networks $G$ and $H$. $\langle k_G \rangle$ and $\langle k_H \rangle$ represent the average degrees of $G$ and $H$, respectively.

\[tab:known-properties\]

The three graph product operators described above are commutative, i.e., the resulting graphs of $G * H$ and $H * G$ (where $*$ is either $\operatorname*{\Box}$, $\operatorname*{\times}$, or $\operatorname*{\boxtimes}$) are isomorphic to each other with appropriate node permutations. Therefore, it is an arbitrary choice which factor network is considered intra- or inter-layer. In addition, these graph product operators are also known to be associative.

One interesting, and quite useful, fact already known about GPMNs is that the spectral properties of their degree, adjacency, and Laplacian matrices are related to those of their factor networks [@macduffee1933theory; @fiedler1973algebraic; @sayama2016estimation]. More specifically, degree and adjacency spectra of CP/DP/SPMNs and Laplacian spectra of CPMNs are characterized exactly by the spectra of their factor networks, as follows [@sayama2016estimation]:

Degree spectra

:   With $\left(d^G_i\right)$ and $\left(d^H_j\right)$ being the degree spectra (node degrees) of factor networks $G$ and $H$, respectively:

    CPMN

    :   $\left(d^G_i + d^H_j\right)$   $\forall i, j$

    DPMN

    :   $\left(d^G_i d^H_j\right)$   $\forall i, j$

    SPMN

    :   $\left(d^G_i + d^H_j + d^G_i d^H_j\right)$   $\forall i, j$

Adjacency spectra

:   With $\left(\lambda^G_i\right)$ and $\left(\lambda^H_j\right)$ being the adjacency 