Extremal properties of evolving networks: local dependence and heavy tails

TL;DR

This paper investigates the extremal properties of evolving networks, focusing on tail and extremal indices of PageRank and Max-Linear models, to predict network growth and node influence distributions.

Contribution

It extends recent results on sums and maxima of non-stationary random variables to the context of random graph evolution, introducing algorithms for predicting extremal behavior.

Findings

The extremal index measures local dependence and clustering in network processes.

The tail index indicates the heaviness of the influence distribution tail.

Algorithms successfully predict the evolution of extremal indices in network growth.

Abstract

A network evolution with predicted tail and extremal indices of PageRank and the Max-Linear Model used as node influence indices in random graphs is considered. The tail index shows a heaviness of the distribution tail. The extremal index is a measure of clustering (or local dependence) of the stochastic process. The cluster implies a set of consecutive exceedances of the process over a sufficiently high threshold. Our recent results concerning sums and maxima of non-stationary random length sequences of regularly varying random variables are extended to random graphs. Starting with a set of connected stationary seed communities as a hot spot and ranking them with regard to their tail indices, the tail and extremal indices of new nodes that are appended to the network may be determined. This procedure allows us to predict a temporal network evolution in terms of tail and extremal indices. The extremal index determines limiting distributions of a maximum of the PageRank and the Max-Linear Model of newly attached nodes. The exposition is provided by algorithms and examples. To validate our theoretical results, our simulation and real data study concerning a linear preferential attachment as a tool for network growth are provided.