Isotropic covariance functions on graphs and their edges

TL;DR

This paper introduces new isotropic covariance functions on graphs with Euclidean edges, utilizing the resistance metric to extend classical covariance models to complex network structures.

Contribution

It develops parametric classes of covariance functions on graphs using the resistance metric, broadening the applicability of spatial models on networks.

Findings

Many classical covariance functions are valid with the resistance metric on Euclidean edge graphs.

Resistance metric offers advantages over geodesic metric for covariance function validity.

The approach extends spatial statistics models to complex network structures.

Abstract

We develop parametric classes of covariance functions on linear networks and their extension to graphs with Euclidean edges, i.e., graphs with edges viewed as line segments or more general sets with a coordinate system allowing us to consider points on the graph which are vertices or points on an edge. Our covariance functions are defined on the vertices and edge points of these graphs and are isotropic in the sense that they depend only on the geodesic distance or on a new metric called the resistance metric (which extends the classical resistance metric developed in electrical network theory on the vertices of a graph to the continuum of edge points). We discuss the advantages of using the resistance metric in comparison with the geodesic metric as well as the restrictions these metrics impose on the investigated covariance functions. In particular, many of the commonly used isotropic covariance functions in the spatial statistics literature (the power exponential, Mat{\'e}rn, generalized Cauchy, and Dagum classes) are shown to be valid with respect to the resistance metric for any graph with Euclidean edges, whilst they are only valid with respect to the geodesic metric in more special cases.