Mean-Field Control on Sparse Graphs: From Local Limits to GNNs via Neighborhood Distributions

• URL: http://arxiv.org/abs/2601.21477v1
• Date: Thu, 29 Jan 2026 09:57:48 GMT
• Title: Mean-Field Control on Sparse Graphs: From Local Limits to GNNs via Neighborhood Distributions
• Authors: Tobias Schmidt, Kai Cui,
• Abstract summary: Mean-field control (MFC) offers a scalable solution to the curse of dimensionality in multi-agent systems.<n>We bridge the gap to real-world network structures by proposing a rigorous framework for MFC on large sparse graphs.
• Score: 5.081469534056712
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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