Degree asymptotics with rates for preferential attachment random graphs

By Erol A. Peköz, Adrian Röllin, Nathan Ross. 2011

79 citations recorded by the source corpus.

We provide optimal rates of convergence to the asymptotic distribution of the (properly scaled) degree of a fixed vertex in two preferential attachment random graph models. Our approach is to show that these distributions are unique fixed points of certain distributional transformations which allows us to obtain rates of convergence using a new variation of Stein's method. Despite the large literature on these models, there is surprisingly little known about the limiting distributions so we also provide some properties and new representations, including an explicit expression for the densities in terms of the confluent hypergeometric function of the second kind.

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/8565b8bc926f892c3dbc838494b381d8aa390a3f

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In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Stein's Method and the Laplace Distribution (2012, background): “The idea of composing distributional transformations appears in [13] as well, and their results also involve a second order characterizing equation.”
Random Graphs and Complex Networks (2016, background): “For m = 1 and δ = 0, Peköz, Röllin and Ross [218] investigate the explicit law of the different vertices, and identify the density to be related to confluent hypergeometric functions of the second kind (also known as the Kummer U function), Interestingly, the precise limit is different (though relat”
Malliavin-Stein method for variance-gamma approximation on Wiener space (2014, methodology): “We emphasize that second-order Stein operators are not commonly used in the literature, although in [24] the authors obtained a second-order Stein operator for the so-called Kummert-U density.”
Stein's method for comparison of univariate distributions (2014, methodology): “As noted in [80], the canonical Stein operator (as given in (7)) is cumbersome.”
On Stein's method for products of normal random variables and zero bias couplings (2013, background): “[35] have recently obtained a similar Stein equation for the family of densities”