To the best of our knowledge, the measures that we introduce for variance and covariance are new in the context of distributions on networks. Mathematically, our variance measure is a special case of the well-known Rao’s quadratic entropy [40, 45] which measures the diversity of distributions on general categorical variables with a notion of dissimilarity between the variables. This correspondence connects our work to the more general setting of diversity measures which are widely used, for instance in ecology [42, 9]. The work of Leinster et al. on maximum diversity distributions [30, 29] is particularly relevant (see Section 5).