scholarly journals Subgraphs in preferential attachment models

2019 ◽  
Vol 51 (03) ◽  
pp. 898-926
Author(s):  
Alessandro Garavaglia ◽  
Clara Stegehuis
Keyword(s):  
Power Law ◽  
Optimization Problem ◽  
Preferential Attachment ◽  
Urn Model ◽  
Expected Number ◽  
Pólya Urn ◽  
Polya Urn ◽  
Pólya Urn Model ◽  
Attachment Model ◽  
Law Degree

AbstractWe consider subgraph counts in general preferential attachment models with power-law degree exponent $\tau > 2$ . For all subgraphs H, we find the scaling of the expected number of subgraphs as a power of the number of vertices. We prove our results on the expected number of subgraphs by defining an optimization problem that finds the optimal subgraph structure in terms of the indices of the vertices that together span it and by using the representation of the preferential attachment model as a Pólya urn model.

Strong convergence of proportions in a multicolor Pólya urn

1997 ◽  
Vol 34 (2) ◽  
pp. 426-435 ◽  
Author(s):  
Raúl Gouet
Keyword(s):  
Strong Convergence ◽  
Convex Combination ◽  
Urn Model ◽  
Pólya Urn ◽  
Polya Urn ◽  
Pólya Urn Model ◽  
Generalized Pólya Urn

We prove strong convergence of the proportions Un/Tn of balls in a multitype generalized Pólya urn model, using martingale arguments. The limit is characterized as a convex combination of left dominant eigenvectors of the replacement matrix R, with random Dirichlet coefficients.

scholarly journals Scale-Free Property for Degrees and Weights in a Preferential Attachment Random Graph Model

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
István Fazekas ◽  
Bettina Porvázsnyik
Keyword(s):  
Asymptotic Behaviour ◽  
Random Graph ◽  
Power Law ◽  
Degree Distribution ◽  
Preferential Attachment ◽  
Graph Model ◽  
Random Graph Model ◽  
Scale Free ◽  
Attachment Model ◽  
Law Degree

A random graph evolution mechanism is defined. The evolution studied is a combination of the preferential attachment model and the interaction of four vertices. The asymptotic behaviour of the graph is described. It is proved that the graph exhibits a power law degree distribution; in other words, it is scale-free. It turns out that any exponent in(2,∞)can be achieved. The proofs are based on martingale methods.

Joint distributions of circular runs of various lengths based on multi-colour Pólya urn model

2012 ◽  
Vol 49 (3) ◽  
pp. 283-300
Author(s):  
Sonali Bhattacharya
Keyword(s):  
Probability Generating Function ◽  
Discrete Distribution ◽  
Time Sharing ◽  
Urn Model ◽  
Joint Distributions ◽  
Pólya Urn ◽  
Start Up ◽  
Polya Urn ◽  
Pólya Urn Model ◽  
First Time

In this paper, we have used Eryilmaz’s (2008) multi-colour Pólya urn model to obtain joint distributions of runs of t-types of exact lengths (k1, k2, …, kt), at least lengths (k1, k2, …, kt), non-overlapping runs of lengths (k1, k2, … kt) and overlapping runs of lengths (k1, k2, … kt) when counting of runs is done in a circular setup. We have also derived joint distributions of longest runs of various types under similar conditions. Distributions of runs have found applications in fields of reliability of consecutive-k-out-of n: F system, consecutive k-out-of-r-from n: F system, start-up demonstration test, molecular biology, radar detection, time sharing systems and quality control. The literature is profound in discussion of marginal distribution and joint distribution of runs of various types under linear and circular setup using techniques like urn model with balls of two or more colours, probability generating function and compounding discrete distribution with suitable beta functions. Through this paper for first time effort been made to discuss joint distributions of runs of various lengths and types using Multi-colour urn model.

A Pólya urn model and the coalescent

1992 ◽  
Vol 29 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Gudrun Trieb
Keyword(s):  
Population Genetics ◽  
Urn Model ◽  
Ewens Sampling Formula ◽  
Sampling Formula ◽  
Pólya Urn ◽  
Polya Urn ◽  
Pólya Urn Model

In recent papers by Hoppe and Donnelly it has been shown that a Pólya urn model generating the Ewens sampling formula (population genetics) parallels a construction of Kingman using a Poisson–Dirichlet ‘paintbox'. Even the jump chain of Kingman's n-coalescent can be constructed using the urn. The properties of a certain process based on the coalescent also are derived. This process was introduced by Hoppe.

Spatial Distribution of Tourism Activities: A Polya Urn Process Model of Rank-Size Distribution

2019 ◽  
Vol 59 (2) ◽  
pp. 231-246 ◽  
Author(s):  
Pong Lung Lau ◽  
Tay T. R. Koo ◽  
Cheng-Lung Wu
Keyword(s):  
Size Distribution ◽  
Power Law ◽  
Process Model ◽  
Word Of Mouth ◽  
Preferential Attachment ◽  
Power Law Distribution ◽  
Habit Persistence ◽  
Pólya Urn ◽  
Polya Urn ◽  
Attachment Process

The power law is considered one of the most enduring regularities in human geography. This article aims to develop an understanding of the circumstances that may result in the power law distribution in the geography of tourism activities. The finite Polya urn process is adopted as a device to model the preferential attachment process in the flow of tourists. The model generates a rank-size distribution of tourism regions along with intuitively appealing parameters. Empirically examined using two independent sets of Australian inbound and outbound tourism data, results show that the rank-size distribution emerging from the finite Polya urn process is a superior fit to the conventional power law curve. This rank-size distribution (termed the Polya urn process model of visitor distribution) is compatible with tourist behaviors such as habit persistence and word-of-mouth effects, and can be adopted by tourism modelers to predict and efficiently summarize the spatiality of tourism.

Strong convergence of proportions in a multicolor Pólya urn

1997 ◽  
Vol 34 (02) ◽  
pp. 426-435 ◽  
Author(s):  
Raúl Gouet
Keyword(s):  
Strong Convergence ◽  
Convex Combination ◽  
Urn Model ◽  
Pólya Urn ◽  
Polya Urn ◽  
Pólya Urn Model ◽  
Generalized Pólya Urn

We prove strong convergence of the proportions Un /Tn of balls in a multitype generalized Pólya urn model, using martingale arguments. The limit is characterized as a convex combination of left dominant eigenvectors of the replacement matrix R, with random Dirichlet coefficients.

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