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

We investigate centrality and root-inference properties in a class of growing\nrandom graphs known as sublinear preferential attachment trees. We show that a\ncontinuous time branching processes called the Crump-Mode-Jagers (CMJ)\nbranching process is well-suited to analyze such random trees, and prove that\nalmost surely, a unique terminal tree centroid emerges, having the property\nthat it becomes more central than any other fixed vertex in the limit of the\nrandom growth process. Our result generalizes and extends previous work\nestablishing persistent centrality in uniform and linear preferential\nattachment trees. We also show that centrality may be utilized to generate a\nfinite-sized $1-\\epsilon$ confidence set for the root node, for any $\\epsilon >\n0$ in a certain subclass of sublinear preferential attachment trees.\n