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 $|S| K^2$ steps it builds an edge-weighted linkage graph $(S, \mathcal{L}, \sigma)$ where $\sigma(\{x, y\})$ is called the in-sway between objects $x$ and $y$. Take $\mathcal{L}_t$ to be the links whose in-sway is at least $t$, and partition $S$ into components of the graph $(S, \mathcal{L}_t)$, 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.

翻译：基于排序的链接是一种根据对象间关系总结集合 $S$ 的新工具。这些对象无需映射为向量，对象间的"相似性"既不必是数值型的，也无需是对称的。对象仅需通过传递性比较器（但无需与整个集合上的任何度量一致）按与自身的相似度对邻近对象进行排序。将此称为 $S$ 上的排序系统。基于排序的链接应用于从排序系统导出的 $K$ 近邻有向图。计算过程发生在一个二维抽象有向单纯复形上，该复形的面由 $S$ 上无向 $K$ 近邻图线图中的点、边和三角构成。通过 $|S| K^2$ 步，算法构建出边加权链接图 $(S, \mathcal{L}, \sigma)$，其中 $\sigma(\{x, y\})$ 称为对象 $x$ 与 $y$ 之间的内摇摆度。取 $\mathcal{L}_t$ 为内摇摆度至少为 $t$ 的链接，并将 $S$ 划分为图 $(S, \mathcal{L}_t)$ 的分量（随 $t$ 变化）。基于排序的链接是从外序有向图范畴到划分集范畴的函子，其实际意义在于：以尊重排序的方式扩充对象集可生成新聚类，且不会"撕裂"原有聚类。这一性质在度量空间情境下的单链接聚类中同样成立，但典型的优化方法则不具备。最后一节提出了若干开放组合问题。