# Average Sensitivity of Spectral Clustering

arXiv: https://arxiv.org/abs/2006.04094 · HTML: https://mlsanity.com/paper/2006.04094/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Jun 2020.

## Built on by (most cited first, 6 of 6)

- [A Tighter Analysis of Spectral Clustering, and Beyond](https://mlsanity.com/paper/2208.01724/) (2022)
- [Mask-GVAE: Blind Denoising Graphs via Partition](https://mlsanity.com/paper/2102.04228/) (2021)
- [A Batch-to-Online Transformation under Random-Order Model](https://mlsanity.com/paper/2306.07163/) (2023)
- [A Sublinear-Time Spectral Clustering Oracle with Improved Preprocessing Time](https://mlsanity.com/paper/2310.17878/) (2023)
- [Cluster Metric Sensitivity to Irrelevant Features](https://mlsanity.com/paper/2402.12008/) (2024)
- [Quantifying uncertainty in spectral clusterings: expectations for perturbed and incomplete data](https://mlsanity.com/paper/2505.17819/) (2025)

## Abstract (arXiv)

Spectral clustering is one of the most popular clustering methods for finding clusters in a graph, which has found many applications in data mining. However, the input graph in those applications may have many missing edges due to error in measurement, withholding for a privacy reason, or arbitrariness in data conversion. To make reliable and efficient decisions based on spectral clustering, we assess the stability of spectral clustering against edge perturbations in the input graph using the notion of average sensitivity, which is the expected size of the symmetric difference of the output clusters before and after we randomly remove edges. We first prove that the average sensitivity of spectral clustering is proportional to λ2/λ32, where λi is the i-th smallest eigenvalue of the (normalized) Laplacian. We also prove an analogous bound for k-way spectral clustering, which partitions the graph into k clusters. Then, we empirically confirm our theoretical bounds by conducting experiments on synthetic and real networks. Our results suggest that spectral clustering is stable against edge perturbations when there is a cluster structure in the input graph.