Variance and Covariance of Distributions on Graphs

We develop a theory to measure the variance and covariance of probability distributions defined on nodes graph, which takes into account distance between nodes. Our approach generalizes usual (co)variance setting weighted graphs retains many its intuitive desired properties. Interestingly, we find that number famous concepts in graph network science can be reinterpreted this as variances covariances particular distributions. As application, define maximum problem with respect effective resistance distance, characterize solutions both numerically theoretically. show how distribution is concentrated boundary illustrate case random geometric graphs. theoretical results are supported by experiments mathematical concepts, where use analytical tools study (co)occurrence scientific papers (network) relations these concepts.