Large deviation principle for the empirical degree measure of   preferential attachment random graphs
  We consider preferential attachment random graphs which may be obtained as follows: It starts with a single node. If a new node appears, it is linked by an edge to one or more existing node(s) with a probability proportional to function of their degree. For a class of linear preferential attachment random graphs we find a large deviation principle (LDP) for the empirical degree measure. In the course of the prove this LDP we establish an LDP for the empirical degree and pair distribution see Theorem 2.3, of the fitness preferential attachment model of random graphs. 
K. Doku-Amponsah, F.O. Mettle and E.N.N. Nortey
math.PR