How hard is it to achieve consensus in a social network under uncertainty? In this paper we model this problem as a social graph of agents where each vertex is initially colored red or blue. The goal of the agents is to achieve consensus, which is when the colors of all agents align. Agents attempt to do this locally through steps in which an agent changes their color to the color of the majority of their neighbors. In real life, agents may not know exactly how many of their neighbors are red or blue, which introduces uncertainty into this process. Modeling uncertainty as perturbations of relative magnitude $1+\varepsilon$ to these color neighbor counts, we show that even small values of $\varepsilon$ greatly hinder the ability to achieve consensus in a social network. We prove theoretically tight upper and lower bounds on the \emph{price of uncertainty}, a metric defined in previous work by Balcan et al. to quantify the effect of uncertainty in network games.

翻译：在不确定性条件下，社会网络达成共识究竟有多困难？本文将这一问题建模为一个社会图，其中每个顶点初始标记为红色或蓝色。所有智能体的目标是达成共识，即所有智能体的颜色趋于一致。智能体通过局部交互尝试实现这一目标：每个智能体将其颜色改为邻居中多数派的颜色。然而在现实中，智能体可能无法精确获知邻居中红色或蓝色的数量，这为共识过程引入了不确定性。我们将不确定性建模为邻居颜色计数上的相对扰动$1+\varepsilon$，并证明即使极小的$\varepsilon$值也会显著阻碍社会网络达成共识的能力。我们给出了关于"不确定性代价"的理论紧确上下界，该指标由Balcan等人先前的工作定义，用于量化网络博弈中不确定性的影响。