  Simplicial complexes constitute the underlying topology of interacting
complex systems including among the others brain and social interaction
networks. They are generalized network structures that allow to go beyond the
framework of pairwise interactions and to capture the many-body interactions
between two or more nodes strongly affecting dynamical processes. In fact, the
simplicial complexes topology allows to assign a dynamical variable not only to
the nodes of the interacting complex systems but also to links, triangles, and
so on. Here we show evidence that the dynamics defined on simplices of
different dimensions can be significantly different even if we compare dynamics
of simplices belonging to the same simplicial complex. By investigating the
spectral properties of the simplicial complex model called "Network Geometry
with Flavor" we provide evidence that the up and down higher-order Laplacians
can have a finite spectral dimension whose value increases as the order of the
Laplacian increases. Finally we discuss the implications of this result for
higher-order diffusion defined on simplicial complexes.
