Graph recovery from graph wave equation

TL;DR

This paper introduces a novel method to recover underlying graph structures from multivariate wave signals by leveraging the graph wave equation, mode extraction via DMD, and applies it to both synthetic and real sensor data.

Contribution

The paper presents a new graph recovery technique using mode extraction from wave signals and modifies DMD for improved accuracy under stationary mode assumptions.

Findings

Successfully recovers path graph from wave signals

Effective on human joint sensor data

Applicable to non-wave signals

Abstract

We propose a method by which to recover an underlying graph from a set of multivariate wave signals that is discretely sampled from a solution of the graph wave equation. Herein, the graph wave equation is defined with the graph Laplacian, and its solution is explicitly given as a mode expansion of the Laplacian eigenvalues and eigenfunctions. For graph recovery, our idea is to extract modes corresponding to the square root of the eigenvalues from the discrete wave signals using the DMD method, and then to reconstruct the graph (Laplacian) from the eigenfunctions obtained as amplitudes of the modes. Moreover, in order to estimate modes more precisely, we modify the DMD method under an assumption that only stationary modes exist, because graph wave functions always satisfy this assumption. In conclusion, we demonstrate the proposed method on the wave signals over a path graph. Since our graph recovery procedure can be applied to non-wave signals, we also check its performance on human joint sensor time-series data.