On the distribution of betweenness centrality in random trees

Betweenness centrality is a quantity that is frequently used to measure how ‘central’ a vertex v is. It is defined as the sum, over pairs of vertices other than v, of the proportions of shortest paths that pass through v. In this paper, we study the distribution of the betweenness centrality in random trees and related, subcritical graph families. Specifically, we prove that the betweenness centrality of the root vertex in a simply generated tree is usually of linear order, but has a mean of order n. We also show that a randomly chosen vertex typically also has linear-order betweenness centrality, and that the maximum betweenness centrality in a simply generated tree is of order n. We obtain limiting distributions for the betweenness centrality of the root vertex and of a randomly chosen vertex, as well as for the maximum betweenness centrality, and we also show that the centroid has positive probability in the limit to be the vertex of maximum betweenness centrality. Some similar results also hold for subcritical graph classes, which will be briefly discussed. Finally, we study random recursive trees and other families of increasing trees, where the situation is quite different: here, the root betweenness centrality is of quadratic order, as is the maximum betweenness centrality. The betweenness centrality of a random vertex, on the other hand, is again of linear order. Again, we also have limiting distributions upon suitable normalisation.