[@nadakuditi2012graph; @peixoto2013eigenvalue; @ghasemian2016detectability]. Compared to them, the example presented in this section above is unique in using graph product as a concise mathematical representation of network topology and focusing specifically on the prediction of epidemic thresholds.

Application II: Propagation in Nontrivial Space and Time {#sec:app2}
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GPMNs can be used to represent dynamical processes taking place in nontrivial spatial and/or temporal structures. A popular way of doing this is to use one of the factor networks as a network-shaped spatial structure, while the other as an identical dynamical network that is embedded inside of each node in space. There are several such multilayer network models developed and used in the literature, including coupled cellular networks [@golubitsky2009bifurcations] and ecological networks on interconnected habitats [@brechtel2016master]. In these cases, network topologies that can be produced by Cartesian products were typically used to couple multiple identical intra-layer networks over space.

Here, we illustrate other uses of GPMNs to represent dynamical processes that involve not just nontrivial space but also time. Specifically, we use graph product to represent various types of weighted random walks on a network. Let $G$ be a weighted, directed network that represents transition likelihoods, and $T$ a chain made of directed 