  Spatial models for areal data are often constructed such that all pairs of
adjacent regions are assumed to have near-identical spatial autocorrelation. In
practice, data can exhibit dependence structures more complicated than can be
represented under this assumption. In this article we develop a new model for
spatially correlated data observed on graphs, which can flexibly represented
many types of spatial dependence patterns while retaining aspects of the
original graph geometry. Our method implies an embedding of the graph into
Euclidean space wherein covariance can be modeled using traditional covariance
functions, such as those from the Mat\'{e}rn family. We parameterize our model
using a class of graph metrics compatible with such covariance functions, and
which characterize distance in terms of network flow, a property useful for
understanding proximity in many ecological settings. By estimating the
parameters underlying these metrics, we recover the "intrinsic distances"
between graph nodes, which assist in the interpretation of the estimated
covariance and allow us to better understand the relationship between the
observed process and spatial domain. We compare our model to existing methods
for spatially dependent graph data, primarily conditional autoregressive models
and their variants, and illustrate advantages of our method over traditional
approaches. We fit our model to bird abundance data for several species in
North Carolina, and show how it provides insight into the interactions between
species-specific spatial distributions and geography.
