  Networked nonlinear dynamics underpin the complex functionality of many
engineering, social, biological, and ecological systems. Monitoring the
networked dynamics via the minimum subset of nodes is essential for a variety
of scientific and operational purposes. When there is a lack of a explicit
model and networked signal space, traditional evolution analysis and non-convex
methods are insufficient. An important data-driven state-of-the-art method use
the Koopman operator to generate a linear evolution model for a vector-valued
observable of original state-space. As a result, one can derive a sampling
strategy via the linear evolution property of observable. However, current
polynomial Koopman operators result in a large sampling space due to: (i) the
large size of polynomial based observables ($O(N^2)$, $N$ number of nodes in
network), and (ii) not factoring in the nonlinear dependency between
observables.
  In this work, to achieve linear scaling ($O(N)$) and a small set of sampling
nodes, we propose to combine a novel Log-Koopman operator and nonlinear Graph
Fourier Transform (NL-GFT) scheme. First, the Log-Koopman operator is able to
reduce the size of observables by transforming multiplicative poly-observable
to logarithm summation. Second, a nonlinear GFT concept and sampling theory are
provided to exploit the nonlinear dependence of observables for Koopman
linearized evolution analysis. Combined, the sampling and reconstruction
algorithms are designed and demonstrated on two established application areas.
The results demonstrate that the proposed Log-Koopman NL-GFT scheme can (i)
linearize unknown nonlinear dynamics using $O(N)$ observables, and (ii) achieve
lower number of sampling nodes, compared with the state-of-the art polynomial
Koopman linear evolution analysis.
