  Network topology is a flourishing interdisciplinary subject that is relevant
for different disciplines including quantum gravity and brain research. The
discrete topological objects that are investigated in network topology are
simplicial complexes. Simplicial complexes generalize networks by not only
taking pairwise interactions into account, but also taking into account
many-body interactions between more than two nodes. Higher-order Laplacians are
topological operators that describe higher-order diffusion on simplicial
complexes and constitute the natural mathematical objects that capture the
interplay between network topology and dynamics. We show that higher-order up
and down Laplacians can have a finite spectral dimension, characterizing the
long time behaviour of the diffusion process on simplicial complexes that
depends on their order $m$. We provide a renormalization group theory for the
calculation of the higher-order spectral dimension of two deterministic models
of simplicial complexes: the Apollonian and the pseudo-fractal simplicial
complexes. We show that the RG flow is affected by the fixed point at zero
mass, which determines the higher-order spectral dimension $d_S$ of the
up-Laplacians of order $m$ with $m\geq 0$.

Disordered Systems and Neural Networks;Statistical Mechanics;General Relativity and Quantum Cosmology;High Energy Physics - Lattice;Physics and Society