This paper considers a social network modeled as an Erdos Renyi random graph. Each individual in the network updates her opinion using the weighted average of the opinions of her neighbors. We explore how an external manipulative agent can drive the opinions of these individuals to a desired state with a limited additive influence on their innate opinions. We show that the manipulative agent can steer the network opinion to any arbitrary value in finite time (i.e., the system is controllable) almost surely when there is no restriction on her influence. However, when the control input is sparsity constrained, the network opinion is controllable with some probability. We lower bound this probability using the concentration properties of random vectors based on the Levy concentration function and small ball probabilities. Further, through numerical simulations, we compare the probability of controllability in Erdos Renyi graphs with that of power-law graphs to illustrate the key differences between the two models in terms of controllability. Our theoretical and numerical results shed light on how the controllability of the network opinion depends on the parameters such as the size and the connectivity of the network, and the sparsity constraints faced by the manipulative agent.

中文翻译：

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使用稀疏控制输入的 Erdos-Renyi 图中网络意见的可控性

本文考虑建模为 Erdos Renyi 随机图的社交网络。网络中的每个人都使用邻居意见的加权平均值来更新自己的意见。我们探索外部操纵代理如何将这些人的意见驱动到理想状态，而对其先天意见的附加影响有限。我们表明，操纵代理几乎可以肯定地在有限时间内将网络意见引导到任意值（即系统是可控的），而她的影响力不受限制。然而，当控制输入是稀疏约束时，网络意见是可控的，有一定的概率。我们使用基于 Levy 集中函数和小球概率的随机向量的集中属性来降低这个概率。更多，通过数值模拟，我们将鄂尔多斯仁义图的可控概率与幂律图的可控概率进行比较，以说明两种模型在可控性方面的主要区别。我们的理论和数值结果阐明了网络意见的可控性如何取决于网络的大小和连通性等参数，以及操纵代理面临的稀疏约束。