We consider the problem of assessing a group of nodes in a network. Our focus is on vitality indices -- a natural class of centrality measures that evaluate the importance of a node by examining the impact of its removal on the network. We conduct a comprehensive analysis of group vitality indices. Specifically, we show that every vitality index admits a unique extension to groups, which can be defined using a group variant of the Shapley value recently proposed in the literature. We also provide an axiomatization of the entire class, along with two specific group vitality indices that satisfy additional normalization conditions. Furthermore, we study the computational properties of all vitality indices, as well as Group Attachment Centrality.

翻译：我们考虑评估网络中一组节点的问题。重点关注活力指数——一种通过考察节点移除对网络的影响来衡量节点重要性的自然中心性度量类别。我们进行了群体活力指数的全面分析。具体而言，我们证明每个活力指数都允许唯一的群体扩展，该扩展可使用文献中最近提出的夏普利值的群体变体来定义。我们还提供了整个类别的公理化，以及两个满足额外标准化条件的特定群体活力指数。此外，我们研究了所有活力指数以及群体附加中心性的计算特性。