Finite-size effects on the convergence time in continuous-opinion dynamics

TL;DR

This paper investigates how the size of a population affects the time it takes for opinions to reach consensus in a continuous-opinion model, revealing different behaviors on various network structures.

Contribution

It demonstrates that finite-size effects on convergence time depend on network topology and provides a mean-field analysis for complete graphs.

Findings

Convergence time increases with system size on lattice networks.

Convergence time is size-independent on certain complex networks unless opinions are copied perfectly.

Mean-field analysis explains the convergence behavior on complete graphs.

Abstract

We study finite-size effects on the convergence time in a continuous-opinion dynamics model. In the model, each individual's opinion is represented by a real number on a finite interval, e.g., $[0,1]$, and a uniformly randomly chosen individual updates its opinion by partially mimicking the opinion of a uniformly randomly chosen neighbor. We numerically find that the characteristic time to the convergence increases as the system size increases according to a particular functional form in the case of lattice networks. In contrast, unless the individuals perfectly copy the opinion of their neighbors in each opinion updating, the convergence time is approximately independent of the system size in the case of regular random graphs, uncorrelated scale-free networks, and complete graphs. We also provide a mean-field analysis of the model to understand the case of the complete graph.