Activity maxima in some models of information networks with random weights and heavy tails

We consider models of information networks described by random graphs and hypergraphs where each node has a random information activity with distribution having a heavy (regularly varying) tail. We derive sufficient conditions under which the maximum of the aggregate activities (over a node and its neighbors or over communities) asymptotically grows in the same way as the maximum of individual activities and the Fréchet limit law holds for them.