  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.
