### Abstract

Motivated by the hierarchial network model of E. Ravasz, A.-L. Barabási, and T. Vicsek, we introduce deterministic scale-free networks derived from a graph directed self-similar fractal Λ. With rigorous mathematical results we verify that our model captures some of the most important features of many real networks: the scale-free and the high clustering properties. We also prove that the diameter is the logarithm of the size of the system. We point out a connection between the power law exponent of the degree distribution and some intrinsic geometric measure theoretical properties of the underlying fractal. Using our (deterministic) fractal Λ we generate random graph sequence sharing similar properties.

Original language | English |
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Pages (from-to) | 651-666 |

Number of pages | 16 |

Journal | Chaos, solitons and fractals |

Volume | 44 |

Issue number | 8 |

DOIs | |

Publication status | Published - Aug 1 2011 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematics(all)
- Physics and Astronomy(all)
- Applied Mathematics

### Cite this

*Chaos, solitons and fractals*,

*44*(8), 651-666. https://doi.org/10.1016/j.chaos.2011.05.012