Optimal Sampling for Dynamic Complex Networks with Graph-Bandlimited Initialization

TL;DR

This paper develops a theoretical framework for optimally sampling and recovering the dynamics of complex networked systems, considering both time and graph domains, with applications in social and biological networks.

Contribution

It introduces a novel theory linking optimal sampling strategies to network properties and nonlinear dynamics, enabling full recovery of networked signals from limited samples.

Findings

Dynamic signals can be recovered if the network is stable and initial conditions are bandlimited.

Sampling locations and rates are determined by graph spectral properties and nonlinear dynamics.

The theory is demonstrated through examples in social growth and biochemical networks.

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 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.