Attachment centrality: Measure for connectivity in networks

Centrality indices aim to quantify the importance of nodes or edges in a network. Much interest has been recently raised by the body of work in which a node's connectivity is understood less as its contribution to the quality or speed of communication in the network and more as its role in enabling communication altogether. Consequently, a node is assessed based on whether or not the network (or part of it) becomes disconnected if this node is removed. While these new indices deliver promising insights, to date very little is known about their theoretical properties. To address this issue, we propose an axiomatic approach. Specifically, we prove that there exists a unique centrality index satisfying a number of desirable properties. This new index, which we call the Attachment centrality, is equivalent to the Myerson value of a certain graph-restricted game. Building upon our theoretical analysis we show that, while computing the Attachment centrality is #P-complete, it has certain computational properties that are more attractive than the Myerson value for an arbitrary game. In particular, it can be computed in chordal graphs in polynomial time.