Normalised Degree Variance

Finding graph indices which are unbiased to network size is of high importance both within a given field and across fields for enhancing the comparability over the cornucopia of modern network science studies as well as in subnetwork comparisons of the same network. The degree variance is an important metric for characterising graph heterogeneity and hub dominance, however this clearly depends on the largest and smallest degrees of the graph which depends on network size. Here, we provide an analytically valid normalisation of degree variance to address outstanding but unnoticed problems with previously proposed formulae. We illustrate its enhanced appropriateness for assessing hierarchical spread of degrees in weighted random networks, random geometric networks, resting-state brain networks and the US airport network. The closed form expression proposed also benefits from high computational efficiency and straightforward mathematical analysis.