Random tree growth by vertex splitting

David, François; Dukes, Mark; Jónsson, Thórdur; Stefánsson, Sigurður Örn

doi:10.1088/1742-5468/2009/04/p04009

Cited by 4 publications

(25 citation statements)

References 27 publications

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“…The methods in [3] were non-rigorous in this case but the results were correct as we confirm here. By Theorem 1.2 the asymptotic vertex degree densities (a i ) ∞ i=1 satisfy…”

Section: Uniform Partitioning Weights Letsupporting

confidence: 60%

“…is diagonalizable. (Appears in Theorem 2.5 in [3].) It was shown that under these assumptions the limits ρ k := lim t→∞ E(n t,k )/t exist for 1 ≤ k ≤ d max and are the unique positive solutions to…”

Section: Introductionmentioning

confidence: 99%

“…The vertex splitting model is a recent model of randomly growing ordered trees introduced in [3]. It is a modification of a model of randomly growing trees encountered in the theory of RNA-folding [4].…”

Section: Introductionmentioning

confidence: 99%

“…We are interested in studying the distribution of vertex degrees in large random trees T grown according to the above rules. More precisely, let n t,k be the number of vertices of degree k in the tree at time t. In [3] the asymptotics of the expected values E(n t,k ) were studied under the following assumptions.…”

Section: Introductionmentioning

confidence: 99%

“…(A2) There is a finite integer d max such that w j,k = 0 if either j or k exceeds d max (no vertices of degree greater than d max are created in the growth process) and w 1,k = w k,1 > 0 for all 2 ≤ k ≤ d max (it is possible to create vertices of degree d max starting from any initial tree). (Corresponds to (1) in Lemma 2.3 in [3].) (A3) w i,dmax+2−i > 0 for some i satisfying 2 ≤ i ≤ d max − 1 (it is possible to split vertices of degree d max ).…”

Section: Introductionmentioning

confidence: 99%

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Almost sure convergence of vertex degree densities in the vertex splitting model

Stefánsson

Thörnblad

2016

Stochastic Models

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ABSTRACT. We study the limiting degree distribution of the vertex splitting model introduced in [3]. This is a model of randomly growing ordered trees, where in each time step the tree is separated into two components by splitting a vertex into two, and then inserting an edge between the two new vertices. Under some assumptions on the parameters, related to the growth of the maximal degree of the tree, we prove that the vertex degree densities converge almost surely to constants which satisfy a system of equations. Using this we are also able to strengthen and prove some previously non-rigorous results mentioned in the literature.

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“…The methods in [3] were non-rigorous in this case but the results were correct as we confirm here. By Theorem 1.2 the asymptotic vertex degree densities (a i ) ∞ i=1 satisfy…”

Section: Uniform Partitioning Weights Letsupporting

confidence: 60%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Almost sure convergence of vertex degree densities in the vertex splitting model

Stefánsson

Thörnblad

2016

Stochastic Models

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Spectral Dimension of Trees with a Unique Infinite Spine

Stefánsson

Zohren

2012

J Stat Phys

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Using generating functions techniques we develop a relation between the Hausdorff and spectral dimension of trees with a unique infinite spine. Furthermore, it is shown that if the outgrowths along the spine are independent and identically distributed, then both the Hausdorff and spectral dimension can easily be determined from the probability generating function of the random variable describing the size of the outgrowths at a given vertex, provided that the probability of the height of the outgrowths exceeding n falls off as the inverse of n. We apply this new method to both critical non-generic trees and the attachment and grafting model, which is a special case of the vertex splitting model, resulting in a simplified proof for the values of the Hausdorff and spectral dimension for the former and novel results for the latter.

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Markov branching in the vertex splitting model

Stefánsson¹

2012

J. Stat. Mech.

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We study a special case of the vertex splitting model which is a recent model of randomly growing trees. For any finite maximum vertex degree D, we find a one parameter model, with parameter α ∈ [0, 1] which has a so-called Markov branching property. When D = ∞ we find a two parameter model with an additional parameter γ ∈ [0, 1] which also has this feature. In the case D = 3, the model bears resemblance to Ford's α-model of phylogenetic trees and when D = ∞ it is similar to its generalization, the αγ-model. For α = 0, the model reduces to the well known model of preferential attachment.In the case α > 0, we prove convergence of the finite volume probability measures, generated by the growth rules, to a measure on infinite trees which is concentrated on the set of trees with a single spine. We show that the annealed Hausdorff dimension with respect to the infinite volume measure is 1/α. When γ = 0 the model reduces to a model of growing caterpillar graphs in which case we prove that the Hausdorff dimension is almost surely 1/α and that the spectral dimension is almost surely 2/(1 + α). We comment briefly on the distribution of vertex degrees and correlations between degrees of neighbouring vertices.

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Random tree growth by vertex splitting

Cited by 4 publications

References 27 publications

Almost sure convergence of vertex degree densities in the vertex splitting model

Almost sure convergence of vertex degree densities in the vertex splitting model

Spectral Dimension of Trees with a Unique Infinite Spine

Markov branching in the vertex splitting model

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