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 root node in growing random tree models. We show that a continuous time branching processes called the Crump-Mode-Jagers (CMJ) branching process is well-suited to analyze such random trees, and establish centrality and root inference properties of sublinear preferential attachment trees. We show that with probability 1, there exists a persistent tree centroid; i.e., a vertex that remains the tree centroid after a finite amount of time. Furthermore, we show that the same centrality criterion produces a finite-sized $1 - \epsilon$ confidence set for the root node, for any $\epsilon > 0$.