# A Characterization of Linkage-Based Hierarchical Clustering

Margareta Ackerman, Shai Ben-David.

Year: 2016, Volume: 17, Issue: 231, Pages: 1−17

#### Abstract

The class of linkage-based algorithms is perhaps the most popular class of hierarchical algorithms. We identify two properties of hierarchical algorithms, and prove that linkage- based algorithms are the only ones that satisfy both of these properties. Our characterization clearly delineates the difference between linkage-based algorithms and other hierarchical methods. We formulate an intuitive notion of locality of a hierarchical algorithm that distinguishes between linkage-based and global hierarchical algorithms like bisecting $k$-means, and prove that popular divisive hierarchical algorithms produce clusterings that cannot be produced by any linkage-based algorithm.