Extremum statistics in scale-free network models

TL;DR

This paper studies the extreme degree nodes in scale-free networks, showing how finite size effects influence their distribution and identifying different statistical behaviors for homogeneous and heterogeneous node models.

Contribution

It demonstrates how finite size truncation affects extremum statistics in scale-free networks and distinguishes the behaviors between homogeneous and heterogeneous node models.

Findings

Finite size truncation governs extreme value scaling in homogeneous networks.

Maxima follow Gumbel statistics in homogeneous models.

Heterogeneous models deviate from Gumbel and Frechet distributions.

Abstract

We investigate the statistics of the most connected nodes in scale-free networks. For a scale-free network model with homogeneous nodes, we show by means of extensive simulations that the exponential truncation--due to the finite size of the network--of the degree distribution governs the scaling of the extreme values. We also find that the distribution of maxima obeys scaling and follows the Gumbel statistics. For a scale-free network model with heterogeneous nodes, we show that scaling no longer holds and that the truncation of the degree distribution no longer controls the maximum distribution. Moreover, we find that neither the Gumbel nor the Frechet statistics describe the data.