Fractal and Transfractal Recursive Scale-Free Nets

TL;DR

This paper introduces a new family of recursive scale-free networks, analyzing their self-similarity, fractal properties, and multiscaling behaviors, including diffusion and resistance, using renormalization techniques.

Contribution

It provides exact analysis of recursive scale-free nets, defining transfinite dimensions, and explores their multiscaling properties and Einstein relations.

Findings

Some nets are fractals with finite fractal dimension.

Small world nets exhibit infinite-dimensional behavior with logarithmic diameter growth.

Diffusion and resistance scale differently for hubs and other nodes, yet Einstein relations hold.

Abstract

We explore the concepts of self-similarity, dimensionality, and (multi)scaling in a new family of recursive scale-free nets that yield themselves to exact analysis through renormalization techniques. All nets in this family are self-similar and some are fractals - possessing a finite fractal dimension - while others are small world (their diameter grows logarithmically with their size) and are infinite-dimensional. We show how a useful measure of "transfinite" dimension may be defined and applied to the small world nets. Concerning multiscaling, we show how first-passage time for diffusion and resistance between hub (the most connected nodes) scale differently than for other nodes. Despite the different scalings, the Einstein relation between diffusion and conductivity holds separately for hubs and nodes. The transfinite exponents of small world nets obey Einstein relations analogous to those in fractal nets.