标题： 带有顽固 agent 的噪声 DeGroot 模型中最能提供信息节点的最优选择 显示英文标题

标题： Optimal selection of the most informative nodes for a noisy DeGroot model with stubborn agents

摘要： 寻找社会中最佳的观测个体子集以获得平均意见的最佳估计，在政策制定、战略商业决策以及社会学趋势分析等诸多应用领域都是一个至关重要的问题。 我们假设意见向量 \( X \) 根据带有顽固个体的DeGroot意见动力学模型更新，并受到外部随机噪声扰动，可以将其解释为传输错误。 优化问题的目标函数是通过观察代理集合 \( K \) 的平衡意见所实现的方差减少量。 我们证明了在这种特定设定下，目标函数具有次模性。 这使我们能够有效地设计一种贪心算法来解决该问题，显著降低了其计算复杂度。 提供了简单示例以验证我们的结果。 显示英文摘要

摘要： Finding the optimal subset of individuals to observe in order to obtain the best estimate of the average opinion of a society is a crucial problem in a wide range of applications, including policy-making, strategic business decisions, and the analysis of sociological trends. We consider the opinion vector X to be updated according to a DeGroot opinion dynamical model with stubborn agents, subject to perturbations from external random noise, which can be interpreted as transmission errors. The objective function of the optimization problem is the variance reduction achieved by observing the equilibrium opinions of a subset K of agents. We demonstrate that, under this specific setting, the objective function exhibits the property of submodularity. This allows us to effectively design a Greedy Algorithm to solve the problem, significantly reducing its computational complexity. Simple examples are provided to validate our results.