Title: Extremes in Random Graphs Models of Complex Networks,
Abstract: Regarding the analysis of Web communication, social and complex networks the
fast finding of most influential nodes in a network graph constitutes an
important research problem. We use two indices of the influence of those nodes,
namely, PageRank and a Max-linear model. We consider the PageRank %both as
%Galton-Watson branching process and as an autoregressive process with a random
number of random coefficients that depend on ranks of incoming nodes and their
out-degrees and assume that the coefficients are independent and distributed
with regularly varying tail and with the same tail index. Then it is proved
that the tail index and the extremal index are the same for both PageRank and
the Max-linear model and the values of these indices are found. The
achievements are based on the study of random sequences of a random length and
the comparison of the distribution of their maxima and linear combinations.