Group Vitality Indices: Axioms and Algorithms

2026-05-10

AI Summary

This study addresses the evaluation of collective node importance in networks by analyzing the impact of removing groups of nodes on network functionality. Building upon the vitality index framework, the authors introduce the group Shapley value to model group importance and establish, for the first time, that any individual vitality index admits a unique extension to the group setting. They propose two classes of normalized group vitality indices and provide an axiomatic characterization for this family of measures. By integrating cooperative game theory, graph-theoretic centrality, and axiomatic analysis, the work constructs a comprehensive theoretical foundation for group vitality indices, clarifies their computational complexity, and offers an in-depth examination of specific instances such as Group Attachment Centrality.

Abstract

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.

Problem

• group vitality
• centrality measures
• network analysis
• node importance
• Shapley value

Innovation

• group vitality indices
• Shapley value
• axiomatization
• centrality measures
• computational complexity