We address the self-stabilizing bit-dissemination problem, designed to capture the challenges of spreading information and reaching consensus among entities with minimal cognitive and communication capacities. Specifically, a group of $n$ agents is required to adopt the correct opinion, initially held by a single informed individual, choosing from two possible opinions. In order to make decisions, agents are restricted to observing the opinions of a few randomly sampled agents, and lack the ability to communicate further and to identify the informed individual. Additionally, agents cannot retain any information from one round to the next. According to a recent publication in SODA (2024), a logarithmic convergence time without memory is achievable in the parallel setting (where agents are updated simultaneously), as long as the number of samples is at least $\Omega(\sqrt{n \log n})$. However, determining the minimal sample size for an efficient protocol to exist remains a challenging open question. As a preliminary step towards an answer, we establish the first lower bound for this problem in the parallel setting. Specifically, we demonstrate that it is impossible for any memory-less protocol with constant sample size, to converge with high probability in less than an almost-linear number of rounds. This lower bound holds even when agents are aware of both the exact value of $n$ and their own opinion, and encompasses various simple existing dynamics designed to achieve consensus. Beyond the bit-dissemination problem, our result sheds light on the convergence time of the "minority" dynamics, the counterpart of the well-known majority rule, whose chaotic behavior is yet to be fully understood despite the apparent simplicity of the algorithm.

翻译：我们研究了自稳定比特传播问题，该问题旨在刻画在认知与通信能力极为有限的实体间传播信息并达成共识所面临的挑战。具体而言，一组由n个智能体构成的群体需要从两个可能意见中采纳正确观点，该观点最初仅由单个知情个体持有。在决策过程中，智能体仅能观察随机采样的少数几个智能体的意见，缺乏进一步通信的能力，也无法识别知情个体。此外，智能体无法在轮次间保留任何信息。根据SODA（2024）的最新研究，在并行设置（即智能体同步更新）下，只要采样数量至少达到Ω(√(n log n))，无需记忆即可实现对数级收敛时间。然而，确定存在高效协议的最小样本量仍是一个具有挑战性的开放问题。作为回答该问题的初步步骤，我们建立了该问题在并行设置下的首个下界。具体而言，我们证明：任何具有常数采样规模的无记忆协议均无法在亚近线性轮数内以高概率收敛。该下界即使在智能体知晓n的精确值及自身意见的情况下依然成立，并涵盖了多种为达成共识而设计的简单现有动力学机制。除比特传播问题外，我们的结果还为“少数派”动力学（即著名的多数规则的对应物）的收敛时间提供了新见解——尽管该算法表面简单，但其混沌行为尚未被完全理解。