Degree Distribution and 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 $H^{(t)}_{a ,k}$ in the Buckley-Osthus model, where $t$ is the number of nodes, $kt$ is the number of edges (so that $k \in \mathbb{N}$), and $a \gt 0$ 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 $a \in \mathbb{N}$ 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 $R(d_1 , t)$ 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 $t$.