+ d^G_i \mu^H_j - \mu^G_i \mu^H_j \right)$   $\forall i, j$

    SPMN

    :   $\left(\mu^G_i + \mu^H_j + \mu^G_i d^H_j + d^G_i \mu^H_j - \mu^G_i \mu^H_j \right)$   $\forall i, j$

Note that these formulae involve two distinct spectra for each factor network, $\left( d \right)$ and $\left( \mu \right)$, whose orderings are independent from each other. This implies that optimizing their orderings can help improve the accuracy of the heuristic approximation. In [@sayama2016estimation], we explored several different ordering methods and found that sorting both spectra in an ascending order achieves the best approximation (we will revisit this issue in the next section). See [@sayama2016estimation] for more details.

In summary, the spectral relationships between GPMNs and their factor networks described above help study structural and dynamical properties of GPMNs, such as degree distributions, bipartiteness, algebraic connectivities, number of connected components, spectral gaps, eigenratios, and so on.

Spectral Properties of Nonsimple GPMNs {#sec:nonsimple}
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In this paper, we aim to extend the definitions of GPMNs to make them capable of capturing a greater variety of complex networks.

The first extension is to consider nonsimple graphs with directed, weighted, signed, and/or self-looped edges. Mathematically, this relaxation is equivalent to considering any arbitrary real-valued square matrices 