We investigate opinion dynamics in a fully-connected system, consisting of $n$ identical and anonymous agents, where one of the opinions (which is called correct) represents a piece of information to disseminate. In more detail, one source agent initially holds the correct opinion and remains with this opinion throughout the execution. The goal for non-source agents is to quickly agree on this correct opinion, and do that robustly, i.e., from any initial configuration. The system evolves in rounds. In each round, one agent chosen uniformly at random is activated: unless it is the source, the agent pulls the opinions of $\ell$ random agents and then updates its opinion according to some rule. We consider a restricted setting, in which agents have no memory and they only revise their opinions on the basis of those of the agents they currently sample. As restricted as it is, this setting encompasses very popular opinion dynamics, such as the voter model and best-of-$k$ majority rules. Qualitatively speaking, we show that lack of memory prevents efficient convergence. Specifically, we prove that no dynamics can achieve correct convergence in an expected number of steps that is sub-quadratic in $n$, even under a strong version of the model in which activated agents have complete access to the current configuration of the entire system, i.e., the case $\ell=n$. Conversely, we prove that the simple voter model (in which $\ell=1$) correctly solves the problem, while almost matching the aforementioned lower bound. These results suggest that, in contrast to symmetric consensus problems (that do not involve a notion of correct opinion), fast convergence on the correct opinion using stochastic opinion dynamics may indeed require the use of memory. This insight may reflect on natural information dissemination processes that rely on a few knowledgeable individuals.

翻译：我们研究了一个由$n$个全同匿名智能体构成的全连接系统中的意见动力学，其中一种意见（称为正确意见）代表待传播的信息片段。具体而言，一个源智能体初始持有正确意见，并在整个运行过程中始终保持该意见。非源智能体的目标是快速就这一正确意见达成共识，并且能够从任意初始配置出发鲁棒地实现该目标。系统按轮次演化。每轮中，均匀随机选择一个智能体被激活：除非该智能体是源，否则它会拉取$\ell$个随机智能体的意见，然后根据某种规则更新自身意见。我们考虑一个受限场景：智能体无记忆，仅依赖当前采样的其他智能体意见来修正自身观点。尽管存在限制，该场景仍涵盖了非常流行的意见动力学模型，如投票者模型和最佳-$k$多数投票规则。定性而言，我们证明缺乏记忆会阻碍高效收敛。具体而言，我们证明，即使在激活智能体可完全访问系统当前全局配置的强模型版本中（即$\ell=n$的情况），没有任何动力学能在期望步数$o(n^2)$内实现正确收敛。反之，我们证明简单的投票者模型（$\ell=1$）能正确解决问题，且几乎达到上述下界。这些结果表明，与对称性共识问题（不涉及正确意见概念）相比，使用随机意见动力学快速收敛到正确意见可能需要记忆机制。这一发现可能反映依赖于少数知情个体的自然信息传播过程的内在特性。