Clustering Signed Networks with the Geometric Mean of Laplacians
This addresses clustering in signed networks, which is incremental as it improves upon existing spectral methods for a specific domain.
The paper tackles the problem of spectral clustering in signed networks by identifying that existing methods fail to recover ground truth clusters in noise-free scenarios, and proposes using the geometric mean of Laplacians, which outperforms prior approaches.
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.