Trees grown under young-age preferential attachment

2020 ◽  
Vol 57 (3) ◽  
pp. 911-927
Author(s):  
Merritt R. Lyon ◽  
Hosam M. Mahmoud
Keyword(s):  
Preferential Attachment ◽  
Stirling Numbers ◽  
Poisson Approximation ◽  
Young Age ◽  
Insertion Depth ◽  
Exact Probability ◽  
Exact Probability Distribution ◽  
Attachment Model ◽  
Poisson Limit ◽  
Stein Method

AbstractWe introduce a class of non-uniform random recursive trees grown with an attachment preference for young age. Via the Chen–Stein method of Poisson approximation, we find that the outdegree of a node is characterized in the limit by ‘perturbed’ Poisson laws, and the perturbation diminishes as the node index increases. As the perturbation is attenuated, a pure Poisson limit ultimately emerges in later phases. Moreover, we derive asymptotics for the proportion of leaves and show that the limiting fraction is less than one half. Finally, we study the insertion depth in a random tree in this class. For the insertion depth, we find the exact probability distribution, involving Stirling numbers, and consequently we find the exact and asymptotic mean and variance. Under appropriate normalization, we derive a concentration law and a limiting normal distribution. Some of these results contrast with their counterparts in the uniform attachment model, and some are similar.

Models of network formation

2018 ◽  
Author(s):  
Mark Newman
Keyword(s):  
Degree Distribution ◽  
Network Formation ◽  
Preferential Attachment ◽  
Network Size ◽  
Large Network ◽  
Citation Networks ◽  
Airline Networks ◽  
Attachment Model ◽  
Delivery Networks

This chapter describes models of the growth or formation of networks, with a particular focus on preferential attachment models. It starts with a discussion of the classic preferential attachment model for citation networks introduced by Price, including a complete derivation of the degree distribution in the limit of large network size. Subsequent sections introduce the Barabasi-Albert model and various generalized preferential attachment models, including models with addition or removal of extra nodes or edges and models with nonlinear preferential attachment. Also discussed are node copying models and models in which networks are formed by optimization processes, such as delivery networks or airline networks.

scholarly journals Blinking statistics and molecular counting in direct stochastic reconstruction microscopy (dSTORM)

2021 ◽  
Author(s):  
Lekha Patel ◽  
David Williamson ◽  
Dylan M Owen ◽  
Edward A K Cohen
Keyword(s):  
Probability Distribution ◽  
Single Molecule ◽  
Immunological Synapse ◽  
Training Data ◽  
Supplementary Information ◽  
Cellular Structures ◽  
Exact Probability ◽  
Stochastic Optical Reconstruction Microscopy ◽  
Exact Probability Distribution ◽  
Optical Reconstruction

Abstract Motivation Many recent advancements in single-molecule localization microscopy exploit the stochastic photoswitching of fluorophores to reveal complex cellular structures beyond the classical diffraction limit. However, this same stochasticity makes counting the number of molecules to high precision extremely challenging, preventing key insight into the cellular structures and processes under observation. Results Modelling the photoswitching behaviour of a fluorophore as an unobserved continuous time Markov process transitioning between a single fluorescent and multiple dark states, and fully mitigating for missed blinks and false positives, we present a method for computing the exact probability distribution for the number of observed localizations from a single photoswitching fluorophore. This is then extended to provide the probability distribution for the number of localizations in a direct stochastic optical reconstruction microscopy experiment involving an arbitrary number of molecules. We demonstrate that when training data are available to estimate photoswitching rates, the unknown number of molecules can be accurately recovered from the posterior mode of the number of molecules given the number of localizations. Finally, we demonstrate the method on experimental data by quantifying the number of adapter protein linker for activation of T cells on the cell surface of the T-cell immunological synapse. Availability and implementation Software and data available at https://github.com/lp1611/mol_count_dstorm. Supplementary information Supplementary data are available at Bioinformatics online.

On improvements of the order of approximation in the Poisson limit theorem

1996 ◽  
Vol 33 (01) ◽  
pp. 146-155 ◽  
Author(s):  
K. Borovkov ◽  
D. Pfeifer
Keyword(s):  
Limit Theorem ◽  
Random Variables ◽  
Poisson Approximation ◽  
Order Of Approximation ◽  
Operator Semigroups ◽  
Bernoulli Random Variables ◽  
Usual Rate ◽  
Poisson Limit ◽  
Poisson Measures ◽  
Special Case

In this paper we consider improvements in the rate of approximation for the distribution of sums of independent Bernoulli random variables via convolutions of Poisson measures with signed measures of specific type. As a special case, the distribution of the number of records in an i.i.d. sequence of length n is investigated. For this particular example, it is shown that the usual rate of Poisson approximation of O(1/log n) can be lowered to O(1/n 2). The general case is discussed in terms of operator semigroups.

scholarly journals On the birthday problem: some generalizations and applications

2003 ◽  
Vol 2003 (60) ◽  
pp. 3827-3840 ◽  
Author(s):  
P. N. Rathie ◽  
P. Zörnig
Keyword(s):  
Probability Distribution ◽  
Random Graph ◽  
Generating Functions ◽  
Hypergeometric Functions ◽  
Exact Probability ◽  
Confluent Hypergeometric Functions ◽  
Birthday Problem ◽  
Exact Probability Distribution

We study the birthday problem and some possible extensions. We discuss the unimodality of the corresponding exact probability distribution and express the moments and generating functions by means of confluent hypergeometric functionsU(−;−;−)which are computable using the software Mathematica. The distribution is generalized in two possible directions, one of them consists in considering a random graph with a single attracting center. Possible applications are also indicated.

Clustering Coefficient of a Spatial Preferential Attachment Model

2018 ◽  
Vol 98 (1) ◽  
pp. 304-307 ◽  
Author(s):  
L. N. Iskhakov ◽  
M. S. Mironov ◽  
L. A. Prokhorenkova ◽  
B. Kamiński ◽  
P. Prałat
Keyword(s):  
Preferential Attachment ◽  
Clustering Coefficient ◽  
Attachment Model

scholarly journals [Poisson Approximation and the Chen-Stein Method]: Rejoinder

1990 ◽  
Vol 5 (4) ◽  
pp. 432-434 ◽  
Author(s):  
Richard Arratia ◽  
Larry Goldstein ◽  
Louis Gordon
Keyword(s):  
Poisson Approximation ◽  
Stein Method

ON SEVERAL PROPERTIES OF A CLASS OF PREFERENTIAL ATTACHMENT TREES—PLANE-ORIENTED RECURSIVE TREES

2020 ◽  
pp. 1-19
Author(s):  
Panpan Zhang
Keyword(s):  
Continuous Time ◽  
Preferential Attachment ◽  
Mass Function ◽  
Probability Mass Function ◽  
Zagreb Index ◽  
Exact Probability ◽  
Probability Mass ◽  
Proper Scaling ◽  
Recursive Trees ◽  
Degree Variable

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.

scholarly journals Nonuniform Distribution of Nodes in the Spatial Preferential Attachment Model

2015 ◽  
Vol 12 (1-2) ◽  
pp. 121-144 ◽  
Author(s):  
Jeannette Janssen ◽  
Paweł Prałat ◽  
Rory Wilson
Keyword(s):  
Preferential Attachment ◽  
Nonuniform Distribution ◽  
Attachment Model
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