The higher-order spectrum of simplicial complexes: a renormalization group approach

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. Higher-order up and down Laplacians can have a finite spectral dimension, characterizing the long time behaviour of the diffusion process on simplicial complexes. Here 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 $m\geq 0$. Finally we discuss how the spectral properties of the higher-order up-Laplacian can change if one considers the simplicial complexes generated by the model "Network Geometry with Flavor". These simplicial complexes are random and display a structural topological phase transition as a function of the parameter $\beta$, which is also reflected in the spectrum of higher-order Laplacians.