In temporal ordered clustering , given a single snapshot of a dynamic network in which nodes arrive at distinct time instants, we aim at partitioning its nodes into $K$ ordered clusters $\mathcal {C}_1 \prec \cdots \prec \mathcal {C}_K$ such that for $i< j$ , nodes in cluster $\mathcal {C}_i$ arrived before nodes in cluster $\mathcal {C}_j$ , with $K$ being a data-driven parameter and not known upfront. Such a problem is of considerable significance in many applications ranging from tracking the expansion of fake news to mapping the spread of information. We first formulate our problem for a general dynamic graph, and propose an integer programming framework that finds the optimal clustering, represented as a strict partial order set, achieving the best precision (i.e., fraction of successfully ordered node pairs) for a fixed density (i.e., fraction of comparable node pairs). We then develop a sequential importance procedure and design unsupervised and semi-supervised algorithms to find temporal ordered clusters that efficiently approximate the optimal solution. To illustrate the techniques, we apply our methods to the vertex copying (duplication-divergence) model which exhibits some edge-case challenges in inferring the clusters as compared to other network models. Finally, we validate the performance of the proposed algorithms on synthetic and real-world networks.

中文翻译：

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动态网络中的时间有序聚类：无监督和半监督学习算法

在 时间有序聚类 ，给定节点在不同时刻到达的动态网络的单个快照，我们的目标是将其节点划分为 $K$ 有序簇 $\mathcal {C}_1 \prec \cdots \prec \mathcal {C}_K$ 使得对于 $i<j$ , 集群中的节点 $\mathcal {C}_i$ 在集群中的节点之前到达 $\mathcal {C}_j$ ， 和 $K$是一个数据驱动的参数，事先不知道。这样的问题在许多应用中都具有重要意义，从跟踪假新闻的扩展到映射信息的传播。我们首先为一般动态图制定我们的问题，并提出一个整数规划框架，该框架可以找到最优聚类，表示为严格的偏序集，在固定密度下实现最佳精度（即成功排序的节点对的分数）（即，可比节点对的分数）。然后，我们开发了一个顺序重要性程序并设计了无监督和半监督算法来找到有效地逼近最优解的时间有序集群。为了说明这些技术，我们将我们的方法应用于顶点复制（重复-发散）模型，与其他网络模型相比，该模型在推断集群方面表现出一些边缘情况挑战。最后，我们验证了所提出算法在合成和现实世界网络上的性能。