Social Influencing and Associated Random Walk Models: Asymptotic Consensus Times on the Complete Graph

TL;DR

This paper analyzes how consensus forms in agent-based models on complete graphs, using a random walk approach to derive formulas for consensus times and exploring effects of external influences and committed agents.

Contribution

It introduces a coarse-graining method to analyze consensus dynamics via a random walk framework, providing general equations and asymptotic solutions for the naming game.

Findings

Derived formulas for expected consensus times.

Analyzed the naming game with external field influence.

Examined the impact of committed agents on consensus.

Abstract

We investigate consensus formation and the asymptotic consensus times in stylized individual- or agent-based models, in which global agreement is achieved through pairwise negotiations with or without a bias. Considering a class of individual-based models on finite complete graphs, we introduce a coarse-graining approach (lumping microscopic variables into macrostates) to analyze the ordering dynamics in an associated random-walk framework. Within this framework, yielding a linear system, we derive general equations for the expected consensus time and the expected time spent in each macro-state. Further, we present the asymptotic solutions of the 2-word naming game, and separately discuss its behavior under the influence of an external field and with the introduction of committed agents.