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Network representations are useful for describing the structure of a largevariety of complex systems. Although most studies of real-world networkssuppose that nodes are connected by only a single type of edge, most naturaland engineered systems include multiple subsystems and layers of connectivity.This new paradigm has attracted a great deal of attention and one fundamentalchallenge is to characterize multilayer networks both structurally anddynamically. One way to address this question is to study the spectralproperties of such networks. Here, we apply the framework of graph quotients,which occurs naturally in this context, and the associated eigenvalueinterlacing results, to the adjacency and Laplacian matrices of undirectedmultilayer networks. Specifically, we describe relationships between theeigenvalue spectra of multilayer networks and their two most natural quotients,the network of layers and the aggregate network, and show the dynamicalimplications of working with either of the two simplified representations. Ourwork thus contributes in particular to the study of dynamical processes whosecritical properties are determined by the spectral properties of the underlyingnetwork.

Dimensionality reduction and spectral properties of multilayer networks