  Many engineering, social, and biological complex systems consist of dynamical
elements connected via a large-scale network. Monitoring the network's dynamics
is essential for a variety of maintenance and scientific purposes. Whilst we
understand how to optimally sample a single dynamic element or a non-dynamic
graph, we do not possess a theory on how to optimally sample networked
dynamical elements. Here, we study nonlinear dynamic graph signals on a fixed
complex network. We define the necessary conditions for optimal sampling in the
combining time- and graph-domain to fully recover the networked dynamics. We
firstly interpret the networked dynamics into a linearized matrix. Then, we
prove that the dynamic signals can be sampled and fully recovered if the
networked dynamics is stable and their initialization is bandlimited in the
graph spectral domain. This new theory directly maps optimal sampling locations
and rates to the graph properties and governing nonlinear dynamics. This can
inform the optimal placement of experimental probes and sensors on dynamical
networks as well as inform the design of each sensor's optimal sampling rate.
We motivate the reader with two examples of recovering the networked dynamics
for: social population growth and networked protein biochemical interactions
with both bandlimited and arbitrary initialization.
