时态有向网络中的动态层次结构
Dynamic hierarchies in temporal directed networks
摘要：
在许多现实世界的系统中，交互的结果通常可以用参与者之间的层次结构来解释。从一个给定的有向网络中发现层次结构可以用公式表示如下：将顶点划分为层次，使得理想情况下只有前向边，即从上层到下层的边。在实践中，理想的情况是不可能的，所以我们最小化后向边缘上的一些惩罚函数。这种惩罚的一个实际选择是痛苦，惩罚取决于违法行为的严重程度。在本文中，我们扩展的定义痛苦的时间网络。在这个设置中，我们给出了一个带有时间戳边缘的有向网络，并且我们允许秩分配随时间变化。我们提出了两种策略来控制个体等级的变化。在我们的第一个变体中，我们通过直接向优化函数添加惩罚来惩罚排名随时间的波动。在我们的第二个变体中，我们允许最多改变一次等级。我们表明，第一个变量可以解决确切的多项式时间，而第二个变量是NP-困难的，实际上是不可逼近的。然而，我们开发了一种迭代法，首先固定变点并优化秩，然后固定秩并优化变点，并重复直到收敛。我们的经验表明，算法在实践中是相当快的，得到的排名是明智的。
英文摘要：
The outcome of interactions in many real-world systems can be often explained by a hierarchy between the participants. Discovering hierarchy from a given directed network can be formulated as follows: partition vertices into levels such that, ideally, there are only forward edges, that is, edges from upper levels to lower levels. In practice, the ideal case is impossible, so instead we minimize some penalty function on the backward edges. One practical option for such a penalty is agony, where the penalty depends on the severity of the violation. In this paper we extend the definition of agony to temporal networks. In this setup we are given a directed network with time stamped edges, and we allow the rank assignment to vary over time. We propose 2 strategies for controlling the variation of individual ranks. In our first variant, we penalize the fluctuation of the rankings over time by adding a penalty directly to the optimization function. In our second variant we allow the rank change at most once. We show that the first variant can be solved exactly in polynomial time while the second variant is NP-hard, and in fact inapproximable. However, we develop an iterative method, where we first fix the change point and optimize the ranks, and then fix the ranks and optimize the change points, and reiterate until convergence. We show empirically that the algorithms are reasonably fast in practice, and that the obtained rankings are sensible.
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