Scaling Laws for Majority-based Opinion Dynamics in the Presence of Stubborn Agents

2026-08-11

AI Summary

This study investigates opinion dynamics under majority-vote rules in networks comprising both stubborn and non-stubborn agents. Employing a 2k-choices majority update rule coupled with probabilistic sampling, the authors analyze the diffusion process via Stein’s method and uncover a sharp phase transition in convergence time as a function of the fraction of stubborn agents. When this fraction is sufficiently large, the system converges in logarithmic time; however, when it is small and near a critical threshold, convergence time grows exponentially. Notably, at the phase transition boundary, the work establishes for the first time a diffusion approximation with polynomial mixing time, thereby characterizing diffusion-dominated dynamical behavior in this regime.

Abstract

In a multi-agent system, there are often stubborn followers of specific opinions or beliefs. Motivated by this observation, in this paper, we aim to understand how stubborn agents affect the distribution of opinions in a network where both stubborn and non-stubborn agents interact with each other. To do so, we assume that all agents have an opinion in the set $\{0,1\}$ and each non-stubborn agent updates its opinion according to the $2k$\textit{-choices rule}, where the agent samples $2k$ neighbours (including both stubborn and non-stubborn neighbours) uniformly at random and adopts the majority opinion among the sampled group of neighbours and itself. We assume that a proportion of agents, $γ_i$, are stubborn followers of opinion $i\in \{0,1\}$. It is natural to expect that the steady-state distribution of the opinions in the network will be dominated by the opinion with the larger proportion of stubborn followers. We show that while this is true, the time to reach steady-state depends heavily on the values of the parameters $γ_0$ and $γ_1$. When the individual values of these parameters, as well as their difference, are small, it can take an exponentially long time (in the network size) to reach the steady-state. In sharp contrast, when at least one of the parameters $γ_0$ and $γ_1$ is large, the network reaches the steady-state in a time that is only logarithmic in the network size. Hence, there exists a sharp phase transition in the network dynamics based on the proportions of stubborn agents. We also characterise the behaviour of the system when the parameters $γ_0$ and $γ_1$ lie on the boundary of the phase transition. In this boundary region, we show using Stein's method that the dynamics are driven by a diffusion process which takes polynomial time to mix.

Problem

• opinion dynamics
• stubborn agents
• scaling laws
• phase transition
• majority rule

Innovation

• scaling laws
• opinion dynamics
• stubborn agents
• phase transition
• Stein's method