RAFlow: Graph-Conditioned Continuous Normalizing Flows for Relational Time-Series Anomaly Detection

Multivariate time-series anomaly detection requires modelling both temporal dynamics and evolving dependencies among correlated entities. Existing graph-based normalizing flow methods incorporate learned relational structures into density estimation, but condition the graph only before the flow transformation, limiting the ability of the likelihood model to adapt continuously to dynamic relational structure. We propose RAFlow, a graph-conditioned continuous normalizing flow framework for relational time-series anomaly detection, where the velocity field of a neural ordinary differential equation (ODE) is jointly conditioned on learned temporal and relational embeddings at every integration step, ensuring inter-entity dependencies continuously shape the density transformation throughout the entire trajectory. Terminal likelihood alone cannot distinguish anomalies that reach similar density regions through irregular latent trajectories; we introduce a path-integral anomaly score that combines terminal log-likelihood with the kinetic energy of the ODE trajectory as complementary detection signals. Theoretically, we prove that RAFlow yields a consistent anomaly score that is stable, robust to measurement noise, and resistant to wild fluctuations under small input variations. Extensive experiments on five benchmark datasets against nineteen baselines show that RAFlow consistently outperforms existing graph-based flow methods and achieves the strongest performance on densely correlated benchmarks, highlighting the effectiveness of continuous graph-conditioned density estimation.