# Mean Field Games on Weighted and Directed Graphs via Colored Digraphons

arXiv: https://arxiv.org/abs/2209.03887 · HTML: https://mlsanity.com/paper/2209.03887/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 28 references that are in the map.
- First posted on arXiv 8 Sep 2022.

## Builds on (most cited first, 8 of 8)

- [Approximately Solving Mean Field Games via Entropy-Regularized Deep Reinforcement Learning](https://mlsanity.com/paper/2102.01585/) (2021)
- [Fictitious Play for Mean Field Games: Continuous Time Analysis and Applications](https://mlsanity.com/paper/2007.03458/) (2020)
- [On the Convergence of Model Free Learning in Mean Field Games](https://mlsanity.com/paper/1907.02633/) (2019)
- [Applications of Deep Learning and Reinforcement Learning to Biological Data](https://mlsanity.com/paper/1711.03985/) (2017)
- [Markov-Nash Equilibria in Mean-Field Games with Discounted Cost](https://mlsanity.com/paper/1612.07878/) (2016)
- [Learning Graphon Mean Field Games and Approximate Nash Equilibria](https://mlsanity.com/paper/2112.01280/) (2021)
- [Finite State Graphon Games with Applications to Epidemics](https://mlsanity.com/paper/2106.07859/) (2021)
- [Graphon-based sensitivity analysis of SIS epidemics](https://mlsanity.com/paper/1912.10330/) (2019)

## Built on by (most cited first, 3 of 3)

- [Graphon Mean-Field Subsampling for Cooperative Heterogeneous Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2602.16196/) (2026)
- [RKHS Representation of Algebraic Convolutional Filters with Integral Operators](https://mlsanity.com/paper/2602.19094/) (2026)
- [Graphon Particle Systems, Part II: Dynamics of Distributed Stochastic Continuum Optimization](https://mlsanity.com/paper/2407.02765/) (2024)

## Abstract (arXiv)

The field of multi-agent reinforcement learning (MARL) has made considerable progress towards controlling challenging multi-agent systems by employing various learning methods. Numerous of these approaches focus on empirical and algorithmic aspects of the MARL problems and lack a rigorous theoretical foundation. Graphon mean field games (GMFGs) on the other hand provide a scalable and mathematically well-founded approach to learning problems that involve a large number of connected agents. In standard GMFGs, the connections between agents are undirected, unweighted and invariant over time. Our paper introduces colored digraphon mean field games (CDMFGs) which allow for weighted and directed links between agents that are also adaptive over time. Thus, CDMFGs are able to model more complex connections than standard GMFGs. Besides a rigorous theoretical analysis including both existence and convergence guarantees, we provide a learning scheme and illustrate our findings with an epidemics model and a model of the systemic risk in financial markets.