in [@sayama2016estimation]), the same methods described in Section \[sec:simple\] can approximate in- and out-strength Laplacian spectra of nonsimple DP/SPMNs most effectively. Examples of approximations are shown in Fig. \[fig:laplacian-approximation-nonsimple\].

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    (a) DPMNs                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  (b) SPMNs

  ![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 