Growing network: Models following nonlinear preferential attachment rule

We investigate the preferential attachment graphs proceeding from the following two assumptions. The first one: the probability that a new vertex connects to a vertex i is proportional to an arbitrary nonnegative function f of a vertex degree k. The second assumption: a new vertex can have a random number of edges. We derive formulas for any f to determine the vertex degree distribution {Qk} in generated graphs. The inverse problem is solved: we have obtained formulas, that allow from a given distribution {Qk} to determine f (the problem of a model calibration). The formulas allowing for any f to calculate the joint distribution of vertex degrees at the ends of randomly selected edge are also obtained. Some other results are presented in the paper.