Extremes in Random Graphs Models of Complex Networks

TL;DR

This paper analyzes the extremal behavior of influence measures like PageRank and Max-linear models in complex networks, revealing that their tail and extremal indices are equivalent under certain heavy-tailed assumptions.

Contribution

It establishes the equivalence of tail and extremal indices for PageRank and Max-linear models in complex networks with heavy-tailed influence coefficients.

Findings

Tail and extremal indices are the same for PageRank and Max-linear models.

The indices are explicitly calculated based on the tail behavior of coefficients.

The study uses random sequences of random length to derive results.

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.