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

Abstract

In this paper, we study some important statistics of the random graph in the Buckley-Osthus model. This model is a modification of the well-known Bollob\'as-Riordan model. We denote the number of nodes by t, the so-called initial attractiveness of a node by a. First, we find a new asymptotic formula for the expectation of the number R(d,t) of nodes of a given degree d in a graph in this model. Such a formula is known for positive integer values of a and d \le t^{1/100(a+1)}. Both restrictions are unsatisfactory from theoretical and practical points of view. We completely remove them. Then we calculate the covariances between any two quantities R(d_1,t), R(d_2,t), and using the second moment method we show that R(d,t) is tightly concentrated around its mean for every possible values of d and t. Furthermore, we study a more complicated statistic of the web graph: X(d_1,d_2,t) is the total number of edges between nodes whose degrees are equal to d_1 and d_2 respectively. We also find an asymptotic formula for the expectation of X(d_1,d_2,t) and prove a tight concentration result. Again, we do not impose any substantial restrictions on the values of d_1, d_2, and t.