We study the minority-opinion dynamics over a fully-connected network of $n$ nodes with binary opinions. Upon activation, a node receives a sample of opinions from a limited number of neighbors chosen uniformly at random. Each activated node then adopts the opinion that is least common within the received sample. Unlike all other known consensus dynamics, we prove that this elementary protocol behaves in dramatically different ways, depending on whether activations occur sequentially or in parallel. Specifically, we show that its expected consensus time is exponential in $n$ under asynchronous models, such as asynchronous GOSSIP. On the other hand, despite its chaotic nature, we show that it converges within $O(\log^2 n)$ rounds with high probability under synchronous models, such as synchronous GOSSIP. Finally, our results shed light on the bit-dissemination problem, that was previously introduced to model the spread of information in biological scenarios. Specifically, our analysis implies that the minority-opinion dynamics is the first stateless solution to this problem, in the parallel passive-communication setting, achieving convergence within a polylogarithmic number of rounds. This, together with a known lower bound for sequential stateless dynamics, implies a parallel-vs-sequential gap for this problem that is nearly quadratic in the number $n$ of nodes. This is in contrast to all known results for problems in this area, which exhibit a linear gap between the parallel and the sequential setting.

翻译：我们研究了在一个具有二元意见的完全连接网络（节点数为$n$）上的少数意见动力学。节点被激活时，会从均匀随机选择的有限数量邻居中获取意见样本，然后采用样本中出现最少的意见。与所有其他已知的共识动力学不同，我们证明这一基本协议在激活方式顺序执行或并行执行时表现出截然不同的行为。具体而言，我们证明在异步模型（如异步GOSSIP）下，其期望共识时间关于$n$呈指数增长；另一方面，尽管其具有混沌特性，我们证明在同步模型（如同步GOSSIP）下，它能在$O(\log^2 n)$轮内以高概率收敛。最后，我们的结果揭示了比特传播问题（此前被引入以模拟生物场景中的信息传播）的本质。具体而言，我们的分析表明，少数意见动力学是并行被动通信环境下首个实现该问题无状态解决方案的协议，能在多对数轮内收敛。结合已知的顺序无状态动力学下界，这暗示该问题在并行与顺序设置之间存在近$n^2$量级的差异。这与该领域所有已知问题结果形成鲜明对比——后者在并行与顺序设置间仅呈现线性差异。