the algebraic properties of Kronecker sum and product, and therefore they also apply to nonsimple counterparts without any modification (we also confirmed this numerically; results not shown).

Some additional considerations are needed for Laplacian spectra of nonsimple DP/SPMNs. As described in Section \[sec:simple\], the original approximation method developed for simple DP/SPMNs [@sayama2016estimation] involved the sorting of degree and Laplacian spectra of factor networks. This requires additional investigation when applied to nonsimple networks, because adjacency and Laplacian matrices of nonsimple networks are generally asymmetric and thus have complex eigenvalues, for which sorting is not obvious.

To address this issue, we have conducted numerical experiments with the following seven different heuristic criteria for sorting Laplacian eigenvalues of factor networks (while node strengths were always sorted in an ascending order):

1.  Random

2.  By real part (ascending)

3.  By real part (descending)

4.  By imaginary part (ascending)

5.  By imaginary part (descending)

6.  By absolute value (ascending)

7.  By absolute value (descending)

Results are summarized in Fig. \[fig:laplacian-approximation-nonsimple-sortings\]. The results clearly show that, when the eigenvalues of Laplacian matrices of factor networks are sorted by their real parts in an ascending order (which is the most natural generalization of the sorting method used 