# Finite-Agent Stochastic Differential Games on Large Graphs: II. Graph-Based Architectures

arXiv: https://arxiv.org/abs/2509.12484 · HTML: https://mlsanity.com/paper/2509.12484/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 53 references that are in the map.
- First posted on arXiv 15 Sep 2025.

## Builds on (most cited first, 25 of 25)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Graph Neural Networks: A Review of Methods and Applications](https://mlsanity.com/paper/1812.08434/) (2018)
- [Convolutional Networks on Graphs for Learning Molecular Fingerprints](https://mlsanity.com/paper/1509.09292/) (2015)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Diffusion-Convolutional Neural Networks](https://mlsanity.com/paper/1511.02136/) (2015)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [NetGAN: Generating Graphs via Random Walks](https://mlsanity.com/paper/1803.00816/) (2018)
- [Deep Learning Approximation for Stochastic Control Problems](https://mlsanity.com/paper/1611.07422/) (2016)
- [A Short Tutorial on The Weisfeiler-Lehman Test And Its Variants](https://mlsanity.com/paper/2201.07083/) (2022)
- [Recent Developments in Machine Learning Methods for Stochastic Control and Games](https://mlsanity.com/paper/2303.10257/) (2023)
- [Global Convergence of Localized Policy Iteration in Networked Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2211.17116/) (2022)
- [Deep neural networks algorithms for stochastic control problems on finite horizon: numerical applications](https://mlsanity.com/paper/1812.05916/) (2018)
- [Deep Fictitious Play for Finding Markovian Nash Equilibrium in Multi-Agent Games](https://mlsanity.com/paper/1912.01809/) (2019)
- [Deep Fictitious Play for Stochastic Differential Games](https://mlsanity.com/paper/1903.09376/) (2019)
- [SparseProp: Efficient Sparse Backpropagation for Faster Training of Neural Networks](https://mlsanity.com/paper/2302.04852/) (2023)
- [Convergence of Deep Fictitious Play for Stochastic Differential Games](https://mlsanity.com/paper/2008.05519/) (2020)
- [A case study on stochastic games on large graphs in mean field and sparse regimes](https://mlsanity.com/paper/2005.14102/) (2020)
- [A brief review of the Deep BSDE method for solving high-dimensional partial differential equations](https://mlsanity.com/paper/2505.17032/) (2025)
- [Multi-Agent Reinforcement Learning with Graph Convolutional Neural Networks for optimal Bidding Strategies of Generation Units in Electricity Markets](https://mlsanity.com/paper/2208.06242/) (2022)
- [Learning from one graph: transductive learning guarantees via the geometry of small random worlds](https://mlsanity.com/paper/2509.06894/) (2025)
- [Finite-Agent Stochastic Differential Games on Large Graphs: I. The Linear-Quadratic Case](https://mlsanity.com/paper/2406.09523/) (2024)

## Built on by (most cited first, 1 of 1)

- [BRAID: Learning Equilibrium Maps in Interdependent Security Games via Weight-Tied Iterative Graph Neural Networks](https://mlsanity.com/paper/2608.14856/) (2026)

## Abstract (arXiv)

We propose a novel neural network architecture, called Non-Trainable Modification (NTM), for computing Nash equilibria in stochastic differential games (SDGs) on graphs. These games model a broad class of graph-structured multi-agent systems arising in finance, robotics, energy, and social dynamics, where agents interact locally under uncertainty. The NTM architecture imposes a graph-guided sparsification on feedforward neural networks, embedding fixed, non-trainable components aligned with the underlying graph topology. This design enhances interpretability and stability, while significantly reducing the number of trainable parameters in large-scale, sparse settings. We theoretically establish a universal approximation property for NTM in static games on graphs and numerically validate its expressivity and robustness through supervised learning tasks. Building on this foundation, we incorporate NTM into two state-of-the-art game solvers, Direct Parameterization and Deep BSDE (backward stochastic differential equation), yielding their sparse variants (NTM-DP and NTM-DBSDE). Numerical experiments on three SDGs across various graph structures demonstrate that NTM-based methods achieve performance comparable to their fully trainable counterparts, while offering improved computational efficiency.