Large deviations for the degree structure in preferential attachment schemes

TL;DR

This paper establishes a large deviation principle for the degree distribution in preferential attachment models, revealing how atypical degree structures, including condensation effects, can occur with finite probability.

Contribution

It introduces an infinite-dimensional large deviation framework for preferential attachment schemes and characterizes the rate function, highlighting the possibility of nonpower law degree distributions.

Findings

Large deviation principle for degree evolution

Law of large numbers for degree distribution

Power law bounds and atypical distributions

Abstract

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.