edges (Fig. \[fig:GPMN-random-walk\], top). Each of these networks in isolation can be considered a representation of either spatial movement without time ($G$), or the flow of time without spatial movement ($T$). Applying different graph product operators to these two factor networks produces a specific spatio-temporal representation of different random walk processes, as follows:

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  $G$ :   
           
  $T$ :   
           
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  \(a) $G \operatorname*{\times}T$ :
  ![Weighted random walk processes represented by GPMNs. Top: Two factor networks. $G$ represents an underlying spatial transition network, while $T$ represents time. (a) Direct product $G \operatorname*{\times}T$, representing random walk with forced temporal progress. (b) Cartesian product $G \operatorname*{\Box}T$, representing random walk within each time segment with occasional stochastic temporal progress. (c) Generalized product $\displaystyle G \operatorname*{\boxtimes}_{\alpha, \beta} T$, representing the combination of (a) and (b). Circles represent layers (time segments).[]{data-label="fig:GPMN-random-walk"}](GPMN-random-walk-DP.pdf "fig:"){width="90.00000%"}
   
  \(b) $G \operatorname*{\Box}T$ :
  ![Weighted random walk processes represented by GPMNs. Top: Two factor networks. $G$ represents an underlying spatial transition network, while $T$ represents time. (a) Direct product $G \operatorname*{\times}T$, representing random walk with forced temporal progress. (b) Cartesian product $G \operatorname*{\Box}T$, representing random walk within each time segment with occasional stochastic temporal progress. (c) Generalized product $\displaystyle G \operatorname*{\boxtimes}_{\alpha, 