  Models for areal data are traditionally defined using the neighborhood
structure of the regions on which data are observed. The unweighted adjacency
matrix of a graph is commonly used to characterize the relationships between
locations, resulting in the implicit assumption that all pairs of neighboring
regions interact similarly, an assumption which may not be true in practice. It
has been shown that more complex spatial relationships between graph nodes may
be represented when edge weights are allowed to vary. Christensen and Hoff
(2023) introduced a covariance model for data observed on graphs which is more
flexible than traditional alternatives, parameterizing covariance as a function
of an unknown edge weights matrix. A potential issue with their approach is
that each edge weight is treated as a unique parameter, resulting in
increasingly challenging parameter estimation as graph size increases. Within
this article we propose a framework for estimating edge weight matrices that
reduces their effective dimension via a basis function representation of of the
edge weights. We show that this method may be used to enhance the performance
and flexibility of covariance models parameterized by such matrices in a series
of illustrations, simulations and data examples.
