as follows:

In- and out-strength spectra (= in- and out-strength sequence)

:   With $\left( d^G_i \right)$ and $\left( d^H_j \right)$ being the in- or out-strength spectra (node strengths) of factor networks $G$ and $H$, respectively:

    GGPMN

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

Adjacency spectra

:   With $\left( \lambda^G_i \right)$ and $\left(
      \lambda^H_j \right)$ being the adjacency spectra (eigenvalues of adjacency matrices) of factor networks $G$ and $H$, respectively:

    GGPMN

    :   $\left( \alpha \lambda^G_i + \alpha \lambda^H_j + \beta \lambda^G_i \lambda^H_j \right)$    $\forall i, j$

Approximated Laplacian spectra

:   With $\left[\left(d^G_i\right),
      \left(\mu^G_i\right)\right]$ and $\left[\left(d^H_j\right),
      \left(\mu^H_j\right)\right]$ being the in- or out-strength/Laplacian spectra of factor networks $G$ and $H$, respectively (all spectra should be sorted in an ascending order of real parts for best approximation):

    GGPMN

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

Details of derivation are given in Appendix. Note that these spectral relationships described in this section hold for both simple and nonsimple GGPMNs.

With these two extensions introduced in Sections \[sec:nonsimple\] and \[sec:generalized-gp\], GPMNs as a modeling framework have gained an enhanced expressive power and 