Spectral dimensions of hierarchical scale-free networks with shortcuts

TL;DR

This paper investigates the spectral dimension of hierarchical scale-free networks with shortcuts, revealing how it varies with network structure and providing exact analytical results through renormalization group methods.

Contribution

It introduces an exact analytical approach to determine the spectral dimension of hierarchical scale-free networks with shortcuts, highlighting the crossover between fractal and non-fractal cases.

Findings

Spectral dimension ranges from 1 to 2 for fractal networks.

Spectral dimension remains at 2 for non-fractal networks.

Crossover behavior is characterized by RG flow analysis.

Abstract

The spectral dimension has been widely used to understand transport properties on regular and fractal lattices. Nevertheless, it has been little studied for complex networks such as scale-free and small world networks. Here we study the spectral dimension and the return-to-origin probability of random walks on hierarchical scale-free networks, which can be either fractals or non-fractals depending on the weight of shortcuts. Applying the renormalization group (RG) approach to the Gaussian model, we obtain the spectral dimension exactly. While the spectral dimension varies between $1$ and $2$ for the fractal case, it remains at $2$, independent of the variation of network structure for the non-fractal case. The crossover behavior between the two cases is studied through the RG flow analysis. The analytic results are confirmed by simulation results and their implications for the architecture of complex systems are discussed.