Consensus Formation in Multi-state Majority and Plurality Models

TL;DR

This paper investigates how consensus forms in multi-state systems under majority and plurality rules, analyzing the dynamics and time scales in mean field and finite dimensions, revealing different spreading behaviors.

Contribution

It introduces a detailed analysis of multi-state consensus models with both majority and plurality rules, highlighting how parameters influence the dynamics and consensus time.

Findings

Consensus time scales as ln N in mean field for both rules.

Majority rule exhibits diffusive or ballistic spreading depending on parameters.

Plurality rule always shows diffusive domain coarsening toward consensus.

Abstract

We study consensus formation in interacting systems that evolve by multi-state majority rule and by plurality rule. In an update event, a group of G agents (with G odd), each endowed with an s-state spin variable, is specified. For majority rule, all group members adopt the local majority state; for plurality rule the group adopts the local plurality state. This update is repeated until a final consensus state is generally reached. In the mean field limit, the consensus time for an N-spin system increases as ln N for both majority and plurality rule, with an amplitude that depends on s and G. For finite spatial dimensions, domains undergo diffusive coarsening in majority rule when s or G is small. For larger s and G, opinions spread ballistically from the few groups with an initial local majority. For plurality rule, there is always diffusive domain coarsening toward consensus.