Hyper-diffusion on multiplex networks

Multiplex networks describe systems whose interactions can be of different nature, and are fundamental to understand complexity beyond the framework simple graphs. Recently it has been pointed out that restricting attention pairwise is also a limitation, as vast majority complex include higher-order strongly affect their dynamics. Here, we propose hyper-diffusion on multiplex networks, dynamical process in which diffusion each single layer coupled with other layers thanks presence occurring when there exists link overlap. We show duplex network described by Hyper-Laplacian strength four-body among every set four replica nodes connected multilink $(1,1)$ tuned parameter $\delta_{11}\ge 0$. The reduces standard lower Laplacian, capturing at two layers, $\delta_{11}=0$. By combining tools spectral graph theory, applied topology science provide general understanding $\delta_{11}>0$, including theoretical bounds Fiedler largest eigenvalue Hyper-Laplacians asymptotic expansion spectrum for $\delta_{11}\ll1$ $\delta_{11}\gg1$. Although does not imply direct ``transfer mass" (i.e. average state conserved quantity dynamics), find dynamics relaxation steady becomes synchronous taken into account localized network.