Large Deviations of the Degree Structure in Preferential Attachment Schemes

Details Large Deviations of the Degree Structure in Preferential Attachment Schemes Large Deviations of the Degree Structure in Preferential Attachment Schemes

2011-05-30

arxiv.org/abs/1105.6047v1

Mathematics

Probability

29 pages, 1 figure

Scientific paper

Preferential attachment schemes where the selection mechanism is 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, includes a term which accounts for the cost of assigning a fraction of the total degree to an 'infinite' degree component. As a consequence of the large deviation results, a sample path a.s. law of large numbers for the degree structure is deduced in terms of a coupled system of ODE's 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 non-power law distributions with finite cost.

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