3 years ago

Normalised Degree Variance.

Keith Smith, Javier Escudero

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 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 random networks of the same size. We strengthen these results with model observations in weighted random networks, random geometric networks and resting-state brain networks, showing that the proposed normalisation is unbiased to both network size and density. The closed form expression proposed also benefits from high computational efficiency and straightforward mathematical analysis. In an application of a subnetwork comparability problem of nationwide and within state US airport networks, the nationwide US airport network is shown to be much more heterogeneous than most within-state networks, illustrating the importance of the increased reliability of this true normalisation.

Publisher URL: http://arxiv.org/abs/1803.03057

DOI: arXiv:1803.03057v3

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