Variance and covariance of distributions on graphs

TL;DR

This paper introduces a new framework for measuring variance and covariance of probability distributions on graph nodes, incorporating node distances, and applies it to analyze network structures and concept co-occurrences.

Contribution

It generalizes variance and covariance to weighted graphs, linking these concepts to graph theory and network science, and explores the maximum variance problem with theoretical and numerical insights.

Findings

Maximum variance distribution concentrates on graph boundaries

Variance and covariance help analyze concept co-occurrence in scientific papers

The approach unifies various graph-theoretic concepts under a probabilistic framework

Abstract

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