for adjacency matrices of factor networks, $A_G$ and $A_H$. Through the rest of the paper, “adjacency matrices” are used in this broader definition. Note that the ways adjacency matrices of graph products are calculated in Eqs. (\[eq:cp\]), (\[eq:dp\]), and (\[eq:sp\]) do not assume graph simplicity, so they can seamlessly apply to nonsimple graphs as is. We call the resulting networks [*nonsimple GPMNs*]{}.

Of particular interest is whether the spectral relationships between simple GPMNs and their factor networks described in Section \[sec:simple\] also apply to nonsimple GPMNs. For asymmetric networks with weighted/signed edges, we define strength (degree) and Laplacian matrices as follows (note that these Laplacian matrices are no longer symmetric or positive-semidefinite in general):

In-strength matrix

:   $D^\mathrm{in} = \mathrm{diag}(A {\mathbbm{1}}_n)$

Out-strength matrix

:   $D^\mathrm{out} = \mathrm{diag}({\mathbbm{1}}^T_n A)$

In-strength Laplacian matrix

:   $L^\mathrm{in} = D^\mathrm{in} - A$

Out-strength Laplacian matrix

:   $L^\mathrm{out} = D^\mathrm{out} - A$

Here $A$ is an $n \times n$ asymmetric, weighted and/or signed adjacency matrix with possible self-loops (non-zero diagonal entries), and ${\mathbbm{1}}_n$ is an all-one column vector of size $n$.

We take note that the analytical characterizations of degree and adjacency spectra of simple CP/DP/SPMNs and Laplacian spectra of CPMNs [@sayama2016estimation] were solely based on 