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

Abstract

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 nin(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 nin(t, d) while d grows up to the moment when nin(t, d) decreases to ln 2t; if d grows even faster, nin(t, d) is zero whp. Furthermore, we study an average number of edges from a vertex of out-degree d1 to a vertex of in-degree d2. In particular, we prove that it grows proportionally to d1d2/t if and to something between and if , tending to the first expression when d1 is small compared to d2 and to the second one when d1 is large; is such that the main term of nin(t, d) is proportional to , is symmetrical for out-degrees. We also give exact formulas for intermediate cases.