🤖 AI Summary
本文研究了在时间变化的意见网络中，如何通过选择少量代理和标量干预来实现资源有限的外部影响。使用仿射自由响应模型，并提出了一种精确稀疏一步最优控制方法。
📝 Abstract
This paper studies resource-limited external influence in time-varying opinion networks when a controller must choose both a small set of agents and a scalar intervention at each update. We use the affine free response, which includes DeGroot and Friedkin--Johnsen dynamics, followed by a direct sparse action. Eliminating the scalar action reduces the one-step problem to a cardinality-constrained support objective whose global optimum is obtained from the largest or smallest residual components. Hence exact sparse one-step-ahead optimal control requires one sort and \(O(|\mathcal T|\log|\mathcal T|)\) operations, where \(\mathcal T\) is the support set.
For periodic affine dynamics, the resulting state-dependent low-rank feedback admits cycle-based ultimate bounds and exact tracking of model-invariant targets. The DeGroot and Friedkin--Johnsen specializations expose the role of target invariance and the affine mismatch term. With several competing external players, each sparse best response retains the same sorting structure and the complete binary--continuous stage game is an exact potential game, so a pure-strategy equilibrium exists at every frozen state. Same-state welfare benchmarks separate loss due to scalar competition from the additional loss due to strategic support selection. Numerical examples validate exact support selection and illustrate switching-order effects, DeGroot/Friedkin--Johnsen tracking, resource tradeoffs, and competitive implementations.
