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 remains unclear how structural errors propagate through the dynamics of continuous generative flow models gaining traction for graph signal generation. This paper analyzes continuous normalized flow models parameterized by GNNs and demonstrates 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, quantifying 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 show that this bound-oriented approach yields generative models that are more robust to structural noise without sacrificing output quality.
Blogger's Review: This paper provides theoretical support for the application of flow models in graph signal generation, particularly through the introduction of stability bounds that enhance model robustness. It lays an important theoretical foundation for future related research and offers new insights for noise handling in practical applications. The experimental results also demonstrate that stability and generation quality are not mutually exclusive, warranting further exploration.