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

2026-07-18

AI Summary

This study addresses the modeling, generation, and prediction of dynamic network trajectories—such as time-varying graph structures observed in electroencephalography, gene regulatory networks, or financial markets. The authors propose embedding sequences of time-varying sparse precision matrices into Euclidean space via the log-Euclidean metric and introduce, for the first time, a conditional flow matching (CFM) model to learn their distribution. A geometry-aware decoder is integrated to guarantee the positive definiteness of generated matrices, and a history-based extrapolation strategy is developed for fine-tuned forecasting. Experiments on EEG motor imagery, chaotic dynamical systems, and gene expression datasets demonstrate that the generated trajectories preserve class-discriminative structures and significantly outperform baseline methods that directly model raw signals in predicting future connectivity patterns.

Abstract

Many complex systems such as brain networks, financial markets, and gene-regulatory circuits are described not by a fixed graph but by one that changes over time. A standard way to summarise such structure at each instant is the sparse precision (inverse-covariance) matrix, and the time-varying graphical lasso (TVGL) turns a multivariate signal into a smooth chain of these matrices. We introduce TVGL-CFM, a single model that learns the distribution of such chains and can both generate new, realistic time-varying network trajectories for a given class and forecast how an observed trajectory will continue. Each precision matrix lives on a curved space of positive-definite matrices, but a log-Euclidean chart flattens an entire trajectory into an ordinary vector space, so a simple conditional flow-matching model can be trained and sampled there while every decoded matrix is guaranteed to be a valid precision matrix. For forecasting we start the flow not from noise but from a rough extrapolation of the recent history, so the model only has to learn a small correction. Across EEG motor-imagery, chaotic systems, and gene-expression data, TVGL-CFM generates trajectories that keep the class-discriminative structure of real data, and it forecasts future connectivity more accurately than raw-signal baselines. Generating the structured precision trajectory directly is therefore more faithful than generating raw signals and estimating connectivity afterwards.

Problem

• dynamic networks
• time-varying trajectories
• precision matrix
• trajectory generation
• forecasting

Innovation

• conditional flow matching
• time-varying graphical lasso
• positive-definite manifold
• trajectory generation
• connectivity forecasting