A preferential attachment model with random initial degrees

In this paper, a random graph process {G(t)} (ta parts per thousand yen1) is studied and its degree sequence is analyzed. Let {W (t) } (ta parts per thousand yen1) be an i.i.d. sequence. The graph process is defined so that, at each integer time t, a new vertex with W (t) edges attached to it, is added to the graph. The new edges added at time t are then preferentially connected to older vertices, i.e., conditionally on G(t-1), the probability that a given edge of vertex t is connected to vertex i is proportional to d (i) (t-1)+delta, where d (i) (t-1) is the degree of vertex i at time t-1, independently of the other edges. The main result is that the asymptotical degree sequence for this process is a power law with exponent tau=min{tau(W),tau(P)}, where tau(W) is the power-law exponent of the initial degrees {W (t) } (ta parts per thousand yen1) and tau(P) the exponent predicted by pure preferential attachment. This result extends previous work by Cooper and Frieze.