A preferential attachment model with random initial degrees

Abstract

In this paper, a random graph process ${G(t)}_{t\ge 1}$ is studied and its degree sequence is analyzed. Let $(W_t{)}_{t\ge 1}$ 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 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 $(W_t{)}_{t\ge 1}$ and ${\tau }_P$ the exponent predicted by pure preferential attachment. This result extends previous work by Cooper and Frieze, which is surveyed.

In the published form of the paper, the proof of Proposition 2.1 is incomplete. This version contains the complete proof