  Networked dynamical systems, i.e., systems of dynamical units coupled via
nontrivial interaction topologies, constitute models of broad classes of
complex systems, ranging from gene regulatory and metabolic circuits in our
cells to pandemics spreading across continents. Most of such systems are driven
by irregular and distributed fluctuating input signals from the environment.
Yet how networked dynamical systems collectively respond to such fluctuations
depends on the location and type of driving signal, the interaction topology
and several other factors and remains largely unknown to date. As a key
example, modern electric power grids are undergoing a rapid and systematic
transformation towards more sustainable systems, signified by high penetrations
of renewable energy sources. These in turn introduce significant fluctuations
in power input and thereby pose immediate challenges to the stable operation of
power grid systems. How power grid systems dynamically respond to fluctuating
power feed-in as well as other temporal changes is critical for ensuring a
reliable operation of power grids yet not well understood. In this work, we
systematically introduce a linear response theory for fluctuation-driven
networked dynamical systems. The derivations presented not only provide
approximate analytical descriptions of the dynamical responses of networks, but
more importantly, allows to extract key qualitative features about
spatio-temporally distributed response patterns. Specifically, we provide a
general formulation of a linear response theory for perturbed networked
dynamical systems, explicate how dynamic network response patterns arise from
the solution of the linearized response dynamics, and emphasize the role of
linear response theory in predicting and comprehending power grid responses on
different temporal and spatial scales and to various types of disturbances.
