Asymptotic results on Hoppe trees and their variations

Hiesmayr, Ella; Işlak, Ümi̇t

doi:10.1017/jpr.2020.12

Cited by 10 publications

(8 citation statements)

References 13 publications

(2 reference statements)

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“…The definition of the WRT model follows the one in [10]: Definition 2.1 (Weighted Recursive Tree). Let (W i ) i≥1 be a sequence of i.i.d.…”

Section: Definitions and Main Resultsmentioning

confidence: 99%

“…The Weighted Recursive Tree model (WRT), first introduced by Borovkov and Vatutin [4], is a recursive tree process (T n , n ∈ N) and a generalisation of the random recursive tree model. Here we consider a variation, first studied by Hiesmayr and Işlak [10], where the first vertex does not necessarily have weight one. Let (W i ) i∈N be a sequence of positive vertex-weights.…”

Section: Introductionmentioning

confidence: 99%

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Fine asymptotics for the maximum degree in weighted recursive trees with bounded random weights

Eslava¹,

Lodewijks²,

Ortgiese³

2021

Preprint

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A weighted recursive tree is an evolving tree in which vertices are assigned random vertex-weights and new vertices connect to a predecessor with a probability proportional to its weight. Here, we study the maximum degree and near-maximum degrees in weighted recursive trees when the vertex-weights are almost surely bounded. We are able to specify higher-order corrections to the first order growth of the maximum degree established in prior work. The accuracy of the results depends on the behaviour of the weight distribution near the largest possible value and in certain cases we manage to find the corrections up to random order. Additionally, we describe the tail distribution of the maximum degree and establish asymptotic normality of the number of vertices with near-maximum degree. Our analysis extends the results proved for random recursive trees (where the weights are constant) to the case of random weights. The main technical result shows that the degrees of several uniformly chosen vertices are asymptotically independent with explicit error corrections.

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“…The definition of the WRT model follows the one in [10]: Definition 2.1 (Weighted Recursive Tree). Let (W i ) i≥1 be a sequence of i.i.d.…”

Section: Definitions and Main Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Fine asymptotics for the maximum degree in weighted recursive trees with bounded random weights

Eslava¹,

Lodewijks²,

Ortgiese³

2021

Preprint

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show abstract

“…In this case, the asymptotic expansion has been obtained by Addario-Berry and Ford [1]. Slight modifications of the uniform recursive tree have also been covered: Hoppe trees, where all weights except w 1 equal 1, have been studied in [25], and another extension, where finitely many weights are different from 1, in [20].…”

Section: Related Work and Commentsmentioning

confidence: 99%

Correction terms for the height of weighted recursive trees

Pain¹,

Sénizergues²

2021

Preprint

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Weighted recursive trees are built by adding successively vertices with predetermined weights to a tree: each new vertex is attached to a parent chosen randomly proportionally to its weight. Under some assumptions on the sequence of weights, the first order for the height of such trees has been recently established in [35] by one of the authors. In this paper, we obtain the second and third orders in the asymptotic expansion of the height of weighted recursive trees, under similar assumptions. Our methods are inspired from those used to prove similar results for branching random walks. Our results also apply to a related model of growing trees, called the preferential attachment tree with additive fitnesses.

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“…Most of the contributions concerning WRTs are very recent, see e.g. [34,24,25,20,33]. In [34] those trees are studied because of their connection to a model of random walk with preferential relocation (a.k.a.…”

Section: Introductionmentioning

confidence: 99%

Geometry of weighted recursive and affine preferential attachment trees

Sénizergues

2021

Electron. J. Probab.

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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 u k of un among {u1, u2, . . . , un−1} with probability proportional to w k , 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 u k is proportional to a k + deg + (u k ), where deg + (u k ) 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.

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Asymptotic results on Hoppe trees and their variations

Cited by 10 publications

References 13 publications

Fine asymptotics for the maximum degree in weighted recursive trees with bounded random weights

Fine asymptotics for the maximum degree in weighted recursive trees with bounded random weights

Correction terms for the height of weighted recursive trees

Geometry of weighted recursive and affine preferential attachment trees

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