One may test various structural/dynamical properties of complex network models or data against GPMN-based approximations obtained by assuming that intra-layer networks are identical across all layers and that inter-layer edges follow certain patterns. There are also other areas of applications of GPMNs, as illustrated later in this paper.

The rest of the paper is structured as follows. In the next section, we first define simple GPMNs and summarize their known spectral properties. In Section \[sec:nonsimple\], we extend the definitions of GPMNs to nonsimple graphs. In Section \[sec:generalized-gp\], we generalize graph product operation. Sections \[sec:app1\], \[sec:app2\], and \[sec:app3\] describe applications of GPMNs. Finally, Section \[sec:conclusions\] concludes this paper with a discussion on directions of future research.

Simple Graph Product Multilayer Networks and Their Spectral Properties {#sec:simple}
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We define [*graph product multilayer networks (GPMNs)*]{} as a particular family of multilayer networks that can be obtained by applying graph product operation(s) to two or more smaller networks. We call those smaller networks [*factor networks*]{}. One of the factor networks is often considered a template of intra-layer networks that uniformly applies to all the layers, while other factor networks are often considered to represent aspects of layers [@kivela2014multilayer] that can be nonlinear and 