as is, while all the inter-layer edges are diagonal, i.e., connecting the same node from one time point to the next. The relative probability of such temporal progress can be adjusted by changing the average edge weight in $T$ (we call it $\langle E_T
      \rangle$) with reference to the average edge weight in $G$ (we call it $\langle E_G \rangle$). This can be a useful random walk model if time has well-defined segments (e.g., days, weeks, months) and many transitions are expected to occur within each segment before moving onto the next segment.

Generalized product $\displaystyle G \operatorname*{\boxtimes}_{\alpha, \beta} T$

:   This represents the combination of the above two random walk processes with their relative weights given by $\alpha$ and $\beta$ (Fig. \[fig:GPMN-random-walk\](c)). Therefore, together with the $\langle E_T \rangle / \langle E_G \rangle$ ratio, this model essentially has three adjustable global parameters. They collectively determine the relative likelihoods of (i) transitions within a single time segment (determined by $\alpha \langle E_G
      \rangle$), (ii) temporal progress without spatial transitions (determined by $\alpha \langle E_T \rangle$), and (iii) temporal progress with spatial transitions (determined by $\beta \langle E_G
      \rangle \langle E_T \rangle $).

Using this framework, interactions between nontrivial space and 