The degree sequence of the preferential attachment model

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 ( d ) of vertices with degree d obeys a power law P ( d ) ∝ d − 3 for all d ≤ 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 ≤ C n 1 / 2 − μ , for any arbitrarily small constant μ > 0 and arbitrarily large constant C > 0 . The answer is near optimal, since the maximum degree is O ( n 1 / 2 ) .