# The position value as a centrality measure in social networks

## TL;DR

This paper explores the position value as a centrality measure in social networks, analyzing its properties, extending previous work on related measures, and comparing it with other centrality concepts.

## Contribution

It extends the analysis of the position value as a centrality measure, characterizes its properties, and compares it with existing attachment centrality measures.

## Key findings

- Position centrality responds more versatile to edge addition or removal.
- The paper characterizes the position attachment centrality derived from the position value.
- Comparisons with Skibski et al.'s attachment centrality highlight differences in response to network changes.

## Abstract

The position value, introduced by Meessen (1988), is a solution concept for cooperative games in which the value assigned to a player depends on the value of the connections or links he has with other players. This concept has been studied by Borm et al. (1992) and characterised by Slikker (2005). In this paper, we analyse the position value from the point of view of the typical properties of a measure of centrality in a social network. We extend the analysis already developed in Gomez et al. (2003) for the Myerson centrality measure, where the symmetric effect on the centralities of the end nodes of an added or removed edge is a fundamental part of its characterisation. However, the Position centrality measure, unlike the Myerson centrality measure, responds in a more versatile way to such addition or elimination. After studying the aforementioned properties, we will focus on the analysis and characterisation of the Position attachment centrality given by the position value when the underlying game is the attachment game. Some comparisons are made with the attachment centrality introduced by Skibski et al. (2019).