Rank-based linkage is a new tool for summarizing a collection S of objects
according to their relationships. These objects are not mapped to vectors, and
``similarity'' between objects need be neither numerical nor symmetrical. All
an object needs to do is rank nearby objects by similarity to itself, using a
Comparator which is transitive, but need not be consistent with any metric on
the whole set. Call this a ranking system on S. Rank-based linkage is applied
to the K-nearest neighbor digraph derived from a ranking system. Computations
occur on a 2-dimensional abstract oriented simplicial complex whose faces are
among the points, edges, and triangles of the line graph of the undirected
K-nearest neighbor graph on S. In β£Sβ£K2 steps it builds an
edge-weighted linkage graph (S,L,Ο) where Ο({x,y})
is called the in-sway between objects x and y. Take Ltβ to be
the links whose in-sway is at least t, and partition S into components of
the graph (S,Ltβ), for varying t. Rank-based linkage is a
functor from a category of out-ordered digraphs to a category of partitioned
sets, with the practical consequence that augmenting the set of objects in a
rank-respectful way gives a fresh clustering which does not ``rip apart`` the
previous one. The same holds for single linkage clustering in the metric space
context, but not for typical optimization-based methods. Open combinatorial
problems are presented in the last section.Comment: 37 pages, 12 figure