Abstract
We study two models of growing recursive trees. For both models, the tree initially contains a single vertex and at each time a new vertex is added to the tree and its parent is chosen randomly according to some rule. In the weighted recursive tree, we choose the parent of among with probability proportional to , where is some deterministic sequence that we fix beforehand. In the affine preferential attachment tree with fitnesses, the probability of choosing any is proportional to , where denotes its current number of children, and the sequence of fitnesses is deterministic and chosen as a parameter of the model.
We show that for any sequence , 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
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
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