Branching structure of uniform recursive trees

We study structural properties of trees grown by preferential attachment. In this mechanism, nodes are added sequentially and attached to existing nodes at a rate that is strictly proportional to the degree. We classify nodes by their depth n, defined as the distance from the root of the tree, and find that the network is strongly stratified. Most notably, the distribution f (n) k of nodes with degree k at depth n has a power-law tail, f (n) k ∼ k −γ(n) . The exponent grows linearly with depth, γ(n) = 2 + n−1 n−1 , where the brackets denote an average over all nodes. Therefore, nodes that are closer to the root are better connected, and moreover, the degree distribution strongly varies with depth. Similarly, the in-component size distribution has a power-law tail and the characteristic exponent grows linearly with depth. Qualitatively, these behaviors extend to a class of networks that grow by a redirection mechanism.

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“…FIG.4: The in-component size distribution at different depths. Shown is the closed form expression(18) for y = 0, 1/2, 1, 3/2, 2.…”

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confidence: 99%

Stratification in the preferential attachment network

Ben‐Naim¹,

Krapivsky²

2009

J. Phys. A: Math. Theor.

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“…, n − 1. The following lemma, given by Feng, Su, and Hu [4], is the essential building block for the next theorem.…”

Section: Branching Structurementioning

confidence: 99%

“…For a natural number n, a tree T n with n labeled vertices, rooted at vertex 1, is a uniform recursive tree (also called random recursive tree) if n = 1 or if T n can be constructed by successively joining the jth vertex to one of the first j − 1 vertices for 2 ≤ j ≤ n (see, for example, [4], [9], or [11]). Let T n denote the set of all uniform recursive trees of size n. Then it is easy to show that |T n | = (n − 1)!…”

Section: Introductionmentioning

confidence: 99%

“…We list some of the works here. On the depth of vertex n, Szymański [15] gave its exact distribution; Devroye [1] and Mahmoud [6] established asymptotic normality of its normalized version; Feng, Su, and Hu [4] proved that it has the same distribution as |G 1 |, and established its strong law of large numbers and law of iterated logarithm. Moreover, in a uniform recursive tree, Meir and Moon [10] obtained the distribution of the number of vertices in each stratum; Moon [11] and Dobrow [2] studied the distance between vertices; the total path length was studied by Mahmoud [6], and Dobrow and Fill [3]; Tetzlaff [16] studied the breakage and restoration; Su, Feng, and Hu [4,14] investigated the branching structure and simple random downward walk.…”

Section: Introductionmentioning

confidence: 99%

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The structure and distances in Yule recursive trees

Feng¹,

Su²

2007

Random Struct Algorithms

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Based on uniform recursive trees, we introduce random trees with the factor of time, which are named Yule recursive trees. The structure and the distance between the vertices in Yule recursive trees are investigated in this paper. For arbitrary time t > 0, we first give the probability that a Yule recursive tree Y t is isomorphic to a given rooted tree γ ; and then prove that the asymptotic distribution of ζ t,m , the number of the branches of size m, is the Poisson distribution with parameter λ = 1/m. Finally, two types of distance between vertices in Yule recursive trees are studied, and some limit theorems for them are established.

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“…We refer to [29] for a survey on some classical results on several statistics of URTs including the height, extremal degrees and internodal distances. Applications of URTs are also rich, see for example, [14], [16], [23], [25] for different approaches in modeling real life problems. Despite the applications we have for URTs, it lacks significant properties: the chance to have a different global recursive construction principle and the chance to have nodes with distinct behaviors.…”

Section: Introductionmentioning

confidence: 99%

Some remarks on biased recursive trees

Hiesmayr,

Işlak

2018

Preprint

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The purpose of this paper is to analyze certain statistics of a recently introduced non-uniform random tree model, biased recursive trees. This model is based on constructing a random tree by establishing a correspondence with non-uniform permutations, biased riffle shuffles. The statistics that are treated include the number of nodes with a given number of descendants, the depth of the tree, and the number of branches. The model yields the uniform recursive trees as a certain limit, some new results for the uniform case are obtained as well.

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Branching structure of uniform recursive trees

Cited by 8 publications

References 10 publications

Stratification in the preferential attachment network

Stratification in the preferential attachment network

The structure and distances in Yule recursive trees

Some remarks on biased recursive trees

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