  The advent of data science has provided an increasing number of challenges
with high data complexity. This paper addresses the challenge of space-time
data where the spatial domain is not a planar surface, a sphere, or a linear
network, but a generalized network (termed a graph with Euclidean edges).
Additionally, data are repeatedly measured over different temporal instants. We
provide new classes of nonseparable space-time stationary covariance functions
where {\em space} can be a generalized network, a Euclidean tree, or a linear
network, and where time can be linear or circular (seasonal). Because the
construction principles are technical, we focus on illustrations that guide the
reader through the construction of statistically interpretable examples. A
simulation study demonstrates that we can recover the correct model when
compared to misspecified models. In addition, our simulation studies show that
we effectively recover simulation parameters. In our data analysis, we consider
a traffic accident dataset that shows improved model performance based on
covariance specifications and network-based metrics.
