whose entries were randomly sampled from a uniform distribution $[-1,1]$, and $A_H=$ a $30
    \times 30$ random matrix whose entries were randomly sampled from a uniform distribution $[-2,2]$. These parameter values were chosen arbitrarily just for illustrative purposes, while the overall trends of results were robust to parameter variations. Following the results shown in Fig. \[fig:laplacian-approximation-nonsimple-sortings\], the factor networks’ node strengths and Laplacian eigenvalues were sorted in ascending order (by their real parts for the latter) before the approximation method was used.[]{data-label="fig:laplacian-approximation-nonsimple"}](fig3-a.pdf "fig:"){width="44.00000%"}                                 \times 30$ random matrix whose entries were randomly sampled from a uniform distribution $[-2,2]$. These parameter values were chosen arbitrarily just for illustrative purposes, while the overall trends of results were robust to parameter variations. Following the results shown in Fig. \[fig:laplacian-approximation-nonsimple-sortings\], the factor networks’ node strengths and Laplacian eigenvalues were sorted in ascending order (by their real parts for the latter) before the approximation method was used.[]{data-label="fig:laplacian-approximation-nonsimple"}](fig3-b.pdf "fig:"){width="44.00000%"}
  ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------ ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

To summarize, the extension of GPMNs to nonsimple networks can be done in a straightforward manner, and the known spectral relationships between GPMNs and their factor networks are also applicable to nonsimple ones with little to no modification.

Generalizing Graph Product {#sec:generalized-gp}
==========================

The second 