On spectral partitioning of signed graphs

TL;DR

This paper advocates for using the standard graph Laplacian over the signed Laplacian in spectral partitioning of signed graphs, highlighting advantages in interpretability and eigenvalue properties.

Contribution

It demonstrates that the standard Laplacian provides more meaningful partitions for signed graphs than the signed Laplacian, challenging existing practices.

Findings

Standard Laplacian yields more meaningful partitions.

Negative eigenvalues facilitate computation of the Fiedler vector.

Signed Laplacian may produce meaningless partitions.

Abstract

We argue that the standard graph Laplacian is preferable for spectral partitioning of signed graphs compared to the signed Laplacian. Simple examples demonstrate that partitioning based on signs of components of the leading eigenvectors of the signed Laplacian may be meaningless, in contrast to partitioning based on the Fiedler vector of the standard graph Laplacian for signed graphs. We observe that negative eigenvalues are beneficial for spectral partitioning of signed graphs, making the Fiedler vector easier to compute.