Root finding algorithms and persistence of Jordan centrality in growing random trees

We consider models of growing random trees {Tf(n):n?1} with model dynamics driven by an attachment function f:Z+?R+. At each stage a new vertex enters the system and connects to v in current tree probability proportional f(degree(v)). The main goal this study is understand performance root finding algorithms. A large body work (e.g., Random Structures Algorithms 50 (2017) 158–172; IEEE Trans. Netw. Sci. Eng. 4 1–12; 52 (2018) 136–157) has emerged last few years using techniques based on Jordan centrality measure (J. Reine Angew. Math. 70 (1869) 185–190) its variants develop Given unlabeled unrooted tree, one computes for fixed budget K outputs optimal vertices (as measured centrality). Under general conditions f, we derive necessary sufficient bounds K(?) order recover at least 1??. For canonical examples such as linear preferential uniform attachment, these results give matching upper lower budget. also prove persistence centers any K, that is, existence almost surely finite time n? n?n? identity K-optimal {Tf(n):n?n?} does not change, thus describing robustness properties measure. Key technical ingredients proofs independent interest include exponential moments limits (appropriately normalized) continuous branching processes within which can be embedded, well rates convergence limits.