Clustering signed networks with the geometric mean of Laplacians

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.