# Clustering Signed Networks with the Geometric Mean of Laplacians

arXiv: https://arxiv.org/abs/1701.00757 · HTML: https://mlsanity.com/paper/1701.00757/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Jan 2017.

## Built on by (most cited first, 11 of 11)

- [MSGNN: A Spectral Graph Neural Network Based on a Novel Magnetic Signed Laplacian](https://mlsanity.com/paper/2209.00546/) (2022)
- [SSSNET: Semi-Supervised Signed Network Clustering](https://mlsanity.com/paper/2110.06623/) (2021)
- [PyTorch Geometric Signed Directed: A Software Package on Graph Neural Networks for Signed and Directed Graphs](https://mlsanity.com/paper/2202.10793/) (2022)
- [Differential geometry with extreme eigenvalues in the positive semidefinite cone](https://mlsanity.com/paper/2304.07347/) (2023)
- [Exact Community Recovery over Signed Graphs](https://mlsanity.com/paper/2202.12255/) (2022)
- [Beyond Spatio-Temporal Representations: Evolving Fourier Transform for Temporal Graphs](https://mlsanity.com/paper/2402.16078/) (2024)
- [Sublinear-Time Clustering Oracle for Signed Graphs](https://mlsanity.com/paper/2206.13813/) (2022)
- [A Thorough View of Exact Inference in Graphs from the Degree-4 Sum-of-Squares Hierarchy](https://mlsanity.com/paper/2102.08019/) (2021)
- [Defensive Alliances in Signed Networks](https://mlsanity.com/paper/2309.06801/) (2023)
- [A Spectral Framework for Tracking Communities in Evolving Networks](https://mlsanity.com/paper/2412.07378/) (2024)
- [Toward Robust Signed Graph Learning through Joint Input-Target Denoising](https://mlsanity.com/paper/2510.22513/) (2025)

## Abstract (arXiv)

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.