Open Access
2023 Degree centrality and root finding in growing random networks
Sayan Banerjee, Xiangying Huang
Author Affiliations +
Electron. J. Probab. 28: 1-39 (2023). DOI: 10.1214/23-EJP930


We consider growing random networks {Gn}n1 where, at each time, a new vertex attaches itself to a collection of existing vertices via a fixed number m1 of edges, with probability proportional to a function f (called attachment function) of their degree. It was shown in [BB21] that such network models exhibit two distinct regimes: (i) the persistent regime, corresponding to i=1f(i)2<, where the top K maximal degree vertices fixate over time for any given K, and (ii) the non-persistent regime, with i=1f(i)2=, where the identities of these vertices keep changing infinitely often over time. In this article, we develop root finding algorithms using the empirical degree structure and local network information based on a snapshot of such a network at some large time. In the persistent regime, the algorithm is purely based on degree centrality, that is, for a given error tolerance ε(0,1), there exists KεN such that for any n1, the confidence set for the root in Gn, which contains the root with probability at least 1ε, consists of the top Kε maximal degree vertices. In particular, the size of the confidence set is stable in the network size. Upper and lower bounds on Kε are explicitly characterized in terms of the error tolerance ε and the attachment function f. In the non-persistent regime, for an appropriate choice of rn at a rate much smaller than the diameter of the network, the neighborhood of radius rn around the maximal degree vertex is shown to contain the root with high probability, and a size estimate for this set is obtained. It is shown that, when f(k)=kα,k1, for any α(0,12], this size grows at a smaller rate than any positive power of the network size.

Funding Statement

SB was supported in part by the NSF-CAREER award (DMS-2141621) and the NSF RTG award (DMS-2134107).


We are grateful to Shankar Bhamidi for continued discussions throughout the project. We thank Pratima Hebbar for very helpful advice. We also thank an anonymous referee and an associate editor whose suggestions substantially improved this article.


Download Citation

Sayan Banerjee. Xiangying Huang. "Degree centrality and root finding in growing random networks." Electron. J. Probab. 28 1 - 39, 2023.


Received: 21 February 2022; Accepted: 7 March 2023; Published: 2023
First available in Project Euclid: 13 March 2023

Digital Object Identifier: 10.1214/23-EJP930

Primary: 05C82 , 60J28 , 60J85

Keywords: attachment functions , continuous time branching processes , degree centrality , network centrality measures , Persistence , root finding algorithms

Vol.28 • 2023
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