### Abstract

Complex networks have been studied extensively owing to their relevance to many real systems such as the world-wide web, the Internet, energy landscapes and biological and social networks. A large number of real networks are referred to as 'scale-free' because they show a power-law distribution of the number of links per node. However, it is widely believed that complex networks are not invariant or self-similar under a length-scale transformation. This conclusion originates from the 'small-world' property of these networks, which implies that the number of nodes increases exponentially with the 'diameter' of the network, rather than the power-law relation expected for a self-similar structure. Here we analyse a variety of real complex networks and find that, on the contrary, they consist of self-repeating patterns on all length scales. This result is achieved by the application of a renormalization procedure that coarse-grains the system into boxes containing nodes within a given 'size'. We identify a power-law relation between the number of boxes needed to cover the network and the size of the box, defining a finite self-similar exponent. These fundamental properties help to explain the scale-free nature of complex networks and suggest a common self-organization dynamics.

Original language | English (US) |
---|---|

Pages (from-to) | 392-395 |

Number of pages | 4 |

Journal | Nature |

Volume | 433 |

Issue number | 7024 |

DOIs | |

State | Published - Jan 27 2005 |

Externally published | Yes |

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### ASJC Scopus subject areas

- General

### Cite this

*Nature*,

*433*(7024), 392-395. https://doi.org/10.1038/nature03248

**Self-similarity of complex networks.** / Song, Chaoming; Havlin, Shlomo; Makse, Hernán A.

Research output: Contribution to journal › Article

*Nature*, vol. 433, no. 7024, pp. 392-395. https://doi.org/10.1038/nature03248

}

TY - JOUR

T1 - Self-similarity of complex networks

AU - Song, Chaoming

AU - Havlin, Shlomo

AU - Makse, Hernán A.

PY - 2005/1/27

Y1 - 2005/1/27

N2 - Complex networks have been studied extensively owing to their relevance to many real systems such as the world-wide web, the Internet, energy landscapes and biological and social networks. A large number of real networks are referred to as 'scale-free' because they show a power-law distribution of the number of links per node. However, it is widely believed that complex networks are not invariant or self-similar under a length-scale transformation. This conclusion originates from the 'small-world' property of these networks, which implies that the number of nodes increases exponentially with the 'diameter' of the network, rather than the power-law relation expected for a self-similar structure. Here we analyse a variety of real complex networks and find that, on the contrary, they consist of self-repeating patterns on all length scales. This result is achieved by the application of a renormalization procedure that coarse-grains the system into boxes containing nodes within a given 'size'. We identify a power-law relation between the number of boxes needed to cover the network and the size of the box, defining a finite self-similar exponent. These fundamental properties help to explain the scale-free nature of complex networks and suggest a common self-organization dynamics.

AB - Complex networks have been studied extensively owing to their relevance to many real systems such as the world-wide web, the Internet, energy landscapes and biological and social networks. A large number of real networks are referred to as 'scale-free' because they show a power-law distribution of the number of links per node. However, it is widely believed that complex networks are not invariant or self-similar under a length-scale transformation. This conclusion originates from the 'small-world' property of these networks, which implies that the number of nodes increases exponentially with the 'diameter' of the network, rather than the power-law relation expected for a self-similar structure. Here we analyse a variety of real complex networks and find that, on the contrary, they consist of self-repeating patterns on all length scales. This result is achieved by the application of a renormalization procedure that coarse-grains the system into boxes containing nodes within a given 'size'. We identify a power-law relation between the number of boxes needed to cover the network and the size of the box, defining a finite self-similar exponent. These fundamental properties help to explain the scale-free nature of complex networks and suggest a common self-organization dynamics.

UR - http://www.scopus.com/inward/record.url?scp=13444263410&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=13444263410&partnerID=8YFLogxK

U2 - 10.1038/nature03248

DO - 10.1038/nature03248

M3 - Article

C2 - 15674285

AN - SCOPUS:13444263410

VL - 433

SP - 392

EP - 395

JO - Nature

JF - Nature

SN - 0028-0836

IS - 7024

ER -