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In this paper, we study some important statistics of the random graph in theBuckley-Osthus model. This model is a modification of the well-knownBollob\'as-Riordan model. We denote the number of nodes by t, the so-calledinitial attractiveness of a node by a. First, we find a new asymptotic formulafor the expectation of the number R(d,t) of nodes of a given degree d in agraph in this model. Such a formula is known for positive integer values of aand d \le t^{1/100(a+1)}. Both restrictions are unsatisfactory from theoreticaland practical points of view. We completely remove them. Then we calculate thecovariances between any two quantities R(d_1,t), R(d_2,t), and using the secondmoment method we show that R(d,t) is tightly concentrated around its mean forevery possible values of d and t. Furthermore, we study a more complicatedstatistic of the web graph: X(d_1,d_2,t) is the total number of edges betweennodes whose degrees are equal to d_1 and d_2 respectively. We also find anasymptotic formula for the expectation of X(d_1,d_2,t) and prove a tightconcentration result. Again, we do not impose any substantial restrictions onthe values of d_1, d_2, and t.

The degree distribution and the number of edges between nodes of given degrees in the Buckley-Osthus model of a random web graph