A preferential attachment model with random initial degrees

Abstract

In this paper, a random graph process G(t)t≥ 1 is studied and its degree sequence is analyzed. Let (Wt)t≥ 1 be an i.i.d. sequence. The graph process is defined so that, at each integer time t, a new vertex, with Wt 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 is connected to vertex i is proportional to di(t-1)+δ, where di(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 τ=\τW, τP\, where τW is the power-law exponent of the initial degrees (Wt)t≥ 1 and τP the exponent predicted by pure preferential attachment. This result extends previous work by Cooper and Frieze, which is surveyed.