Stabilizing Filters for Graph Signal Generation

Generating signals on graphs requires permutation-equivariant models that exhibit stability with respect to relative structural perturbations. While recent graph-aware Schrödinger bridge models incorporate topology information directly into their reference dynamics, it is unclear how perturbations of the graph propagate through these dynamics and affect the resulting generated distributions. In this paper, we analyze the structural stability of graph-aware continuous-time generative models whose drift combines a graph filter with a learned graph neural network. We derive explicit Wasserstein stability bounds that quantify the effect of relative graph perturbations on the generated distributions. Motivated by these bounds, we introduce a principled framework for designing stable graph filters that preserve the smoothing behavior of graph heat diffusion, while boosting structural stability. Experiments on synthetic and fMRI signals show our stable filters enhance structural robustness while matching or exceeding the generative quality of the heat equation baseline.

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.

Sampling stochastic signals supported on a graph underlies many graph machine learning tasks, including recommender systems, forecasting in financial markets, and wireless network optimization. In these settings, the target signals are realizations of unknown conditional distributions. However, prevailing approaches rely mostly on intricate, application-tailored designs that often regress to a conditional mean instead of sampling from the conditional law. This paper unifies such problems as conditional graph signal generative modeling and tackles them with a single denoising diffusion framework. We learn a reverse diffusion process, parametrized by graph neural networks (GNNs), that draws graph signals conditioned directly on the graph topology and on node-feature side information. The reverse process is realized by a novel architecture, the U-Graph Neural Network (U-GNN), which generalizes the image-convolutional U-Net to graph-structured signals. The U-GNN performs multi-resolution encoder--decoder processing in which pooling and unpooling reduce to a learned node selection, expressed by nested selection matrices, and a zero-padded lifting of coarse signals back to the full node set. The graph convolutions are carried out on the original graph, with a stride that sets their hop reach, so the U-GNN bypasses explicit graph coarsening at every resolution. We demonstrate our method on two generative tasks: stock price forecasting and optimal wireless resource allocation, with extensive numerical results in both domains.