AbstractThe web may be viewed as a directed graph each of whose vertices is a static HTML Web page, and each of whose edges corresponds to a hyperlink from one Web page to another. In this paper we study the model of random graph (i.e., scale‐free random graph process) introduced by Barabási and Albert and formalized by Bollobás, Riordan, Spencer, and Tusnády, which obey two properties known as power laws for vertex degrees and “small‐world phenomenon.” In the paper we give some exact formulas and fair estimations for expectations of the number of vertices of given degree and the degree of given vertex. Moreover, based on Hájek‐Rényi inequality we give the rate of concentration of the number of vertices of given degree. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005