PolarFlow: Spectral Geometric Flow Matching for Directed Graph Generation via Polar Decomposition

Directed graphs model asymmetric, ordered relationships in biology, transportation, social networks, and electronic design automation, making their generation important for analysis, simulation and design. Recent discrete diffusion and flow-matching methods generate directed graphs effectively, but their edge-wise categorical formulations model every ordered node pair, requiring dense predictions during training and repeated categorical updates during generation. This motivates a fundamental question: can comparable generative quality be achieved through a more computationally efficient formulation that better exploits the structure of directed adjacency matrices? Moreover, existing directed-graph benchmarks cover a limited range of network structures, hindering a comprehensive evaluation of graph generation. We introduce *PolarFlow*, a continuous geometric flow-matching framework based on the polar decomposition of the adjacency matrix and truncated spectral coordinates. Its representation consists of four coupled coordinate components: a Stiefel frame and rotation angles that encode the orthogonal factor, together with a second Stiefel frame and nonnegative eigenvalues that characterize the positive-semidefinite factor. PolarFlow learns a joint velocity field along geometry-aware conditional paths on this product space, reconstructs a continuous adjacency-score matrix, and discretizes it into a binary directed graph. By replacing discrete edge-wise generation with joint continuous transport on this product space, PolarFlow offers a computationally efficient alternative while retaining the global structure of the adjacency matrix. To broaden evaluation, we introduce complementary Circuit and Twitter benchmarks spanning engineered functional and naturally evolved social networks. Across synthetic and real-world datasets, PolarFlow achieves generative quality comparable to competitive discrete methods while substantially reducing training and sampling time.