  We explore how to study dynamical interactions between brain regions using
functional multilayer networks whose layers represent the different frequency
bands at which a brain operates. Specifically, we investigate the consequences
of considering the brain as a multilayer network in which all brain regions can
interact with each other at different frequency bands, instead of as a
multiplex network, in which interactions between different frequency bands are
only allowed within each brain region and not between them. We study the second
smallest eigenvalue of the combinatorial supra-Laplacian matrix of the
multilayer network in detail, and we thereby show that the heterogeneity of
interlayer edges and, especially, the fraction of missing edges crucially
modify the spectral properties of the multilayer network. We illustrate our
results with both synthetic network models and real data sets obtained from
resting state magnetoencephalography. Our work demonstrates an important issue
in the construction of frequency-based multilayer brain networks.
