Data-Driven Edge-Resolved Perturbation Propagation in Nonlinear Dynamical Networks

Analysis of perturbation propagation in nonlinear networks has traditionally relied on explicit governing equations. When the governing equations are unknown, resolving source-specific perturbation responses into contributions from individual interactions and organizing them into source-connected propagation structures from observed trajectories remains challenging. To address this challenge, we introduce a data-driven method for finite-time perturbation analysis using observed trajectories and known network structure. A graph-constrained neural vector field is learned from the trajectories, and its time-varying Jacobian is evaluated along an observed unperturbed reference trajectory. The resulting Jacobian drives non-autonomous variational dynamics to infer source-specific first-order perturbation responses. We then apply variation of constants to decompose the response of each affected non-source node into time-integrated incoming-edge contributions and evaluate these contributions at the node’s inferred arrival time. These arrival-conditioned contributions are combined with arrival ordering to construct predecessor graphs and extract source-connected propagation paths. Across three nonlinear network systems, our method recovers state-dependent Jacobian structure, affected-node sets, propagation timing, and arrival-conditioned edge contributions. The resulting predecessor graphs closely agree with those constructed from the ground-truth Jacobian and retain most of the corresponding contribution mass.