A note on the consensus time of mean-field majority-rule dynamics

TL;DR

This paper investigates how the time to reach consensus in mean-field majority-rule opinion dynamics depends on population size, revealing two regimes based on the group size distribution, supported by analytical and numerical evidence.

Contribution

It identifies two distinct regimes of consensus time dependence in mean-field majority-rule models based on group size distribution, extending prior understanding.

Findings

Logarithmic consensus time when group size distribution has finite mean.

Power-law relation between consensus time and population size when mean group size diverges.

Numerical simulations confirm analytical predictions.

Abstract

In this work, it is pointed out that in the mean-field version of majority-rule opinion dynamics, the dependence of the consensus time on the population size exhibits two regimes. This is determined by the size distribution of the groups that, at each evolution step, gather to reach agreement. When the group size distribution has a finite mean value, the previously known logarithmic dependence on the population size holds. On the other hand, when the mean group size diverges, the consensus time and the population size are related through a power law. Numerical simulations validate this semi-quantitative analytical prediction.