Undecided State Dynamics with Stubborn Agents

TL;DR

This work studies consensus in population protocols under a biased undecided-state dynamics with a stubborn Opinion 1, identifying a phase transition at $p_s = 1 - x_1/x_2$ that governs which opinion ultimately wins. The authors introduce a weighted bias $\Delta_w(t)$ and employ martingale methods (Azuma) and drift theorems to obtain sharp convergence-time bounds: $O(n \log n)$ when $p$ deviates from $p_s$ by $\Omega(\sqrt{(\log n)/n})$, and $O(n \log^2 n)$ in the critical window where either opinion may survive. A coupling/monotonicity framework is developed to compare configurations and transfer high-probability results across different $p$ values and initial states. These results advance understanding of robustness and resilience in consensus processes under minority stubbornness, with potential implications for designing fault-tolerant distributed systems. The work also opens questions about tightening the near-threshold runtime and extending the analysis to more complex bias and multi-opinion settings.

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 $x_1 = Θ(n)$ 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.

Figures (2)

• Figure 1: Overview over the results (not to scale). The x-coordinate indicates $x_1$ from $1$ to $x_2 + o(n)$ -- we omit larger values of $x_1$. The y-coordinate indicates the stubbornness $p$ from $0$ to $1$. The black diagonal line represents $p=p_s$ and the dashed blue lines around it represent $p = p_s \pm \Theta(\sqrt{n^{-1} \cdot \log n})$. At $p=0$, they coincide with the known bounds from DBLP:conf/dna/CondonHKM17: Opinion 1 wins w.h.p. for $x_1 \geq x_2 + \Omega(\sqrt{n\log n})$, Opinion 2 wins w.h.p. for $x_1 \leq x_2 - \Omega(\sqrt{n\log n})$. The inner opaque rectangular area corresponds to \ref{['thm:main-theorem']}. The remaining colored areas correspond to \ref{['thm:coupling-theorem']}. The area colored in green represents where Opinion 1 wins w.h.p. in $O(n\log n)$ interactions. The area colored in red represents where Opinion 2 wins w.h.p. in $O(n\log n)$ interactions. In the blue area, either Opinion wins w.h.p. in $O(n\log^2n)$ interactions.
• Figure 2: An overview over all possible interactions. All but one interaction have a deterministic outcome which we indicated by $\mapsto_1$. The interaction $(1,2)$ produces Opinion $1$ with probability $p$ and $\bot$ with $1-p$. We indicated this by $\mapsto_p \set{1,\bot}$.

Theorems & Definitions (35)

• theorem 1
• theorem 2
• lemma 1
• proof
• lemma 2
• proof
• lemma 3
• proof
• proof : Proof of Statement \ref{['thm:main-theorem:1']}
• lemma 4