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：基于秩的连接是一种新工具，用于根据对象间的关系来总结集合 $S$ 中的对象。这些对象不被映射为向量，且对象间的“相似性”既无须是数值的，也无须是对称的。每个对象只需使用一个可传递的比较器（Comparator）对邻近对象按其与自身的相似性进行排序，而该比较器无需与整个集合上的任何度量保持一致。将此称为 $S$ 上的排序系统。基于秩的连接应用于从排序系统导出的 $K$ 近邻有向图。计算发生在一个二维抽象定向单纯复形上，该复形的面包括 $S$ 的无向 $K$ 近邻图线图中的点、边和三角形。通过 $|S| K^2$ 步，它构建了一个边加权的连接图 $(S, \mathcal{L}, \sigma)$，其中 $\sigma(\{x, y\})$ 称为对象 $x$ 与 $y$ 之间的内摆度（in-sway）。令 $\mathcal{L}_t$ 为内摆度至少为 $t$ 的连接，并对于不同的 $t$ 将 $S$ 划分为图 $(S, \mathcal{L}_t)$ 的连通分量。基于秩的连接是从出序有向图范畴到分割集范畴的一个函子，其实际意义是：以尊重秩的方式扩充对象集会得到一个新的聚类，且不会“撕裂”先前的聚类。在度量空间情境下，单链接聚类也具有相同性质，但典型的基于优化的方法则不然。最后一节给出了若干开放的组合问题。