extremely significant differences among the conditions ($p=3.75          ![Comparison of approximation performance among seven heuristic eigenvalue-sorting criteria (see text for details) for approximating in-strength Laplacian spectra of nonsimple (a) DPMNs and (b) SPMNs. “$\uparrow$” and “$\downarrow$” denote ascending and descending orders, respectively. The root mean square error (RMSE) per eigenvalue was used as a performance metric, in which an error was measured by the absolute value of the difference between true and approximated eigenvalues. Results were collected from one hundred independent tests for each condition, and their distributions were shown in box-whisker plots. Two adjacency matrices of factor networks used were: $A_G=$ a random matrix whose entries were randomly sampled from a uniform distribution $[-1,1]$, and $A_H=$ a random matrix whose entries were randomly sampled from a uniform distribution $[-2,2]$. The sizes of $A_G$ and $A_H$ were randomly and independently set between 40 and 60. These parameter values were chosen arbitrarily just for illustrative purposes, while the overall trends of results were robust to parameter variations. ANOVA and Tukey/Bonferroni posthoc tests showed extremely significant differences among the conditions ($p=3.75
    \times 10^{-331}$ for DPMNs; $p=2.13 \times 10^{-324}$ for SPMNs). Results for out-strength Laplacian spectra showed similar trends.[]{data-label="fig:laplacian-approximation-nonsimple-sortings"}](fig2-a.pdf "fig:"){width="44.00000%"}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 \times 10^{-331}$ for DPMNs; 