---
abstract: |
    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.\
    Keywords: graph product, multilayer networks, degree/adjacency/Laplacian spectra, epidemic thresholds, propagation, self-similar networks
author:
- |
    Hiroki Sayama$^{1,2,3,4}$\
    $^1$ Center for Collective Dynamics of Complex Systems,\
    Binghamton University, Binghamton, New York 13902, USA\
    $^2$ Max Planck Institute for the Physics of Complex Systems,\
    D-01187 Dresden, Germany\
    $^3$ Center for Complex Network Research and Department of Physics,\
    Northeastern University, Boston, Massachusetts 02115, USA\
    $^4$ Faculty of Commerce, Waseda University, Shinjuku, Tokyo 169-8050, Japan
bibliography:
- 'sayama.bib'
title: 'Graph Product Multilayer Networks: Spectral Properties and Applications'
---

Introduction {#sec:intro}
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Multilayer networks 