Unique superdiffusion induced by directionality in multiplex networks

The multilayer network framework has served to describe and uncover a number of novel and
unforeseen physical behaviors and regimes in interacting complex systems. However, the majority
of existing studies are built on undirected multilayer networks while most complex systems in
nature exhibit directed interactions. Here, we propose a framework to analyze diffusive dynamics
on multilayer networks consisting of at least one directed layer. We rigorously demonstrate that
directionality in multilayer networks can fundamentally change the behavior of diffusive
dynamics: from monotonic (in undirected systems) to non-monotonic diffusion with respect to
the interlayer coupling strength. Moreover, for certain multilayer network configurations, the
directionality can induce a unique superdiffusion regime for intermediate values of the interlayer
coupling, wherein the diffusion is even faster than that corresponding to the theoretical limit for
undirected systems, i.e. the diffusion in the integrated network obtained from the aggregation of
each layer. We theoretically and numerically show that the existence of superdiffusion is fully
determined by the directionality of each layer and the topological overlap between layers. We
further provide a formulation of multilayer networks displaying superdiffusion. Our results
highlight the significance of incorporating the interacting directionality in multilevel networked
systems and provide a framework to