Degree Distribution and Number of Edges between Nodes of Given Degrees in the Buckley–Osthus Model of a Random Web Graph

Abstract

Abstract In this paper, we study some important statistics of the random graph H (t) a,k in the Buckley–Osthus model, where t is the number of nodes, kt is the number of edges (so that ), and a>0 is the so-called initial attractiveness of a node. This model is a modification of the well-known Bollobás–Riordan model. 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 and d⩽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) and R(d 2, t), and using the second-moment method we show that R(d, t) is tightly concentrated around its mean for all 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.