In this paper, we study the problem of discovering a timeline of events in a temporal network. We model events as dense subgraphs that occur within intervals of network activity. We formulate the event discovery task as an optimization problem, where we search for a partition of the network timeline into k non-overlapping intervals, such that the intervals span subgraphs with maximum total density. The output is a sequence of dense subgraphs along with corresponding time intervals, capturing the most interesting events during the network lifetime. A naïve solution to our optimization problem has polynomial but prohibitively high running time. We adapt existing recent work on dynamic densest subgraph discovery and approximate dynamic programming to design a fast approximation algorithm. Next, to ensure richer structure, we adjust the problem formulation to encourage coverage of a larger set of nodes. This problem is NP-hard; however, we show that on static graphs a simple greedy algorithm leads to approximate solution due to submodularity. We extend this greedy approach for temporal networks, but we lose the approximation guarantee in the process. Finally, we demonstrate empirically that our algorithms recover solutions with good quality.

中文翻译：

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在时间网络中查找事件：分割遇到最密集的子图发现

在本文中，我们研究了在时间网络中发现事件时间线的问题。我们将事件建模为在网络活动间隔内发生的密集子图。我们将事件发现任务公式化为一个优化问题，在其中我们将网络时间轴的一部分搜索为k非重叠间隔，以使间隔跨越具有最大总密度的子图。输出是一系列密集的子图以及相应的时间间隔，可捕获网络生命周期中最有趣的事件。对于我们的优化问题而言，幼稚的解决方案具有多项式，但是运行时间过长。我们采用现有的有关动态最密子图发现和近似动态编程的最新工作来设计快速近似算法。接下来，为了确保结构更丰富，我们调整问题的表述以鼓励覆盖更大的节点集。这个问题是NP-硬; 但是，我们表明，在静态图上，由于子模量的原因，一种简单的贪心算法导致了近似解。我们将这种贪婪方法扩展到时间网络，但是在此过程中我们失去了近似保证。最后，我们凭经验证明我们的算法可以回收高质量的解决方案。