Diffusion on asymmetric fractal networks

TL;DR

This paper introduces a renormalization method to compute the spectral dimension of deterministic self-similar networks, enabling detailed analysis of microstructural effects on diffusion and challenging existing scaling laws.

Contribution

A new general renormalization approach for calculating spectral dimensions of complex fractal networks with arbitrary structures.

Findings

Spectral dimension can be precisely calculated for diverse self-similar networks.

Asymmetric and non-recurrent trees can violate the Alexander Orbach scaling law.

The method allows detailed microstructural impact analysis on diffusive transport.

Abstract

We derive a renormalization method to calculate the spectral dimension $\bar{d}$ of deterministic self-similar networks with arbitrary base units and branching constants. The generality of the method allows the affect of a multitude of microstructural details to be quantitatively investigated. In addition to providing new models for physical networks, the results allow precise tests of theories of diffusive transport. For example, the properties of a class of non-recurrent trees ($\bar{d}>2$) with asymmetric elements and branching violate the Alexander Orbach scaling law.