Node centrality based on its edges importance: The Position centrality

We propose a novel family of node centralities in social networks, named family of position centralities, which explicitly takes into account the importance of the links to assess the centrality of the nodes that support them through the Position value (Meessen, 1988). Our proposal shares with the family of Myerson centralities (Gómez et al., 2003) that it is a game-theoretic family of measures that allows to consider the functionality of the network modelled by a symmetric cooperative game. We prove that, like the Myerson centrality measures, every Position centrality measure also satisfies essential properties expected of a centrality measure. We analyse in detail the main differences between the Myerson and the position families of centrality measures. Specifically, we study the differences regarding the connection structures that share dividends and the fairness and stability properties. Along this analysis we consider the case of hub-and-spoke clusters, a prevalent model for studying transportation networks. Finally, a characterisation of the Position Attachment centrality is given, which is the Position centrality obtained when the functionality of the network is modelled by the attachment game. Some comparisons are made with the Attachment centrality introduced by Skibski et al. (2019), which is the analogue member of the family of Myerson centralities.