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Heterogeneous Mean Field Game Framework for LEO Satellite-Assisted V2X Networks

Kangkang Sun, Jianhua Li, Xiuzhen Chen, Minyi Guo

Abstract

Coordinating mixed fleets of $10^4$ to $10^5$ vehicles, passenger cars, freight trucks, and autonomous vehicles, under stringent delay constraints is a central scalability bottleneck in next-generation V2X networks. Heterogeneous mean field games (HMFG) offer a principled coordination framework, yet a fundamental design question lacks theoretical guidance: how many agent types $K$ should be used for a fleet of size $N$? The core challenge is a two-sided trade-off that existing theory does not resolve: increasing $K$ reduces type-discretization error but simultaneously starves each class of the samples needed for reliable mean-field approximation. We resolve this trade-off by deriving an explicit $\varepsilon$-Nash error decomposition driven by a Wasserstein-based heterogeneity measure, and prove that the unique error-minimizing type count satisfies $K^*(N)=Θ(N^{1/3})$ in the canonical one-dimensional queue setting. We further establish a heterogeneity-aware convergence condition for G-prox PDHG and extend the framework to temporal-graph LEO satellite backhaul dynamics with provable robustness guarantees. A perhaps surprising consequence is that even for $N=10^5$ vehicles, only about 28 type classes suffice, cube-root compression rather than per-vehicle modeling, so type-granularity selection is largely a set-once design decision. Experiments validate the scaling law, achieve $2.3\times$ faster PDHG convergence at $K=5$, and deliver up to $29.5\%$ lower delay and $60\%$ higher throughput compared with homogeneous baselines.

Heterogeneous Mean Field Game Framework for LEO Satellite-Assisted V2X Networks

Abstract

Coordinating mixed fleets of to vehicles, passenger cars, freight trucks, and autonomous vehicles, under stringent delay constraints is a central scalability bottleneck in next-generation V2X networks. Heterogeneous mean field games (HMFG) offer a principled coordination framework, yet a fundamental design question lacks theoretical guidance: how many agent types should be used for a fleet of size ? The core challenge is a two-sided trade-off that existing theory does not resolve: increasing reduces type-discretization error but simultaneously starves each class of the samples needed for reliable mean-field approximation. We resolve this trade-off by deriving an explicit -Nash error decomposition driven by a Wasserstein-based heterogeneity measure, and prove that the unique error-minimizing type count satisfies in the canonical one-dimensional queue setting. We further establish a heterogeneity-aware convergence condition for G-prox PDHG and extend the framework to temporal-graph LEO satellite backhaul dynamics with provable robustness guarantees. A perhaps surprising consequence is that even for vehicles, only about 28 type classes suffice, cube-root compression rather than per-vehicle modeling, so type-granularity selection is largely a set-once design decision. Experiments validate the scaling law, achieve faster PDHG convergence at , and deliver up to lower delay and higher throughput compared with homogeneous baselines.

Paper Structure

This paper contains 58 sections, 13 theorems, 55 equations, 8 figures, 7 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. MFG Applications and Primal-Dual Solvers in Wireless and V2X
  4. HMFG Well-Posedness, Propagation of Chaos, and Empirical Convergence Rates
  5. Quantization Theory, Graphon Limits, and Finite Type versus Continuum Models
  6. Optimization-Driven Learning and LEO-Assisted Communication
  7. System Model and HMFG Formulation
  8. Scenario Description
  9. Data Queue Dynamics
  10. Cost Function Formulation
  11. LEO Satellite-Assisted Backhaul Model
  12. Heterogeneous Mean Field Game Formulation
  13. Problem Formulation
  14. Heterogeneity Measure and Error Decomposition
  15. Quantifying Heterogeneity
  16. ...and 43 more sections

Key Result

Lemma 1

Under Assumption ass:param, suppose $b_1$ is $L_b$-Lipschitz in $(q, \bm{\rho})$ uniformly over $a \in A$. Then: where $\mathrm{Var}(\theta) = \frac{1}{K}\sum_k (\theta_k - \bar{\theta})^2$ is the empirical variance of type parameters, $\bar{\theta} = \frac{1}{K}\sum_k \theta_k$, and $H_K^{(0)} = \frac{1}{K(K-1)}\sum_{k\neq k'} W_2(\rho_0^{(k)}, \rho_0^{(k')})$ is the initial heterogeneity. $\bla

Figures (8)

  • Figure 1: Illustrative V2X-MEC scenario for heterogeneous mean field resource allocation. The diverse vehicle population is partitioned into $K$ types with distinct data loads and preferences. Per-type mean fields $\rho^{(k)}$ capture class-wise congestion, while cross-type coupling enters through interference $I(t)$ and the computational price $\Phi(t)$.
  • Figure 2: Scaling-law validation (Corollary \ref{['cor:scaling']}). Left axis: continuous $K^*(N)$ (solid red) and rounded integer $\hat{K}$ (blue circles); log-log slope $\approx 1/3$. Right axis: minimum achievable error $\mathcal{E}^*(N)$ (dashed); slope $\approx -1/6$.
  • Figure 3: $\varepsilon$-Nash approximation error vs. fleet size $N$ (Theorem \ref{['thm:error_decomp']}). Proposed $K^*(N)$ policy dominates all fixed-$K$ baselines. Error bands indicate $\pm 1$ standard deviation across 25 Monte Carlo trials.
  • Figure 4: PDHG residual convergence validation (Theorem \ref{['thm:convergence']}). Log-scale residual vs. iteration count for adaptive step (Algorithm \ref{['alg:step_adapt']}) and two fixed-step baselines across $K \in \{1, 3, 5\}$. Adaptive rule satisfies \ref{['eq:corrected_condition']} automatically and achieves the fastest residual reduction.
  • Figure 5: Scalability validation (Proposition \ref{['prop:complexity']}). Per-iteration wall-clock runtime vs. fleet size $N$ (log-log scale). Proposed HMFG with adaptive $K^*(N)$ grows as $O(N^{2/3})$; naive $N$-body solver grows quadratically. Horizontal dashed line indicates $N$-independent ideal complexity.
  • ...and 3 more figures

Theorems & Definitions (43)

  • Remark 1: Reduction to Homogeneous MFG
  • Definition 1: Heterogeneity Measure
  • Remark 2
  • Lemma 1: Heterogeneity Measure under Parameterization
  • proof
  • Lemma 2: Empirical Wasserstein Rates on a Bounded Interval
  • proof
  • Remark 3: Reconciliation with qiao2025hmfg
  • Theorem 1: Error Decomposition
  • proof
  • ...and 33 more