The degree sequence of the preferential attachment model

Abstract

For the preferential attachment model of random networks, Bollobás et al. proved that when the number $n$ of vertices is large enough, the proportion $P\left(d\right)$ of vertices with degree $d$ obeys a power law $P\left(d\right)\propto d^{-3}$ for all $d\le n^{1/15}$. They asked that how far the result can be extended to degrees $d>n^{1/15}$. We give an answer by extending the range to all $d\le C{n}^{1/2-\mu }$, for any arbitrarily small constant $\mu >0$ and arbitrarily large constant $C>0$. The answer is near optimal, since the maximum degree is $O\left(n^{1/2}\right)$.