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

Abstract

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