Stabilizing Filters for Graph Signal Generation

Generating graph signals is an important task in machine learning, and incorporating the underlying graph into a generative model enables structurally-informed distribution modeling. However, said graph is often noisy or imperfectly estimated, and structural errors can affect the generative process and the resulting signal distribution. While recent work has established stability guarantees for graph neural ordinary differential equations, the robustness of graph-aware reference dynamics, as used in topological Schrödinger bridge models, remains largely unexplored. Here, we investigate the structural stability of continuous-time generative models whose dynamics combine a graph filter with a learned graph neural network. We first derive a Wasserstein stability bound that quantifies the effect of graph perturbations on the generated signal distributions. Informed by this analysis, we then propose a principled framework for designing graph filters that balance structural stability with induced signal smoothing. We characterize the optimal filter design and provide an explicit family of polynomial filters with provable stability guarantees. Experiments on synthetic graphs and real-world fMRI signals support our theoretical findings, illustrating the enhanced structural stability of the proposed filters while preserving generative performance.