Isotropic covariance functions on graphs and their edges

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, Matérn, 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.