# Scaling laws for diffusion on (trans)fractal scale-free networks

## TL;DR

This paper explores how diffusion processes and spectral properties behave on fractal and transfractal scale-free networks, revealing that similar scaling laws emerge governed by their fractal dimensions.

## Contribution

It demonstrates that diffusion and spectral properties follow similar scaling laws on both fractal and transfractal networks, controlled by their respective dimensions.

## Key findings

- Scaling laws for random walks are governed by fractal dimensions.
- Laplacian spectra exhibit similar scaling behaviors in fractal and transfractal networks.
- Dynamic processes are influenced by topological features controlled by network parameters.

## Abstract

Fractal (or transfractal) features are common in real-life networks and are known to influence the dynamic processes taking place in the network itself. Here we consider a class of scale-free deterministic networks, called $(u,v)$-flowers, whose topological properties can be controlled by tuning the parameters $u$ and $v$; in particular, for $u>1$, they are fractals endowed with a fractal dimension $d_f$, while for $u=1$, they are transfractal endowed with a transfractal dimension $\tilde{d}_f$. In this work we investigate dynamic processes (i.e., random walks) and topological properties (i.e., the Laplacian spectrum) and we show that, under proper conditions, the same scalings (ruled by the related dimensions), emerge for both fractal and transfractal.