Sampling and Inference of Networked Dynamics Using Log-Koopman Nonlinear Graph Fourier Transform

Monitoring the networked dynamics via the subset of nodes is essential for a variety of scientific and operational purposes. When there is a lack of an explicit model and networked signal space, traditional observability analysis and non-convex methods are insufficient. Current data-driven Koopman linearization, although derives a linear evolution model for selected vector-valued observable of original state-space, may result in a large sampling set due to: (i) the large size of polynomial based observables (<inline-formula><tex-math notation="LaTeX">$O(N^2)$</tex-math></inline-formula>, <inline-formula><tex-math notation="LaTeX">$N$</tex-math></inline-formula> number of nodes in network), and (ii) not factoring in the nonlinear dependency betweenobservables. In this work, to achieve linear scaling (<inline-formula><tex-math notation="LaTeX">$O(N)$</tex-math></inline-formula>) and a small set of sampling nodes, wepropose 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, anonlinear GFT concept and sampling theory are provided to exploit the nonlinear dependence of observables for observability analysis using Koopman evolution model. The results demonstrate that the proposed Log-Koopman NL-GFT scheme can (i) linearize unknownnonlinear dynamics using <inline-formula><tex-math notation="LaTeX">$O(N)$</tex-math></inline-formula> observables, and (ii) achieve lower number of sampling nodes, compared with the state-of-the art polynomial Koopman based observability analysis.