Intersecting geodesics and centrality in graphs

Abstract

In a graph, vertices that are more central are often placed at the intersection of geodesics between other pairs of vertices. This model can be applied to organizational networks, where we assume the flow of information follows shortest paths of communication and there is a required action (i.e., signature or approval) by each person located on these paths. The number of actions a person must perform is linked to both the topology of the network as well as their location within it. The number of expected actions that a person must perform can be quantified by betweenness centrality. The betweenness centrality of a vertex $v$ is the ratio of shortest paths between all other pairs of vertices $u$ and $w$ in which $v$ appears to the total number of shortest paths from $u$ to $w$. We precisely compute the betweenness centrality for vertices in several families of graphs motivated by different organizational networks.