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 any protocol with constant sample size requires asymptotically an almost-linear number of rounds to converge, with high probability. 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：我们研究自稳定比特传播问题，旨在刻画具有最低认知与通信能力的实体在信息传播及达成共识过程中面临的挑战。具体而言，由$n$个智能体组成的群体需从两种备选观点中选择采纳正确观点——最初仅由单个知情个体持有。在决策过程中，智能体仅能观测随机抽取的少数个体的观点，既无进一步通信能力，也无法识别知情个体。此外，智能体不具备跨轮次保留任何信息的能力。根据SODA 2024年近期发表的研究，在并行更新场景（所有智能体同步更新）下，当日志采样数至少为$\Omega(\sqrt{n \log n})$时，可实现无记忆机制的对数级收敛时间。然而，确定高效协议所需的最小样本量仍是悬而未决的难题。作为初步探索，我们首次建立了该问题在并行场景下的下界：证明任何采用常数采样规模的协议，以高概率需要渐进近线性的轮次才能收敛。该下界在智能体已知精确群体规模$n$及自身观点的条件下依然成立，并涵盖了多种旨在达成共识的简单现有动力学机制。除比特传播问题外，我们的研究成果还揭示了著名多数决规则的对应——少数派动力学的收敛时间特性。尽管该算法形式简洁，但其混沌行为至今尚未被完全理解。