更快的痛苦之路：在有向图中发现层次结构
Faster way to agony: Discovering hierarchies in directed graphs
摘要：
许多现实世界的现象表现出强烈的层次结构。因此，在许多现实世界中的有向社交网络顶点不发挥平等的作用。相反，顶点形成层次结构，使得边缘主要从上层到下层出现。从这样的图中发现层次结构是一个具有挑战性的问题，已经引起了人们的关注。形式上，给定一个有向图，我们希望将顶点划分为级别，使得理想情况下只有从上层到下层的边。从计算的角度来看，理想的情况是当底层有向图是非循环的。在这种情况下，我们可以将顶点划分为层次结构，使得从上层到下层只有边。在实践中，图很少是非循环的，因此我们需要惩罚违反层次结构的边。一种实用的方法是痛苦，其中每个违反边都根据违反的严重程度受到惩罚。计算痛苦的最快算法需要$O(nm^2)$时间。本文提出了一种计算痛苦的算法，即$O(m^2)$，它具有较好的理论界。我们还表明，在实践中得到的界限是悲观的，我们可以使用我们的算法来计算大型数据集的痛苦。此外，我们的算法可以被用作任何时间的算法。
英文摘要：
Many real-world phenomena exhibit strong hierarchical structure. Consequently, in many real-world directed social networks vertices do not play equal role. Instead, vertices form a hierarchy such that the edges appear mainly from upper levels to lower levels. Discovering hierarchies from such graphs is a challenging problem that has gained attention. Formally, given a directed graph, we want to partition vertices into levels such that ideally there are only edges from upper levels to lower levels. From computational point of view, the ideal case is when the underlying directed graph is acyclic. In such case, we can partition the vertices into a hierarchy such that there are only edges from upper levels to lower edges. In practice, graphs are rarely acyclic, hence we need to penalize the edges that violate the hierarchy. One practical approach is agony, where each violating edge is penalized based on the severity of the violation. The fastest algorithm for computing agony requires $O(nm^2)$ time. In the paper we present an algorithm for computing agony that has better theoretical bound, namely $O(m^2)$. We also show that in practice the obtained bound is pessimistic and that we can use our algorithm to compute agony for large datasets. Moreover, our algorithm can be used as any-time algorithm.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
