How can we naturally order and organize graph Laplacian eigenvectors?

TL;DR

This paper addresses the challenge of organizing graph Laplacian eigenvectors by introducing a novel method based on Ramified Optimal Transport Theory to define natural distances and embed eigenvectors into a low-dimensional space.

Contribution

It proposes a new approach to order and organize graph Laplacian eigenvectors using optimal transport distances, improving their interpretability.

Findings

Effective eigenvector organization demonstrated on synthetic graphs.

Successful embedding of eigenvectors into low-dimensional Euclidean space.

Enhanced understanding of eigenvector relationships in complex graphs.

Abstract

When attempting to develop wavelet transforms for graphs and networks, some researchers have used graph Laplacian eigenvalues and eigenvectors in place of the frequencies and complex exponentials in the Fourier theory for regular lattices in the Euclidean domains. This viewpoint, however, has a fundamental flaw: on a general graph, the Laplacian eigenvalues cannot be interpreted as the frequencies of the corresponding eigenvectors. In this paper, we discuss this important problem further and propose a new method to organize those eigenvectors by defining and measuring "natural" distances between eigenvectors using the Ramified Optimal Transport Theory followed by embedding them into a low-dimensional Euclidean domain. We demonstrate its effectiveness using a synthetic graph as well as a dendritic tree of a retinal ganglion cell of a mouse.