TVGL-CFM:Generating and Forecasting Time-Varying Trajectories of Dynamic Networks with Conditional Flow Matching

Many complex systems, including brain networks, financial markets, and gene-regulatory circuits, are better described by interaction structures that evolve over time than by a single fixed graph. The time-varying graphical lasso (TVGL) estimates this structure from multivariate signals as a temporally coherent sequence of sparse precision matrices. We introduce TVGL-CFM, a unified generative framework that learns distributions over complete SPD precision-matrix trajectories without requiring a pre-specified graph, supporting both class-conditional generation and history-conditioned forecasting. An SPD trajectory with T windows lies on the product Riemannian manifold (S++^p)^T. We construct a global log-Euclidean diffeomorphism from this product space to a Euclidean sequence space, enabling a non-autoregressive conditional flow-matching model with a Transformer backbone to generate all windows jointly and decode them to SPD matrices without post-hoc projection. For forecasting, we use two distinct data-dependent couplings so that the flow transforms an informative prior into a coherent future block. Across EEG motor-imagery data and three nonlinear dynamical systems, TVGL-CFM preserves class-discriminative dependency structure and forecasts future connectivity more accurately than several strongly matched baselines, opening new possibilities for generative dynamic graph models.