Fractal and Transfractal Scale-Free Networks

TL;DR

This paper reviews recent advances in understanding self-similarity in complex networks, exploring its implications for transport, diffusion, and topological properties like degree distribution and modularity.

Contribution

It provides a comprehensive review of self-similarity in complex networks and its relation to various topological and dynamical properties.

Findings

Self-similarity influences transport and diffusion in networks

Fractal and transfractal structures exhibit distinct topological features

Self-similar networks display specific degree distributions and correlations

Abstract

Self-similarity is a property of fractal structures, a concept introduced by Mandelbrot and one of the fundamental mathematical results of the 20th century. The importance of fractal geometry stems from the fact that these structures were recognized in numerous examples in Nature, from the coexistence of liquid/gas at the critical point of evaporation of water, to snowflakes, to the tortuous coastline of the Norwegian fjords, to the behavior of many complex systems such as economic data, or the complex patterns of human agglomeration. Here we review the recent advances in self-similarity of complex networks and its relation to transport, diffusion, percolations and other topological properties such us degree distribution, modularity, and degree-degree correlations.