\beta} T$, representing the combination of (a) and (b). Circles represent layers (time segments).[]{data-label="fig:GPMN-random-walk"}](GPMN-random-walk-CP.pdf "fig:"){width="90.00000%"}
   
  \(c) $\displaystyle G \operatorname*{\boxtimes}_{\alpha, \beta} T$ :
  ![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-GP.pdf "fig:"){width="90.00000%"}
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Direct product $G \operatorname*{\times}T$

:   This represents weighted random walk with forced temporal progress (Fig. \[fig:GPMN-random-walk\](a)). Each layer contains no intra-layer edges, and all the edges connect a node in one layer to another node in the subsequent layer, indicating that no entities can stay within the same time point when they make transitions. This is the most natural representation of “vanilla” random walk in which time is explicitly represented by $T$.

Cartesian product $G \operatorname*{\Box}T$

:   This represents weighted random walk taking place within each time segment (layer) with occasional stochastic temporal progress without spatial transition (Fig. \[fig:GPMN-random-walk\](b)). Each layer contains the original $G$ 