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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We are grateful to J. Brujić for many discussions. This work is supported by the National Science Foundation, Materials Theory. S.H. thanks the Israel Science Foundation and ONR for support.
The authors declare that they have no competing financial interests.
Additional information relating to (a) the box covering method, (b) scale-free tree structure, (c) internet, (d) protein-protein interaction networks, (e) random scale-free networks, (f) the Barabasi-Albert model and the Erdos-Renyi random graph at criticality, and (g) cellular networks. The file contains Supplementary Figures 1-7, Supplementary Table 1 and Supplementary References. (PDF 2123 kb)
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Song, C., Havlin, S. & Makse, H. Self-similarity of complex networks. Nature 433, 392–395 (2005). https://doi.org/10.1038/nature03248
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