Fractal and transfractal recursive scale-free nets

Hernán D. Rozenfeld, Shlomo Havlin, Daniel Ben-Avraham

Research output: Contribution to journalArticlepeer-review

187 Scopus citations


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 hubs (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.

Original languageEnglish
Article number175
JournalNew Journal of Physics
StatePublished - 28 Jun 2007


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