A preferential attachment model with random initial degrees

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 of vertex t 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=min{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.

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