Explicit characterization of moments of balanced triangular Pólya urns by an elementary approach

Zhang, Panpan; Chen, Chen; Mahmoud, Hosam M.

doi:10.1016/j.spl.2014.09.016

Cited by 13 publications

(11 citation statements)

References 6 publications

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“…With the proper normalization, a limit distribution follows from results in [4], [8], and [14], represented in terms of the standard Riemann gamma function. Proposition 2.…”

Section: The Outdegree Of the North Polementioning

confidence: 99%

Degrees in random self-similar bipolar networks

Chen

Mahmoud

2016

J. Appl. Probab.

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We investigate several aspects of a self-similar evolutionary process that builds a random bipolar network from building blocks that are themselves small bipolar networks. We characterize admissible outdegrees in the history of the evolution. We obtain the limit distribution of the polar degrees (when suitably scaled) characterized by its sequence of moments. We also obtain the asymptotic joint multivariate normal distribution of the number of nodes of small admissible outdegrees. Five possible substructures arise, and each has its own parameters (mean vector and covariance matrix) in the multivariate distribution. Several results are obtained by mapping bipolar networks into Pólya urns.

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“…With the proper normalization, a limit distribution follows from results in [4], [8], and [14], represented in terms of the standard Riemann gamma function. Proposition 2.…”

Section: The Outdegree Of the North Polementioning

confidence: 99%

Degrees in random self-similar bipolar networks

Chen

Mahmoud

2016

J. Appl. Probab.

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“…for n ≥ j. We appeal to the recent results in [19] to determine the exact distribution of the number of k-cliques incident with the node labeled j as the network ages, proving Proposition 1.…”

Section: Degree Profile Of An Apollonian Networkmentioning

confidence: 76%

“…A proper normalization gives rise to a limit distribution following from the results in [7] and [19], represented in terms of the standard Riemann's gamma function.…”

mentioning

confidence: 99%

The degree profile and weight in Apollonian networks and k-trees

Zhang

Mahmoud

2016

Adv. Appl. Probab.

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We investigate the degree profile and total weight in Apollonian networks. We study the distribution of the degrees of vertices as they age in the evolutionary process. Asymptotically, the (suitably-scaled) degree of a node with a fixed label has a MittagLeffler-like limit distribution. The degrees of nodes of later ages have different asymptotic distributions, influenced by the time of their appearance. The very late arrivals have a degenerate distribution. The result is obtained via triangular Pólya urns. Also, via the Bagchi-Pal urn, we show that the number of terminal nodes asymptotically follows a Gaussian law. We prove that the total weight of the network asymptotically follows a Gaussian law, obtained via martingale methods. Similar results carry over to the sister structure of the k-trees, with minor modification in the proof methods, done mutatis mutandis.

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“…Matrix (3)) is triangular, so the Pólya urn associated with this kind of replacement matrix is called triangular Pólya urn. Triangular urns are well studied, and the moments of white balls are explicitly characterized in [42,Thm. 3.1].…”

Section: Momentsmentioning

confidence: 99%

On Several Properties of a Class of Preferential Attachment Trees—plane-Oriented Recursive Trees

Zhang¹

2020

Prob. Eng. Inf. Sci.

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In this paper, several properties of a class of trees presenting preferential attachment phenomenon—plane-oriented recursive trees (PORTs) are uncovered. Specifically, we investigate the degree profile of a PORT by determining the exact probability mass function of the degree of a node with a fixed label. We compute the expectation and the variance of degree variable via a Pólya urn approach. In addition, we study a topological index, Zagreb index, of this class of trees. We calculate the exact first two moments of the Zagreb index (of PORTs) by using recurrence methods. Lastly, we determine the limiting degree distribution in PORTs that grow in continuous time, where the embedding is done in a Poissonization framework. We show that it is exponential after proper scaling.

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Explicit characterization of moments of balanced triangular Pólya urns by an elementary approach

Cited by 13 publications

References 6 publications

Degrees in random self-similar bipolar networks

Degrees in random self-similar bipolar networks

The degree profile and weight in Apollonian networks and k-trees

On Several Properties of a Class of Preferential Attachment Trees—plane-Oriented Recursive Trees

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