Large degrees in scale-free inhomogeneous random graphs

TL;DR

This paper analyzes the maximum degree distribution in scale-free inhomogeneous random graphs, showing it converges to a Frechet distribution, and proves the consistency of the Hill estimator for tail index estimation.

Contribution

It establishes the limiting distribution of maximum degrees in such graphs and proves the Hill estimator's consistency for tail exponent estimation.

Findings

Maximum degree distribution converges to Frechet distribution.

Point process of degrees converges to a Poisson process.

Hill estimator is consistent for tail index estimation.

Abstract

We consider a class of scale-free inhomogeneous random graphs, which includes some long-range percolation models. We study the maximum degree in such graphs in a growing observation window and show that its limiting distribution is Frechet. We achieve this by proving convergence of the underlying point process of the degrees to a certain Poisson process. Estimating the index of the power-law tail for the typical degree distribution is an important question in statistics. We prove consistency of the Hill estimator for the inverse of the tail exponent of the typical degree distribution.