We are interested in, firstly, measuring the theoretical performance of hierarchical clustering algorithms which depend on a scale parameter; and secondly, in improving the State-of-the-Art of this family of hierarchical clustering with respect to our new measure. Single-Linkage is perhaps the simplest and the most famous algorithm belonging to this family. Nowadays, the State-of-the-Art clustering algorithm, HDBSCAN, works in a similar way to Single-Linkage (with some refinements we will study later). Herein, we explain why the percolation phenomenon is omnipresent behind this family of clustering algorithms. Previously, we defined an index which we have named percolation rate to measure the theoretical capability of algorithms to identify different high-density levels. In this paper, we show that using hypergraphs is a natural way to generalize Single-Linkage with higher-order interactions (not just ‘single’). New high-order connected components on hypergraphs, we call K-polyhedra, have much better percolation rates than the classic ‘robustification’ of Single-Linkage (used e.g. by HDBSCAN), the K-Robust Single-Linkage components. We investigate in detail the important cases of \(\mathbb R^2\) and \(\mathbb R^3\) for \(K \in \{1,2,3\}\) .