The spectral dimension of simplicial complexes: a renormalization group theory

TL;DR

This paper employs renormalization group techniques to compute the spectral dimension of the graph Laplacian in certain non-amenable simplicial complexes, revealing how topology and randomness influence diffusion properties.

Contribution

It introduces a renormalization group approach to analyze the spectral dimension of simplicial complexes, specifically Apollonian and pseudo-fractal networks, and examines the effects of topology and randomness.

Findings

Spectral dimension scales with topological dimension as d→∞

Randomness reduces the spectral dimension in network structures

Renormalization group provides a new method for spectral analysis

Abstract

Simplicial complexes are increasingly used to study complex system structure and dynamics including diffusion, synchronization and epidemic spreading. The spectral dimension of the graph Laplacian is known to determine the diffusion properties at long time scales. Using the renormalization group here we calculate the spectral dimension of the graph Laplacian of two classes of non-amenable $d$ dimensional simplicial complexes: the Apollonian networks and the pseudo-fractal networks. We analyse the scaling of the spectral dimension with the topological dimension $d$ for $d\to \infty$ and we point out that randomness such as the one present in Network Geometry with Flavor can diminish the value of the spectral dimension of these structures.