# Measuring the stability of spectral clustering

## TL;DR

This paper introduces new measures for the stability of spectral clustering based on the minimal perturbation needed to cause ambiguity in the spectral gap, and proposes algorithms to compute these measures.

## Contribution

It defines structured and unstructured distances to ambiguity for spectral clustering stability and develops a two-level iterative algorithm to compute these measures.

## Key findings

- Structured and unstructured stability indicators can differ significantly.
- Maximal stability criterion may suggest different cluster counts depending on the measure.
- Numerical experiments demonstrate the effectiveness of the proposed stability measures.

## Abstract

As an indicator of the stability of spectral clustering of an undirected weighted graph into $k$ clusters, the $k$th spectral gap of the graph Laplacian is often considered. The $k$th spectral gap is characterized in this paper as an unstructured distance to ambiguity, namely as the minimal distance of the Laplacian to arbitrary symmetric matrices with vanishing $k$th spectral gap. As a conceptually more appropriate measure of stability, the structured distance to ambiguity of the $k$-clustering is introduced as the minimal distance of the Laplacian to Laplacians of graphs with the same vertices and edges but with weights that are perturbed such that the $k$th spectral gap vanishes. To compute a solution to this matrix nearness problem, a two-level iterative algorithm is proposed that uses a constrained gradient system of matrix differential equations in the inner iteration and a one-dimensional optimization of the perturbation size in the outer iteration. The structured and unstructured distances to ambiguity are compared on some example graphs. The numerical experiments show, in particular, that selecting the number $k$ of clusters according to the criterion of maximal stability can lead to different results for the structured and unstructured stability indicators.