Self-similarity of complex networks


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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Figure 1: The renormalization procedure applied to complex networks.
Figure 2: Self-similar scaling in complex networks.
Figure 3: Different averaging techniques lead to qualitatively different results.


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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.

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Correspondence to Hernán A. Makse.

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The authors declare that they have no competing financial interests.

Supplementary information

Supplementary Information

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).

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