In this article, we introduce our study of some important statistics of the random graph in the directed preferential attachment model introduced by B. Bollobás, C. Borgs, J. Chayes, and O. Riordan. First, we find a new asymptotic formula for the expectation of the number _n__in(_t_, _d_) of nodes of a given in-degree _d_ in a graph in this model with _t_ edges, which covers all possible degrees. The out-degree distribution in the model is symmetrical to the in-degree distribution. Then we prove tight concentration for _n__in(_t_, _d_) while _d_ grows up to the moment when _n__in(_t_, _d_) decreases to ln ²_t_; if _d_ grows even faster, _n__in(_t_, _d_) is zero whp. Furthermore, we study an average number of edges from a vertex of out-degree _d_₁ to a vertex of in-degree _d_₂. In particular, we prove that it grows proportionally to _d_₁_d_₂/_t_ if

and to something between

and

, tending to the first expression when _d_₁ is small compared to _d_₂ and to the second one when _d_₁ is large;

is such that the main term of _n__in(_t_, _d_) is proportional to

is symmetrical for out-degrees. We also give exact formulas for intermediate cases.

The Degree Distribution and the Number of Edges Between Nodes of given Degrees in Directed Scale-Free Graphs

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