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April 2013 Large deviations for the degree structure in preferential attachment schemes
Jihyeok Choi, Sunder Sethuraman
Ann. Appl. Probab. 23(2): 722-763 (April 2013). DOI: 10.1214/12-AAP854


Preferential attachment schemes, where the selection mechanism is linear and possibly time-dependent, are considered, and an infinite-dimensional large deviation principle for the sample path evolution of the empirical degree distribution is found by Dupuis–Ellis-type methods. Interestingly, the rate function, which can be evaluated, contains a term which accounts for the cost of assigning a fraction of the total degree to an “infinite” degree component, that is, when an atypical “condensation” effect occurs with respect to the degree structure.

As a consequence of the large deviation results, a sample path a.s. law of large numbers for the degree distribution is deduced in terms of a coupled system of ODEs from which power law bounds for the limiting degree distribution are given. However, by analyzing the rate function, one can see that the process can deviate to a variety of atypical nonpower law distributions with finite cost, including distributions typically associated with sub and superlinear selection models.


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Jihyeok Choi. Sunder Sethuraman. "Large deviations for the degree structure in preferential attachment schemes." Ann. Appl. Probab. 23 (2) 722 - 763, April 2013.


Published: April 2013
First available in Project Euclid: 12 February 2013

zbMATH: 1273.60031
MathSciNet: MR3059274
Digital Object Identifier: 10.1214/12-AAP854

Primary: 60F10
Secondary: 05C80

Keywords: Condensation , degree distribution , large deviations , Law of Large Numbers , power law , preferential attachment , Random graphs , time-dependent

Rights: Copyright © 2013 Institute of Mathematical Statistics

Vol.23 • No. 2 • April 2013
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