# Three-state opinion dynamics in modular networks

## TL;DR

This paper investigates opinion dynamics in modular networks with community interactions, negative and positive influences, inflexible agents, and indecision noise, revealing complex phase transitions and nonmonotonic ordering behaviors.

## Contribution

It introduces a coupled mean-field approximation capturing complex opinion transition phenomena in community-structured networks with various interaction types.

## Key findings

- Identifies continuous and discontinuous phase transitions in opinion states.
- Shows nonmonotonic global ordering as indecision noise increases.
- Demonstrates the dual role of neutrality in minority opinion dominance.

## Abstract

In this work we study the opinion evolution in a community-based population with intergroup interactions. We address two issues. First, we consider that such intergroup interactions can be negative with some probability $p$. We develop a coupled mean-field approximation that still preserves the community structure and it is able to capture the richness of the results arising from our Monte Carlo simulations: continuous and discontinuous order-disorder transitions as well as nonmonotonic ordering for an intermediate community strength. In the second part, we consider only positive interactions, but with the presence of inflexible agents holding a minority opinion. We also consider an indecision noise: a probability $q$ that allows the spontaneous change of opinions to the neutral state. Our results show that the modular structure leads to a nonmonotonic global ordering as $q$ increases. This inclination toward neutrality plays a dual role: a moderated propensity to neutrality helps the initial minority to become a majority, but this noise-driven opinion switching becomes less pronounced if the agents are too susceptible to become neutral.