spectra (eigenvalues of adjacency matrices) of factor networks $G$ and $H$, respectively:

    CPMN

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

    DPMN

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

    SPMN

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

Laplacian spectra

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

    CPMN

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

In contrast to these, no exact formula is known for the characterization of Laplacian spectra of DP/SPMNs using their factor networks’ Laplacian spectra. This is because the Laplacian matrices of DP/SPMNs cannot be expressed simply by using the Laplacian matrices of their factor networks, but instead, they involve the Kronecker products of degree and Laplacian matrices of the factor networks [@sayama2016estimation]. Such a coupling between degree and Laplacian matrices makes it hard to obtain Laplacian spectra of DP/SPMNs analytically.

However, we have recently reported [@sayama2016estimation] that the Laplacian spectra of DP/SPMNs can still be approximated heuristically using degree and Laplacian spectra of their factor networks, as follows:

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 degree/Laplacian spectra of factor networks $G$ and $H$, respectively:

    DPMN

    :   $\left(\mu^G_i d^H_j 