Clustering Signed Networks with the Geometric Mean of Laplacians

TL;DR

This paper introduces a novel spectral clustering method for signed networks using the geometric mean of Laplacians, which outperforms existing approaches especially in noise-free scenarios, and provides an efficient computational scheme.

Contribution

It proposes using the geometric mean of Laplacians for signed network clustering, addressing limitations of previous arithmetic mean methods, and offers an efficient eigenvector computation scheme.

Findings

Geometric mean approach outperforms existing methods in noise-free cases

Eigenvectors of the geometric mean can be computed efficiently for sparse matrices

The method is applicable to large sparse signed networks

Abstract

Signed networks allow to model positive and negative relationships. We analyze existing extensions of spectral clustering to signed networks. It turns out that existing approaches do not recover the ground truth clustering in several situations where either the positive or the negative network structures contain no noise. Our analysis shows that these problems arise as existing approaches take some form of arithmetic mean of the Laplacians of the positive and negative part. As a solution we propose to use the geometric mean of the Laplacians of positive and negative part and show that it outperforms the existing approaches. While the geometric mean of matrices is computationally expensive, we show that eigenvectors of the geometric mean can be computed efficiently, leading to a numerical scheme for sparse matrices which is of independent interest.