Polya Urn Models

Mahmoud, Hosam M.

doi:10.1201/9781420059847

Cited by 324 publications

(304 citation statements)

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“…(See, e.g., Theorem 3.2 in [Mah09] or Section 4.2 in [JK77]). Thus, the limit of A n /(A n + B n ) exists almost surely, and it takes value 1/2 with probability 0 for any starting point of the process T .…”

Section: Persistent Hubmentioning

confidence: 99%

Preferential attachment combined with random number of choices

Malyshkin¹

2018

Internet Mathematics

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Abstract. We study the asymptotic behavior of the maximal degree in the degree distribution in an evolving tree model combining local choice and Mori's preferential attachment. In the considered model, the random graph is constructed in the following way. At each step, a new vertex is introduced. Then, we connect it with one of d possible neighbors, which are sampled from the set of the existing vertices with probability proportional to their degrees plus some parameter β > −1. The number d will be randomly selected and the vertex with the largest degree is chosen. It is known that the maximum of the degree distribution for non-random d > 2 has linear behavior and, for d = 2, asymptotically equals to n/ ln n up to a constant factor. We prove that if Ed < 2 + β, the maximal degree has sublinear behavior with the power Ed/(2 + β) (as in the preferential attachment without choice), if Ed > 2 + β, it has linear behavior and if Ed = 2 + β the maximal degree is of order n/ ln n. The proof combines standard preferential attachment approaches with martingales and stochastic approximation techniques. We also use stochastic approximation results to get a multidimensional central limit theorem for numbers of vertices with fixed degrees.

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Section: Persistent Hubmentioning

confidence: 99%

Preferential attachment combined with random number of choices

Malyshkin¹

2018

Internet Mathematics

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“…balls, each being red with probability θ and black otherwise. This is easily verified by a direct calculation, see [12], [18 −→ θ, where R n is the number of red balls drawn in the first n draws. To see the representation above, with θ ∼ U(0, 1), it thus suffices to show [12], [18,Exercise 3.4].…”

Section: A Related Constructionmentioning

confidence: 85%

An example of graph limits of growing sequences of random graphs

Janson

Severini

2013

Journal of Combinatorics

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In this paper, we consider a class of growing random graphs obtained by creating vertices sequentially one by one. At each step, we uniformly choose the neighbors of the newly created vertex; its degree is a random variable with a fixed but arbitrary distribution, depending on the number of existing vertices. Examples from this class turn out to be the Erdős-Rényi random graph, a natural random threshold graph, etc. By working with the notion of graph limits, we define a kernel which, under certain conditions, is the limit of the growing random graph. Moreover, for a subclass of models, the growing graph on any given n vertices has the same distribution as the random graph with n vertices that the kernel defines. The motivation stems from a model of graph growth whose attachment mechanism does not require information about properties of the graph at each iteration.

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“…, c K ) and replacement matrix A = (a i,j ) 1≤i,j≤K . For a general account on Pólya urns covering many of its applications in the analysis of random trees, we refer to Mahmoud's book [27]. The following lemma coincides with Theorem 7 in Dobrow and Smythe [9] in the case of integer-valued β.…”

Section: Preliminariesmentioning

confidence: 87%

On martingale tail sums for the path length in random trees

Sulzbach

2016

Random Struct Algorithms

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For a martingale (Xn) converging almost surely to a random variable X, the sequence (Xn − X) is called martingale tail sum. Recently, Neininger [Random Structures Algorithms, 46 (2015), 346-361] proved a central limit theorem for the martingale tail sum of Régnier's martingale for the path length in random binary search trees. Grübel and Kabluchko [to appear in Annals of Applied Probability, (2016)] gave an alternative proof also conjecturing a corresponding law of the iterated logarithm. We prove the central limit theorem with convergence of higher moments and the law of the iterated logarithm for a family of trees containing binary search trees, recursive trees and plane-oriented recursive trees.

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Polya Urn Models

Cited by 324 publications

References 0 publications

Preferential attachment combined with random number of choices

Preferential attachment combined with random number of choices

An example of graph limits of growing sequences of random graphs

On martingale tail sums for the path length in random trees

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