This is an extended abstract of our article entitled Higher-Order Spectral Clustering for Geometric Graphs, and which is devoted to clustering geometric graphs.

While the standard spectral clustering is often not effective for geometric graphs, we present an effective generalization, which we call higher-order spectral clustering.

It resembles in concept the classical spectral clustering method but uses for partitioning the eigenvector associated with a higher-order eigenvalue. We establish the weak consistency of this algorithm for a wide class of geometric graphs which we call Soft Geometric Block Model. A small adjustment of the algorithm provides strong consistency. We also show that our method is effective in numerical experiments even for graphs of modest size.

Reference of the full article: Konstantin Avrachenkov, Andrei Bobu, Maximilien Dreveton. Higher-Order Spectral Clustering for Geometric Graphs. \textit{Journal of Fourier Analysis and Applications}, 27(2), 1-29, 2021.