Correction terms for the height of weighted recursive trees

Pain, Michel; Sénizergues, Delphin

doi:10.48550/arxiv.2101.01156

Cited by 3 publications

(3 citation statements)

References 31 publications

(51 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Beyond the initial work of Borovkov and Vatutin and also Hiesmayr and Işlak studying the height, depth and size of branches of the WRT model, other properties such as the degree distribution, large and maximum degrees, and weighted profile and height of the tree have been studied. Mailler and Uribe Bravo [15], as well as Sénizergues [17] and Sénizergues and Pain [16] study the weighted profile and height of the WRT model. Mailler and Uribe Bravo consider random vertex-weights with particular distributions, whereas Sénizergues and Pain allow for a more general model with both sequences of deterministic as well as random weights.…”

Section: Introductionmentioning

confidence: 99%

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

Eslava¹,

Lodewijks²,

Ortgiese³

2021

Preprint

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 99%

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

Eslava¹,

Lodewijks²,

Ortgiese³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…After the introduction of the WRT model by Borovkov and Vatutin, Hiesmayr and Işlak study the height, depth and size of the tree branches of this model. Mailler and Uribe Bravo [11], as well as Sénizergues [13] and Sénizergues and Pain [12] study the weighted profile and height of the WRT model. Mailler and Uribe Bravo consider random vertex-weights with particular distributions, whereas Sénizergues and Pain allow for a more general model with both sequences of deterministic as well as random weights.…”

Section: Introductionmentioning

confidence: 99%

Location of high-degree vertices in weighted recursive graphs with bounded random weights

Lodewijks¹

2021

Preprint

View full text Add to dashboard Cite

We study the asymptotic logarithmic growth rate of the label size of vertices that attain the maximum degree in weighted recursive graphs (WRG) when the weights are independent, identically distributed, almost surely bounded random variables, and as a result confirm a conjecture by Lodewijks and Ortgiese [10]. WRGs are a generalisation of the random recursive tree (RRT) and directed acyclic graph model (DAG), in which vertices are assigned vertexweights and where new vertices attach to m ∈ N predecessors with a probability proportional to the vertex-weight of the predecessor. Prior work established the asymptotic growth rate of the maximum degree of the WRG model and here we study the asymptotic logarithmic growth rate of the location, that is, the label size of the vertices that attain the maximum degree. We show that there exists a critical exponent γm, such that the typical age/label size of the maximum degree vertex equals n γm(1+o(1)) almost surely as n, the size of the graph, tends to infinity. These results extend and improve on the asymptotic behaviour of the location of the maximum degree, formerly only known for the RRT model, to the more general weighted multigraph case of the WRG model. Moreover, in the Weighted Recursive Tree (WRT) model, that is, the WRG model with m = 1, we are able to provide bounds on the fluctuations around the exponent γ 1 under additional assumptions on the vertex-weight distribution. These results are novel even for the RRT model. The approach in this paper combines a refined version of the approach developed for studying the maximum degree of the WRG model with more precise union bounds.

show abstract

“…random variables, under an 8-th moment condition. Sénizergues [15] proved results about its degree distribution and height for a deterministic sequence of weights under very general assumptions (with more refined results about the height -maximum distance of a node to the root-in Pain and Sénizergues [13]). Finally, Iyer [9] and Lodewijks and Ortgiese [11] prove asymptotic results on, respectively the degree distribution and the largest degree of the wrrt with i.i.d.…”

Section: (A2)mentioning

confidence: 90%

Large deviations principle for a stochastic process with random reinforced relocations

Boci¹,

Mailler²

2021

Preprint

View full text Add to dashboard Cite

Stochastic processes with random reinforced relocations have been introduced in the physics literature to model animal foraging behaviour. Such a process evolves as a Markov process, except at random relocation times, when it chooses a time at random in its whole past according to some "memory kernel", and jumps to its value at that random time.We prove a quenched large deviations principle for the value of the process at large times. The difficulty in proving this result comes from the fact that the process is not Markov because of the relocations. Furthermore, the random inter-relocation times act as a random environment.We also prove a partial large deviations principle for the occupation measure of the process in the case when the memory kernel is decreasing. We use a similar approach to prove large deviations principle for the profile of the weighted random recursive tree (wrrt) in the case when older nodes are more likely to attract new links than more recent nodes.

show abstract

Correction terms for the height of weighted recursive trees

Cited by 3 publications

References 31 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

Location of high-degree vertices in weighted recursive graphs with bounded random weights

Large deviations principle for a stochastic process with random reinforced relocations

Contact Info

Product

Resources

About