We study an optimal intervention problem on the linear threshold model (LTM) in which a social planner aims to design minimal-cost interventions that modify the agents' thresholds, under the constraint that at least a predefined fraction of agents reaches a given state after a finite number of iterations. While this problem is known to be NP-hard and its exact solution requires full knowledge of the network structure, we focus on approximate solutions for large-scale networks and assume that the planner has only statistical knowledge of the network. In particular, we build on a local mean-field approximation of the LTM that is known to hold true on large-scale random networks, and reformulate the optimal intervention problem as a linear program with an infinite set of constraints. We then show how to approximate the solutions of the latter problem by standard linear programs with finitely many constraints. Finally, our approach is validated through numerical experiments on real-world networks and compared both with optimal seeding and state-of-the-art algorithms for the least-cost influence.

翻译：我们研究了线性阈值模型上的最优干预问题，其中社会规划者旨在设计最小成本干预，通过调整代理人的阈值，使得在有限次迭代后至少达到预定比例的代理人达到给定状态。尽管已知该问题为NP难问题，且其精确解需要完整的网络结构知识，但我们聚焦于大规模网络的近似解，并假设规划者仅拥有网络的统计知识。具体地，我们基于线性阈值模型的局部平均场近似（该近似在大规模随机网络上已被证实成立），将最优干预问题重新表述为带有无限约束集的线性规划。随后，我们展示了如何通过带有有限约束的标准线性规划来近似该问题的解。最后，通过在真实世界网络上的数值实验验证了我们的方法，并与最优播种及最廉价影响力的最新算法进行了比较。