### Abstract

Fractal scaling and self-similar connectivity behaviour of scale-free (SF) networks are reviewed and investigated in diverse aspects. We first recall an algorithm of box-covering that is useful and easy to implement in SF networks, the so-called random sequential box-covering. Next, to understand the origin of the fractal scaling, fractal networks are viewed as comprising of a skeleton and shortcuts. The skeleton, embedded underneath the original network, is a spanning tree specifically based on the edge-betweenness centrality or load. We show that the skeleton is a non-causal tree, either critical or supercritical. We also study the fractal scaling property of the k-core of a fractal network and find that as k increases, not only does the fractal dimension of the &-core change but also eventually the fractality no longer holds for large enough k. Finally, we study the self-similarity, manifested as the scale-invariance of the degree distribution under coarse-graining of vertices by the box-covering method. We obtain the condition for self-similarity, which turns out to be independent of the fractality, and find that some non-fractal networks are self-similar. Therefore, fractality and self-similarity are disparate notions in SF networks.

Original language | English |
---|---|

Article number | 177 |

Journal | New Journal of Physics |

Volume | 9 |

DOIs | |

Publication status | Published - 2007 Jun 28 |

### ASJC Scopus subject areas

- Physics and Astronomy(all)

## Fingerprint Dive into the research topics of 'Fractality and self-similarity in scale-free networks'. Together they form a unique fingerprint.

## Cite this

*New Journal of Physics*,

*9*, [177]. https://doi.org/10.1088/1367-2630/9/6/177