Bu{\ss} et al [KDD 2020] recently proved that the problem of computing the betweenness of all nodes of a temporal graph is computationally hard in the case of foremost and fastest paths, while it is solvable in time O(n 3 T 2 ) in the case of shortest and shortest foremost paths, where n is the number of nodes and T is the number of distinct time steps. A new algorithm for temporal betweenness computation is introduced in this paper. In the case of shortest and shortest foremost paths, it requires O(n + M ) space and runs in time where M is the number of temporal edges, thus significantly improving the algorithm of Bu{\ss} et al in terms of time complexity (note that T is usually large). Experimental evidence is provided that our algorithm performs between twice and almost 250 times better than the algorithm of Bu{\ss} et al. Moreover, we were able to compute the exact temporal betweenness values of several large temporal graphs with over a million of temporal edges. For such size, only approximate computation was possible by using the algorithm of Santoro and Sarpe [WWW 2022]. Maybe more importantly, our algorithm extends to the case of restless walks (that is, walks with waiting constraints in each node), thus providing a polynomial-time algorithm (with complexity O(nM )) for computing the temporal betweenness in the case of several different optimality criteria. Such restless computation was known only for the shortest criterion (Rymar et al [JGAA 2023]), with complexity O(n 2 M T 2 ). We performed an extensive experimental validation by comparing different waiting constraints and different optimisation criteria. Moreover, as a case study, we investigate six public transit networks including Berlin, Rome, and Paris. Overall we find a general consistency between the different variants of betweenness centrality. However, we do measure a sensible influence of waiting constraints, and note some cases of low correlation for certain pairs of criteria in some networks.

翻译：Bu{\ss}等人[KDD 2020]近期证明，在时序图中计算所有节点的中介中心性，在最早路径和最快路径情形下是计算困难的，而在最短路径和最短最早路径情形下可在O(n³T²)时间内求解，其中n为节点数，T为不同时间步数。本文提出一种新的时序中介中心性计算算法。在最短路径和最短最早路径情形下，该算法需要O(n+M)空间，运行时间为O(nM)，其中M为时序边数量，从而在时间复杂度上显著改进了Bu{\ss}等人的算法（需注意T通常较大）。实验证据表明，我们的算法性能比Bu{\ss}等人的算法提升2倍至近250倍。此外，我们成功计算了多个包含超百万条时序边的大型时序图的精确中介中心性值。对于此等规模的数据，此前仅能使用Santoro和Sarpe[WWW 2022]的算法进行近似计算。更重要的是，我们的算法可扩展至无休游走情形（即在每个节点具有等待约束的游走），从而为多种不同最优性准则下的时序中介中心性计算提供了多项式时间算法（复杂度为O(nM)）。此类无休计算此前仅针对最短路径准则已知（Rymar等人[JGAA 2023]，复杂度为O(n²MT²)）。我们通过比较不同等待约束和不同优化准则进行了广泛的实验验证。此外，作为案例研究，我们分析了包括柏林、罗马和巴黎在内的六个公共交通网络。总体而言，我们发现中介中心性的不同变体间具有普遍一致性。然而，我们确实观测到等待约束的显著影响，并注意到在某些网络中特定准则对之间存在低相关性情况。