April 2022 Correlated randomly growing graphs
Miklós Z. Rácz, Anirudh Sridhar
Author Affiliations +
Ann. Appl. Probab. 32(2): 1058-1111 (April 2022). DOI: 10.1214/21-AAP1703


We introduce a new model of correlated randomly growing graphs and study the fundamental questions of detecting correlation and estimating aspects of the correlated structure. The model is simple and starts with any model of randomly growing graphs, such as uniform attachment (UA) or preferential attachment (PA). Given such a model, a pair of graphs (G1,G2) is grown in two stages: until time t they are grown together (i.e., G1=G2), after which they grow independently according to the underlying growth model.

We show that whenever the seed graph has an influence in the underlying graph growth model—this has been shown for PA and UA trees and is conjectured to hold broadly—then correlation can be detected in this model, even if the graphs are grown together for just a single time step. We also give a general sufficient condition (which holds for PA and UA trees) under which detection is possible with probability going to 1 as t. Finally, we show for PA and UA trees that the amount of correlation, measured by t, can be estimated with vanishing relative error as t.

Funding Statement

The research of M.Z.R. was supported in part by NSF Grant DMS 1811724 and by a Princeton SEAS Innovation Award.
The research of A.S. was supported in part by NSF Grant DMS 1811724.


We thank the anonymous reviewers and an anonymous Associate Editor for helpful comments and feedback which helped improve the manuscript.


Download Citation

Miklós Z. Rácz. Anirudh Sridhar. "Correlated randomly growing graphs." Ann. Appl. Probab. 32 (2) 1058 - 1111, April 2022. https://doi.org/10.1214/21-AAP1703


Received: 1 July 2020; Revised: 1 May 2021; Published: April 2022
First available in Project Euclid: 28 April 2022

MathSciNet: MR4414701
zbMATH: 1502.60011
Digital Object Identifier: 10.1214/21-AAP1703

Primary: 05C80 , 60C05

Keywords: correlated random graphs , influence of the seed , preferential attachment , Random graphs , randomly growing graphs , uniform attachment

Rights: Copyright © 2022 Institute of Mathematical Statistics


This article is only available to subscribers.
It is not available for individual sale.

Vol.32 • No. 2 • April 2022
Back to Top