The betweenness centrality of a graph vertex measures how often this vertex is visited on shortest paths between other vertices of the graph. In the analysis of many real-world graphs or networks, betweenness centrality of a vertex is used as an indicator for its relative importance in the network. In particular, it is among the most popular tools in social network analysis. In recent years, a growing number of real-world networks is modeled as temporal graphs, where we have a fixed set of vertices and there is a finite discrete set of time steps and every edge might be present only at some time steps. While shortest paths are straightforward to define in static graphs, temporal paths can be considered "optimal" with respect to many different criteria, including length, arrival time, and overall travel time (shortest, foremost, and fastest paths). This leads to different concepts of temporal betweenness centrality and we provide a systematic study of temporal betweenness variants based on various concepts of optimal temporal paths. Computing the betweenness centrality for vertices in a graph is closely related to counting the number of optimal paths between vertex pairs. We show that counting foremost and fastest paths is computationally intractable (#P-hard) and hence the computation of the corresponding temporal betweenness values is intractable as well. For shortest paths and two selected special cases of foremost paths, we devise polynomial-time algorithms for temporal betweenness computation. Moreover, we also explore the distinction between strict (ascending time labels) and non-strict (non-descending time labels) time labels in temporal paths. In our experiments with established real-world temporal networks, we demonstrate the practical effectiveness of our algorithms, compare the various betweenness concepts, and derive recommendations on their practical use.

中文翻译：

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时间介数的算法方面

图顶点的介数中心性衡量在图的其他顶点之间的最短路径上访问该顶点的频率。在许多现实世界的图或网络的分析中，顶点的介数中心性被用作其在网络中相对重要性的指标。特别是，它是社交网络分析中最受欢迎的工具之一。近年来，越来越多的现实世界网络被建模为时间图，其中我们有一组固定的顶点和一组有限的离散时间步，每条边可能只出现在某些时间步。虽然在静态图中可以直接定义最短路径，但相对于许多不同的标准，包括长度、到达时间和总旅行时间（最短、最重要、和最快的路径）。这导致了时间中介中心性的不同概念，我们提供了基于最佳时间路径的各种概念的时间中介变体的系统研究。计算图中顶点的介数中心性与计算顶点对之间的最佳路径数密切相关。我们表明，计算最重要和最快的路径在计算上是难以处理的（#P-hard），因此相应的时间介数值的计算也是难以处理的。对于最短路径和最重要路径的两个选定的特殊情况，我们设计了多项式时间算法来进行时间中介计算。此外，我们还探讨了时间路径中严格（上升时间标签）和非严格（非下降时间标签）时间标签之间的区别。