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 price of uncertainty, a metric defined by Balcan et al. to quantify the effect of uncertainty in network games.

翻译：在不确定性条件下，社会网络达成共识的难度有多大？本文将该问题建模为智能体构成的社会图，其中每个顶点初始被标记为红色或蓝色。智能体的目标是达成共识，即所有智能体的颜色保持一致。智能体通过局部步骤尝试实现这一目标：在每一步中，智能体将其颜色更改为其邻居中多数节点的颜色。在现实场景中，智能体可能无法精确获知其邻居中红色或蓝色节点的具体数量，这为共识过程引入了不确定性。通过将不确定性建模为对颜色邻居计数的相对幅度 $1+\varepsilon$ 扰动，我们证明即使 $\varepsilon$ 取值很小，也会显著阻碍社会网络达成共识的能力。我们为不确定性代价——这一由 Balcan 等人提出用于量化网络博弈中不确定性影响的度量指标——给出了理论上的紧致上下界。