For the modeling of large networks a random graph is proposed, the generation of which is based on the use of nonlinear preferential attachment rule. The analytical solution for the vertex degree distribution is given as well as the method for the calibration of graph generator according to the node degree distributions in real networks. Possibilities of structural identification are demonstrated using the proposed graph for a network of actors acted the same films, the Internet, networks of roads, networks of complementary goods in online store.