Divide-and-rule policy in the Naming Game

TL;DR

This paper extends the Naming Game model to multiple opinions, analyzing how a dominant opinion emerges through phase transitions in large populations and random networks, using mean-field theory and simulations.

Contribution

It introduces a recursive formula for multi-opinion evolution and explores the critical threshold for opinion dominance in complex networks.

Findings

Largest committed group triggers phase transition

Critical size leads to opinion dominance

Recursive formula enables analysis of any scenario

Abstract

The Naming Game is a classic model for studying the emergence and evolution of language within a population. In this paper, we extend the traditional Naming Game model to encompass multiple committed opinions and investigate the system dynamics on the complete graph with an arbitrarily large population and random networks of finite size. For the fully connected complete graph, the homogeneous mixing condition enables us to use mean-field theory to analyze the opinion evolution of the system. However, when the number of opinions increases, the number of variables describing the system grows exponentially. To mitigate this, we focus on a special scenario where the largest group of committed agents competes with a motley of committed groups, each of which is smaller than the largest one, while initially, most of uncommitted agents hold one unique opinion. This scenario is chosen for its recurrence in diverse societies and its potential for complexity reduction by unifying agents from smaller committed groups into one category. Our investigation reveals that when the size of the largest committed group reaches the critical threshold, most of uncommitted agents change their beliefs to this opinion, triggering a phase transition. Further, we derive the general formula for the multi-opinion evolution using a recursive approach, enabling investigation into any scenario. Finally, we employ agent-based simulations to reveal the opinion evolution and dominance transition in random graphs. Our results provide insights into the conditions under which the dominant opinion emerges in a population and the factors that influence these conditions.