The spectral dimension of simplicial complexes: a renormalization group theory
This work provides insights into diffusion properties in complex systems, but it is incremental as it applies known methods to specific network classes.
The authors tackled the problem of determining the spectral dimension of graph Laplacians for non-amenable simplicial complexes, specifically Apollonian and pseudo-fractal networks, using renormalization group theory, and found that randomness can reduce the spectral dimension in such structures.
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.