  Higher-order networks are gaining significant scientific attention due to
their ability to encode the many-body interactions present in complex systems.
However, higher-order networks have the limitation that they only capture
many-body interactions of the same type. To address this limitation, we present
a mathematical framework that determines the topology of higher-order multiplex
networks and illustrates the interplay between their topology and dynamics.
Specifically, we examine the diffusion of topological signals associated not
only to the nodes but also to the links and to the higher-dimensional simplices
of multiplex simplicial complexes. We leverage on the ubiquitous presence of
the overlap of the simplices to couple the dynamics among multiplex layers,
introducing a definition of multiplex Hodge Laplacians and Dirac operators. We
show that the spectral properties of these operators determine higher-order
diffusion on higher-order multiplex networks and encode their multiplex Betti
numbers. Our numerical investigation of the spectral properties of synthetic
and real (connectome, microbiome) multiplex simplicial complexes indicates that
the coupling between the layers can either speed up or slow down the
higher-order diffusion of topological signals. This mathematical framework is
very general and can be applied to study generic higher-order systems with
interactions of multiple types. In particular, these results might find
applications in brain networks which are understood to be both multilayer and
higher-order.
