Bures-Wasserstein Flow Matching for Graph Generation

Abstract: Graph generation has emerged as a critical task in fields ranging from molecule design to drug discovery. Contemporary approaches, notably diffusion and flow-based models, have achieved solid graph generative performance through constructing a probability path that interpolates between a reference distribution and the data distribution. However, these methods typically model the evolution of individual nodes and edges independently and use linear interpolations to build the path assuming that the data lie in Euclidean space. We show that this is suboptimal given the intrinsic non-Euclidean structure and interconnected patterns of graphs, and it poses risks to the sampling convergence. To build a better probability path, we model the joint evolution of the nodes and edges by representing graphs as connected systems parameterized by Markov random fields (MRF). We then leverage the optimal transport displacement between MRF objects to design the probability path for graph generation. Based on this, we introduce BWFlow, a flow-matching framework for graph generation that respects the underlying geometry of graphs and provides smooth velocities in the probability path. The novel framework can be adapted to both continuous and discrete flow-matching algorithms. Experimental evaluations in plain graph generation and 2D/3D molecule generation validate the effectiveness of BWFlow in graph generation with competitive performance, stable training, and guaranteed sampling convergence.