  This paper aims to establish theoretical foundations of graph product
multilayer networks (GPMNs), a family of multilayer networks that can be
obtained as a graph product of two or more factor networks. Cartesian, direct
(tensor), and strong product operators are considered, and then generalized. We
first describe mathematical relationships between GPMNs and their factor
networks regarding their degree/strength, adjacency, and Laplacian spectra, and
then show that those relationships can still hold for nonsimple and generalized
GPMNs. Applications of GPMNs are discussed in three areas: predicting epidemic
thresholds, modeling propagation in nontrivial space and time, and analyzing
higher-order properties of self-similar networks. Directions of future research
are also discussed.
