Order statistics of observed network degrees

TL;DR

This paper analyzes the extreme values of normalized network degrees, modeling them as Beta-distributed variables, and explores their statistical properties to understand network degree distributions.

Contribution

It introduces a normalized degree concept and derives properties of degree extremes modeled by Beta distributions, facilitating comparison across different networks.

Findings

Derived simplified formulas for means and variances of degree extremes.

Verified theoretical properties through simulations.

Discussed implications for power-law degree distributions.

Abstract

This article discusses the properties of extremes of degree sequences calculated from network data. We introduce the notion of a normalized degree, in order to permit a comparison of degree sequences between networks with differing numbers of nodes. We model each normalized degree as a bounded continuous random variable, and determine the properties of the ordered k-maxima and minima of the normalized network degrees when they comprise a random sample from a Beta distribution. In this setting, their means and variances take a simplified form given by their ordering, and we discuss the relation of these quantities to other prescribed decays such as power laws. We verify the derived properties from simulated sets of normalized degrees, and discuss possible extensions to more flexible classes of distributions.