Betweenness centrality profiles in trees

Betweenness centrality of a vertex in a graph measures the fraction of shortest paths going through the vertex. This is a basic notion for determining the importance of a vertex in a network. The k-betweenness centrality of a vertex is defined similarly, but only considers shortest paths of length at most k. The sequence of k-betweenness centralities for all possible values of k forms the betweenness centrality profile of a vertex. We study properties of betweenness centrality profiles in trees. We show that for scale-free random trees, for fixed k, the expectation of k-betweenness centrality strictly decreases as the index of the vertex increases. We also analyze worst-case properties of profiles in terms of the distance of profiles from being monotone, and the number of times pairs of profiles can cross. This is related to whether k-betweenness centrality, for small values of k, may be used instead of having to consider all shortest paths. Bounds are given that are optimal in order of magnitude. We also present some experimental results for scale-free random trees.