Stability of Flow Models for Graph Signals

As generative models like continuous normalizing flows (CNFs) gain popularity for graph-structured data, a critical question arises: how do structural errors in the underlying graph topology propagate through the generative dynamics? While GNNs are known for their stability, it remains unclear if these properties hold when GNNs are used to parameterize the time-dependent vector fields of generative flows, particularly when node indexing is arbitrary.

The authors analyze GNN-parameterized CNFs, establishing that they preserve permutation equivariance in both continuous-time ordinary differential equations and their discrete numerical approximations (e.g., Euler and Heun methods). By leveraging the relative perturbation model for graph shift operators, they derive explicit Wasserstein stability bounds that characterize how graph noise impacts the final generated probability distribution. Based on these theoretical insights, they propose a regularized flow matching strategy that explicitly penalizes the spatial Lipschitz constant of the vector field during training to promote structural robustness.

The study demonstrates that the stability of the generative vector field is intrinsically linked to the graph spectrum, allowing for a graph-aware regularization approach. Experiments on synthetic stochastic block model graphs and real-world fMRI signals on brain connectomes show that this bound-oriented regularization significantly improves robustness to structural noise without degrading the quality of the generated signals.

This work provides a formal theoretical foundation for the stability of graph-based generative models. By moving beyond empirical performance to provide explicit bounds, the authors offer a principled way to design more reliable generative systems for applications where the graph topology is noisy or imperfect, such as in neuroscience or distributed sensor networks.