  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.
