June 2022 Root finding algorithms and persistence of Jordan centrality in growing random trees
Sayan Banerjee, Shankar Bhamidi
Author Affiliations +
Ann. Appl. Probab. 32(3): 2180-2210 (June 2022). DOI: 10.1214/21-AAP1731


We consider models of growing random trees {Tf(n):n1} with model dynamics driven by an attachment function f:Z+R+. At each stage a new vertex enters the system and connects to a vertex v in the current tree with probability proportional to f(degree(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 K(ε) in order to recover the root with probability at least 1ε. 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 n such that for nn the identity of the K-optimal Jordan centers in {Tf(n):nn} 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 {Tf(n):n1} can be embedded, as well as rates of convergence results to these limits.

Funding Statement

S. Bhamidi was partially supported by NSF Grants DMS-1613072, DMS-1606839 and ARO Grant W911NF-17-1-0010.


We acknowledge valuable feedback from two anonymous referees which led to major improvements in the presentation and clarity of this article.


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Sayan Banerjee. Shankar Bhamidi. "Root finding algorithms and persistence of Jordan centrality in growing random trees." Ann. Appl. Probab. 32 (3) 2180 - 2210, June 2022. https://doi.org/10.1214/21-AAP1731


Received: 1 July 2020; Revised: 1 May 2021; Published: June 2022
First available in Project Euclid: 29 May 2022

MathSciNet: MR4430011
zbMATH: 1503.05107
Digital Object Identifier: 10.1214/21-AAP1731

Primary: 05C80 , 05C85
Secondary: 60J80 , 60J85

Keywords: centrality measures , continuous time branching processes , Jordan centrality , Malthusian rate of growth , Random trees , recursive distributional equations , stable age distribution theory , temporal networks

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.32 • No. 3 • June 2022
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