product operators: Cartesian, direct (tensor), and strong products. We describe fundamental mathematical relationships between GPMNs and their factor networks regarding their degree/strength, adjacency, and Laplacian spectra (i.e., eigenvalues of degree/strength, adjacency, and Laplacian matrices). These relationships are exact, except for Laplacian ones of direct and strong products, while those Laplacian spectra can also be approximated using heuristic methods [@sayama2016estimation]. We also extend the definitions of GPMNs to nonsimple networks with directed, weighted, signed, and/or self-looped edges, and then generalize graph product operation to arbitrary linear combination of Cartesian and direct products. We show that the previously reported spectral relationships between GPMNs and their factor networks are still maintained in nonsimple and generalized cases. In the latter half of this paper, we demonstrate the effectiveness of GPMNs through three applications: prediction of epidemic thresholds, modeling of propagation in nontrivial space and time, and analysis of higher-order properties of self-similar networks.

It should be noted here that most real-world complex networks are not GPMNs, and therefore their structure and dynamics would not be fully captured within the GPMN framework. However, GPMNs may still be used as a reference or surrogate model to which real-world multilayer network structure and dynamics can be compared. 