Title: Trajectory-Dependent Node Influence and the Boundaries of Static Controllability

Abstract:

Determining how specific nodes influence the overall behavior of complex networks is a frequent objective in data mining. Current methodologies typically depend on static-graph centrality metrics or control-theoretic frameworks, such as controllability Gramians, which presuppose linear, time-invariant system dynamics. However, empirically estimated systems are predominantly nonlinear and time-varying. To address this discrepancy, we introduce "emergent contribution (EC)," a finite-horizon metric quantifying a node’s dynamical leverage. This measure is defined as the metric-weighted energy of the node’s impulse response, integrated along the system’s trajectory. Derived from the Jacobians of any differentiable model, EC is independent of the specific estimator used and converges precisely to average controllability when applied to linear, time-invariant systems. This study characterizes the conditions under which EC and average controllability align or diverge.

By utilizing a controlled synthetic dataset with known ground-truth contributions, we developed a phase diagram that maps variations in nonlinearity, regime structure, persistence, and perturbation amplitude. The results indicate that both EC and average controllability correspond with ground truth and each other under static conditions or during smoothly drifting dynamics. Significant divergence arises during persistent regime switching, peaking when persistent sign reversals occur; this discrepancy vanishes if such sign reversals are eliminated. Furthermore, at extreme perturbation amplitudes, both metrics degrade, highlighting the constraints of local linearization.

We situated five estimated real-world systems from various domains within this phase space. This positioning acts as a diagnostic tool to determine when EC yields insights superior to static controllability, thereby validating its extra computational expense. A detailed examination of one such case, involving a twenty-seed retraining ensemble, uncovered a robust dissociation between variance and leverage. This revealed nodes whose perturbations spread widely despite exhibiting low within-system variance—a phenomenon neither static centralities nor variance-based summaries could detect.

Source: arXiv Generated at: 2026-06-03 00:00:00 UTC