Analysis of centrality in sublinear preferential attachment trees via the CMJ branching process

We investigate centrality properties and the existence of a finite confidence set for the rootnode in growing random tree models. We show that a continuous time branching processescalled the Crump-Mode-Jagers (CMJ) branching process is well-suited to analyze such randomtrees, and establish centrality and root inference properties of sublinear preferential attachmenttrees. We show that with probability 1, there exists a persistent tree centroid; i.e., a vertex thatremains the tree centroid after a finite amount of time. Furthermore, we show that the samecentrality criterion produces a finite-sized 1− ǫ confidence set for the root node, for any ǫ > 0.