\operatorname*{\times}H$

:   A network in which two nodes $(g, h)$ and $(g', h')$ are connected if and only if $(g, g') \in
      E_G$ and $(h, h') \in E_H$. Its adjacency matrix is given by $$A_{G \operatorname*{\times}H} = A_G \operatorname*{\otimes}A_H. \label{eq:dp}$$

Strong product $G \operatorname*{\boxtimes}H$

:   A network that is obtained as the sum of $G \operatorname*{\Box}H$ and $G \operatorname*{\times}H$. Its adjacency matrix is given by $$A_{G \operatorname*{\boxtimes}H} = A_G \operatorname*{\oplus}A_H + A_G \operatorname*{\otimes}A_H. \label{eq:sp}$$

Examples are shown in Figure \[fig:GPMN-examples\]. We call these three types of GPMNs [*Cartesian product multilayer networks (CPMNs)*]{}, [*direct product multilayer networks (DPMNs)*]{}, and [ *strong product multilayer networks (SPMNs)*]{}. The number of nodes is $|V_G| |V_H|$ for all of these three graph products. CPMNs guarantee that all inter-layer edges are diagonal and layer-coupled (i.e., independent of the intra-layer nodes), and therefore, all CPMNs are also multiplex networks (in the sense that all layers share the same intra-layer node set and the inter-layer edges are all diagonal) [@kivela2014multilayer]. Some other known topological properties are summarized in Table \[tab:known-properties\].

  ------ -- ----- -----------------------------------------------------
  $G:$            $\displaystyle A_G = \begin{pmatrix}0 & 1 & 0 & 0\\
                  1 & 0 & 1 & 1\\
                  0 & 1 & 0 & 