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Spectral clustering is a well-known technique which identifies $k$ clustersin an undirected graph with weight matrix $W\in\mathbb{R}^{n\times n}$ byexploiting its graph Laplacian $L(W)$, whose eigenvalues $0=\lambda_1\leq\lambda_2 \leq \dots \leq \lambda_n$ and eigenvectors are related to the $k$clusters. Since the computation of $\lambda_{k+1}$ and $\lambda_k$ affects thereliability of this method, the $k$-th spectral gap $\lambda_{k+1}-\lambda_k$is often considered as a stability indicator. This difference can be seen as anunstructured distance between $L(W)$ and an arbitrary symmetric matrix$L_\star$ with vanishing $k$-th spectral gap. A more appropriate structureddistance to ambiguity such that $L_\star$ represents the Laplacian of a graphhas been proposed by Andreotti et al. (2021). Slightly differently, we considerthe objective functional $F(\Delta)=\lambda_{k+1}\left(L(W+\Delta)\right)-\lambda_k\left(L(W+\Delta)\right)$,where $\Delta$ is a perturbation such that $W+\Delta$ has non-negative entriesand the same pattern of $W$. We look for an admissible perturbation$\Delta_\star$ of smallest Frobenius norm such that $F(\Delta_\star)=0$. Inorder to solve this optimization problem, we exploit its low rank underlyingstructure. We formulate a rank-4 symmetric matrix ODE whose stationary pointsare the optimizers sought. The integration of this equation benefits from thelow rank structure with a moderate computational effort and memory requirement,as it is shown in some illustrative numerical examples.

A low rank ODE for spectral clustering stability