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

2025-09-15

AI Summary

This paper addresses the efficient computation of Nash equilibria in graph-structured stochastic differential games (SDGs). Existing methods suffer from parameter redundancy, poor interpretability, and weak stability on large-scale sparse graphs. To overcome these limitations, we propose a non-trainable modification (NTM) neural architecture that explicitly encodes graph topological priors into fixed, sparse, non-trainable modules—ensuring structural alignment and substantial parameter compression. We instantiate NTM within two solver frameworks: NTM-DP, integrated into the direct parameterization paradigm, and NTM-DBSDE, embedded in the deep backward stochastic differential equation (BSDE) framework. Experiments across three canonical graph SDG benchmarks demonstrate that NTM achieves comparable accuracy and robustness to fully parameterized models while reducing trainable parameters by up to ~70%. Moreover, NTM delivers superior computational efficiency and enhanced interpretability, offering a principled trade-off between expressivity, efficiency, and transparency in graph-structured SDG learning.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Game Theory and Economic Paradigms: Adversarial Learning
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

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, 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.

Problem

• Computing Nash equilibria in graph-based stochastic differential games
• Reducing trainable parameters in large-scale sparse multi-agent systems
• Enhancing interpretability and stability of neural network game solvers

Innovation

• Graph-based sparsification with non-trainable components
• Universal approximation for static graph games
• Integration with state-of-the-art game solvers