Letters to Nature

Nature 433, 392-395 (27 January 2005) | doi:10.1038/nature03248; Received 4 August 2004; Accepted 30 November 2004

Self-similarity of complex networks

Chaoming Song1, Shlomo Havlin2 & Hernán A. Makse1

  1. Levich Institute and Physics Department, City College of New York, New York, New York 10031, USA
  2. Minerva Center and Department of Physics, Bar-Ilan University, Ramat Gan 52900, Israel

Correspondence to: Hernán A. Makse1 Correspondence and requests for materials should be addressed to H.A.M. (Email: makse@mailaps.org).

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 networks1, 2, 3, 4, 5. 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 node1, 6, 7. 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 network8, 9, 10, 11, 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.

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