On the structure and values of betweenness centrality in dense betweenness-uniform graphs

Abstract

Betweenness centrality is a network centrality measure based on the amount of shortest paths passing through a given vertex. A graph is betweenness-uniform (BUG) if all vertices have an equal value of betweenness centrality. In this contribution, we focus on betweenness-uniform graphs with betweenness centrality below one. We disprove a conjecture about the existence of a BUG with betweenness value $\alpha$ for any rational number $\alpha$ from the interval $(\sfrac{3}{4}, \infty)$ by showing that only very few betweenness centrality values below $\sfrac{6}{7}$ are attained for at least one BUG. Furthermore, among graphs with diameter at least three, there are no betweenness-uniform graphs with a betweenness centrality smaller than one. In graphs of smaller diameter, the same can be shown under a uniformity condition on the components of the complement.