TVGL-CFM: Generating and forecasting time-varying trajectories of dynamic networks with conditional flow matching

Many complex systems, including brain networks, financial markets, and generegulatory 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 (Sp ++)T, and we construct a global log-Euclidean diffeomorphism from this product space to a Euclidean sequence space, letting a non-autoregressive conditional flow-matching model with a Transformer backbone generate all windows jointly and decode to SPD matrices with no post-hoc projection. For forecasting, we use two distinct data dependant couplings so 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 up new possibilities for generative dynamic graph models.