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

Nature
 Pub Date:
 January 2005
 DOI:
 10.1038/nature03248
 arXiv:
 arXiv:condmat/0503078
 Bibcode:
 2005Natur.433..392S
 Keywords:

 Condensed Matter  Disordered Systems and Neural Networks
 EPrint:
 28 pages, 12 figures, more informations at http://www.jamlab.org