  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.
