Measuring stability of spectral clustering

As an indicator of the stability of spectral clustering of an undirected weighted graph into k clusters, the kth spectral gap of the graph Laplacian is often considered. The k-th spectral gap is characterized here as an unstructured distance to ambiguity, namely as the minimal distance of the Laplacian to arbitrary symmetric matrices with vanishing kth spectral gap. As a 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 the same graph 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