Mean-Field Control on Sparse Graphs: From Local Limits to GNNs via Neighborhood Distributions
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| Format: | Preprint |
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2026
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| _version_ | 1866917231347105792 |
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| author | Schmidt, Tobias Cui, Kai |
| author_facet | Schmidt, Tobias Cui, Kai |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_21477 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Mean-Field Control on Sparse Graphs: From Local Limits to GNNs via Neighborhood Distributions Schmidt, Tobias Cui, Kai Multiagent Systems Artificial Intelligence Machine Learning Optimization and Control 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. |
| title | Mean-Field Control on Sparse Graphs: From Local Limits to GNNs via Neighborhood Distributions |
| topic | Multiagent Systems Artificial Intelligence Machine Learning Optimization and Control |
| url | https://arxiv.org/abs/2601.21477 |