Activity maxima in free-scale random networks with heavy tails

The oriented power-law randomgraphs are considered as themodels of information networks, where each node has a random information activity whose distribution has heavy (regularly varying) tail. The model of a random graph, in which incoming degrees of vertices are independent and have distribution with power tail, is used. Sufficient conditions have been got under which themaximumtotal activity (of the node and its incoming neighbors) increases asymptotically as well as the maximum of individual activities, and therefore, for them, Frechet limit law is hold.