  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.
