein the competition of the two types of opinions is taken into account. To address this issue, we consider two specific models (model I and model II) involving different opinion-updating dynamics. During the opinion-updating process, the agents either interact with their neighbors under a majority rule with probability $1-q$, or make their own decisions with free will (model I) or according to the massive opinion (model II) with probability $q$. We investigate the difference of the average numbers of the two opinions as a function of $q$ in the steady state. We find that the location of the order-disorder phase transition point may be shifted according to the involved dynamics, giving rise to either smooth or harsh conditions to achieve an ordered state. For the practical case with a finite population size, we conclude that there always exists a threshold for $q$ below which a full consensus phase emerges. Our analytical estimations are in good agreement with simulation results.