Group Vitality Indices: Axioms and Algorithms
For network analysts, this work offers a principled framework to extend node-level vitality measures to groups, but the contribution is primarily theoretical and incremental.
This paper studies group vitality indices for assessing groups of nodes in networks, showing that every vitality index has a unique group extension via a Shapley-like value. It provides an axiomatization of the class and analyzes computational properties.
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.