Undecided State Dynamics with Stubborn Agents

TL;DR

This paper studies a variant of the Undecided State Dynamics in opinion formation, revealing a phase transition influenced by stubbornness that determines whether agents reach consensus on Opinion 1 or 2.

Contribution

It introduces a model with stubborn agents in USD and characterizes the phase transition in consensus behavior based on stubbornness levels.

Findings

Consensus on Opinion 1 occurs when stubbornness exceeds a threshold.

Consensus on Opinion 2 occurs when stubbornness is below the threshold.

Near the threshold, opinions coexist with high probability.

Abstract

In the classical Approximate Majority problem with two opinions there are agents with Opinion 1 and with Opinion 2. The goal is to reach consensus and to agree on the majority opinion if the bias is sufficiently large. It is well known that the problem can be solved efficiently using the Undecided State Dynamics (USD) where an agent interacting with an agent of the opposite opinion becomes undecided. In this paper, we consider a variant of the USD with a preferred Opinion 1. That is, agents with Opinion 1 behave stubbornly -- they preserve their opinion with probability $p$ whenever they interact with an agent having Opinion 2. Our main result shows a phase transition around the stubbornness parameter $p \approx 1-x_1/x_2$. If $x_1 = \Theta(n)$ and $p \geq 1-x_1/x_2 + o(1)$, then all agents agree on Opinion 1 after $O(n\cdot \log n)$ interactions. On the other hand, for $p \leq 1-x_1/x_2 - o(1)$, all agents agree on Opinion 2, again after $O(n\cdot \log n)$ interactions. Finally, if $p \approx 1-x_1/x_2$, then all agents do agree on one opinion after $O(n\cdot \log^2 n)$ interactions, but either of the two opinions can survive. All our results hold with high probability.