Dynamic perturbation spreading in networks

TL;DR

This paper develops analytic estimators for the peak response times and amplitudes of units in linear dynamical networks following perturbations, revealing ballistic spreading in diffusive coupling networks.

Contribution

It introduces a novel method to approximate peak responses in linear systems using expectation values, applicable near stable fixed points, and links response dynamics to network coupling types.

Findings

Estimators become exact in weak coupling limit.

Perturbations spread ballistically in diffusive networks.

Analytic expressions relate peak response to system's Jacobian.

Abstract

Understanding how local perturbations induce the transient dynamics of a network of coupled units is essential to control and operate such systems. Often a perturbation initiated in one unit spreads to other units whose dynamical state they transiently alter. The maximum state changes at those units and the timings of these changes constitute key characteristics of such transient response dynamics. However, even for linear dynamical systems it is not possible to analytically determine time and amplitude of the maximal response of a unit to a perturbation. Here, we propose to extract approximate peak times and amplitudes from effective expectation values used to characterize the typical time and magnitude of the response of a unit by interpreting the system's response as a probability distribution over time. We derive analytic estimators for the peak response based on these expectation value measures in linearized systems operating close to a stable fixed point. These estimators can be expressed in terms of the inverse of the system's Jacobian. We obtain identical results with different approximations for the response dynamics, indicating that these estimators become exact in the limit of weak coupling. Furthermore, the results suggest that perturbations spread ballistically in networks with diffusive coupling.