Stability of Flow Models for Graph Signals

2026-07-08

AI Summary

This work addresses the instability of existing continuous flow models in graph signal generation under structural perturbations, which renders generated signals highly sensitive to noise. To mitigate this issue, the authors propose a continuous normalizing flow model parameterized by graph neural networks and derive, for the first time, explicit stability bounds for both the underlying continuous dynamical system and its discrete numerical approximation. Building on these theoretical guarantees, they introduce a regularized flow matching strategy that enhances robustness by constraining the spatial Lipschitz constant of the vector field. Experiments on stochastic block models and real-world brain connectome graphs demonstrate that the proposed method significantly improves stability against structural perturbations while preserving high-quality signal generation.

Abstract

Generating signals on graphs requires permutation-equivariant models that exhibit stability with respect to relative structural perturbations. While favorable stability properties of Graph Neural Networks (GNNs) have been well documented, it is unclear how structural errors propagate through the dynamics of continuous generative flow models that are gaining traction for graph signal generation. In this paper, we analyze continuous normalized flow models parameterized by GNNs and show that permutation equivariance is preserved for both the resulting continuous-time ordinary differential equations and their discrete numerical approximations used as graph signal samplers. Our primary contribution is to derive explicit stability bounds on the generated probability distributions, which quantify how relative graph perturbations affect the final sampled signals. Motivated by these theoretical bounds, we introduce a stability-promoting regularized flow matching strategy that actively penalizes the spatial Lipschitz constant of the vector field during model training. Experiments using synthetic smooth signals on stochastic block model graphs and real-world fMRI signals on brain connectomes demonstrate that this bound-oriented approach yields generative models that are more robust to structural noise, without sacrificing output quality.

Problem

• graph signal generation
• structural perturbations
• flow models
• stability
• Graph Neural Networks

Innovation

• graph signal generation
• continuous normalizing flows
• stability analysis
• permutation equivariance
• Lipschitz regularization