Group Vitality Indices: Axioms and Algorithms

TL;DR

This paper introduces and analyzes group vitality indices as a class of centrality measures assessing node importance by their impact on network connectivity, providing axioms, algorithms, and computational insights.

Contribution

It extends vitality indices to groups using a Shapley value variant, offers axiomatization, and studies their computational properties.

Findings

Unique extension of vitality indices to groups via a Shapley value variant

Axiomatization of the entire class of group vitality indices

Analysis of computational complexity of vitality indices and 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.