Mean-field control (MFC) offers a scalable solution to the curse of dimensionality in multi-agent systems but traditionally hinges on the restrictive assumption of exchangeability via dense, all-to-all interactions. In this work, we bridge the gap to real-world network structures by proposing a rigorous framework for MFC on large sparse graphs. We redefine the system state as a probability measure over decorated rooted neighborhoods, effectively capturing local heterogeneity. Our central contribution is a theoretical foundation for scalable reinforcement learning in this setting. We prove horizon-dependent locality: for finite-horizon problems, an agent's optimal policy at time t depends strictly on its (T-t)-hop neighborhood. This result renders the infinite-dimensional control problem tractable and underpins a novel Dynamic Programming Principle (DPP) on the lifted space of neighborhood distributions. Furthermore, we formally and experimentally justify the use of Graph Neural Networks (GNNs) for actor-critic algorithms in this context. Our framework naturally recovers classical MFC as a degenerate case while enabling efficient, theoretically grounded control on complex sparse topologies.

翻译：平均场控制（MFC）为多智能体系统中的维度灾难问题提供了可扩展的解决方案，但传统方法依赖于通过密集的全连接交互实现可交换性这一严格假设。本文通过提出适用于大规模稀疏图的严格MFC框架，弥合了理论与现实网络结构之间的鸿沟。我们将系统状态重新定义为带标记的根邻域上的概率测度，从而有效捕捉局部异质性。本工作的核心贡献在于为此场景下的可扩展强化学习建立了理论基础。我们证明了与时间范围相关的局部性：对于有限时间范围问题，智能体在时刻 t 的最优策略严格依赖于其 (T-t) 跳邻域。这一结果使得无限维控制问题变得可处理，并支撑了在邻域分布提升空间上的新型动态规划原理。此外，我们从理论和实验上论证了在此背景下使用图神经网络（GNNs）实现演员-评论员算法的合理性。我们的框架自然地将经典MFC作为退化情况包含在内，同时实现了在复杂稀疏拓扑上高效且理论坚实的控制。