Normalised Degree Variance

TL;DR

This paper introduces a new, analytically valid normalisation for degree variance in networks, making it unbiased to size and density, and applicable across various network types and real-world data.

Contribution

The authors propose a novel normalisation method for degree variance that is valid for all networks, computationally efficient, and robust to size and density variations.

Findings

Normalised degree variance is unbiased to network size and density.

The new normalisation is consistent across different network models and real-world networks.

It reveals greater degree heterogeneity in biological networks like brain connectomes and food webs.

Abstract

Finding graph indices which are unbiased to network size and density is of high importance both within a given field and across fields for enhancing comparability of modern network science studies. The degree variance is an important metric for characterising network degree heterogeneity. Here, we provide an analytically valid normalisation of degree variance to replace previous normalisations which are either invalid or not applicable to all networks. It is shown that this normalisation provides equal values for graphs and their complements; it is maximal in the star graph (and its complement); and its expected value is constant with respect to density for Erd\"os-R\'enyi (ER) random graphs of the same size. We strengthen these results with model observations in ER random graphs, random geometric graphs, scale-free networks, random hierarchy networks and resting-state brain networks, showing that the proposed normalisation is generally less affected by both network size and density than previous normalisation attempts. The closed form expression proposed also benefits from high computational efficiency and straightforward mathematical analysis. Analysis of 184 real-world binary networks across different disciplines shows that normalised degree variance is not correlated with average degree and is robust to node and edge subsampling. Comparisons across subdomains of biological networks reveals greater degree heterogeneity among brain connectomes and food webs than in protein interaction networks.