Models of preferred joining of vertices in random graphs

Several dynamical models of random graphs are considered. The first model of preferred joining for Web appeared in 1999 in paper by A.-L. Barabas and R. Albert. Then different modifications of this model were proposed. Two models of preferred joining are considered. Namely, the preferred joining of the vertex with minimal or maximal degree. First case is the generalization of urn model defined by U. Azar, A.Z. Broder, A.R. Carlin and I. Upfal (1999). The second case is the generalization of Mori model of the tree preferred joining. The procedure of vertex joining affects the asymptotic properties of the distribution of vertices degrees of growing graph. The asymptotic properties of the structure of the tree of preferred joining are studied. There are three different types of asymptotic behavior. It is the complete description of possible asymptotic behavior of maximal vertex degree.