Space-time data on a network are typically dependent in such a way that observations that are closer in space and/or in time tend to behave more similarly. The covariance function of such a stochastic process captures the small-scale variation over space-time in a parsimonious way and plays a central role in prediction and regression-type parameter estimation. I construct brand-new space-time models on the network using a Hilbert space embedding technique and a kernel convolution-based approach.