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2021 Geometry of weighted recursive and affine preferential attachment trees
Delphin Sénizergues
Author Affiliations +
Electron. J. Probab. 26: 1-56 (2021). DOI: 10.1214/21-EJP640

Abstract

We study two models of growing recursive trees. For both models, the tree initially contains a single vertex u1 and at each time n2 a new vertex un is added to the tree and its parent is chosen randomly according to some rule. In the weighted recursive tree, we choose the parent uk of un among {u1,u2,,un1} with probability proportional to wk, where (wn)n1 is some deterministic sequence that we fix beforehand. In the affine preferential attachment tree with fitnesses, the probability of choosing any uk is proportional to ak+deg+(uk), where deg+(uk) denotes its current number of children, and the sequence of fitnesses (an)n1 is deterministic and chosen as a parameter of the model.

We show that for any sequence (an)n1, the corresponding preferential attachment tree has the same distribution as some weighted recursive tree with a random sequence of weights (with some explicit distribution). We then prove almost sure scaling limit convergences for some statistics associated with weighted recursive trees as time goes to infinity, such as degree sequence, height, profile and also the weak convergence of some measures carried on the tree. Thanks to the connection between the two models, these results also apply to affine preferential attachment trees.

Acknowledgments

The author would like to thank the multiple anonymous referees for their numerous comments and suggestions that helped improve the presentation of this paper. He would also like to thank Philippe Marchal whose remarks led to an improvement in the generality of Proposition 5.1(ii).

Citation

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Delphin Sénizergues. "Geometry of weighted recursive and affine preferential attachment trees." Electron. J. Probab. 26 1 - 56, 2021. https://doi.org/10.1214/21-EJP640

Information

Received: 30 May 2020; Accepted: 4 May 2021; Published: 2021
First available in Project Euclid: 2 June 2021

Digital Object Identifier: 10.1214/21-EJP640

Subjects:
Primary: 05C05, 60J05

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