Optimal sampling for dynamic complex networks with graph-bandlimited initialization

Abstract

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 and discretize a single continuous 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 and inputs (which are unknown) are 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 placement of experimental probes and sensors on dynamical networks especially for the case where the employed sensors are difficult to change. Also, this guides the design of each sensor's optimal sampling rate for further digital signal processing. 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 and inputs.