Real-world networks display a strong heterogeneity that is reflected in a heavy-tailed distribution of node influence indices. The PageRank and the Max-Linear Model may be used as node influence indices of random graphs. The present paper aims to summarize shortly some recent author's results with regard to extremal properties of maxima and sums of non-stationary random length sequences and their application to evolving networks. Under the extremal properties we understand the tail and extremal indices. The evolution of the random network by the preferential attachment is considered since it allows us to model heavy-tailed distributed node influence indices.