#### Vol. 12, No. 1, 2019

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Intersecting geodesics and centrality in graphs

### Emily Carter, Bryan Ek, Danielle Gonzalez, Rigoberto Flórez and Darren A. Narayan

Vol. 12 (2019), No. 1, 31–43
##### 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.

##### Keywords
betweenness centrality, shortest paths, distance
##### Mathematical Subject Classification 2010
Primary: 05C12, 05C82
##### Milestones
Received: 4 March 2017
Revised: 26 July 2017
Accepted: 20 January 2018
Published: 31 May 2018

Communicated by Kenneth S. Berenhaut
##### Authors
 Emily Carter School of Mathematical Sciences Rochester Institute of Technology Rochester, NY United States Bryan Ek Department of Mathematics Rutgers University Piscataway, NJ United States Danielle Gonzalez Department of Software Engineering Rochester Institute of Technology Rochester, NY United States Rigoberto Flórez Department of Mathematics and Computer Science The Citadel Charleston, SC United States Darren A. Narayan School of Mathematical Sciences Rochester Institute of Technology Rochester, NY United States