Abstract
We study two models of growing recursive trees. For both models, the tree initially contains a single vertex u1 and at each time n≥2 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,…,un−1} with probability proportional to wk, where (wn)n≥1 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)n≥1 is deterministic and chosen as a parameter of the model. We show that for any sequence (an)n≥1, 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.
Highlights
The uniform recursive tree has been introduced in the 70’s as an example of random graph constructed by addition of vertices: starting from a tree containing a single vertex, the vertices arrive one by one and the n-th vertex picks its parent uniformly at random from the n − 1 already present vertices
We consider a generalization of the uniform recursive tree called the weighted recursive tree (WRT), which was introduced in [8] in 2006
Using the connection from preferential attachment trees to weighted recursive trees given by Theorem 1.1, it includes the case of preferential attachment trees with constant fitnesses, for which similar results were proved, in [46] for the height and in [30, 29] for the asymptotic https://www.imstat.org/ejp behaviour of the profile (1.8)
Summary
The uniform recursive tree has been introduced in the 70’s as an example of random graph constructed by addition of vertices: starting from a tree containing a single vertex, the vertices arrive one by one and the n-th vertex picks its parent uniformly at random from the n − 1 already present vertices. The two papers [25, 20] study the properties of some more general models of increasing trees, and their results apply to WRTs when the sequence of weights is i.i.d. and identify in that case the asymptotic degree distribution in the tree. We shall see that using a de Finetti-type argument, a PAT can be seen as a WRT with a random sequence of weights that almost surely decays like a power of n This enables us to translate all of the results obtained for WRTs to corresponding results for PATs, and prove asymptotics for degrees, height and profile of the tree. This relates in various ways to other results that can be found in the literature associated to preferential attachment trees or to urn models, contained in [36, 27, 41, 26, 42, 40, 39, 1]
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