Self-similarity of complex networks

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 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.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

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.

References

  1. 1

    Albert, R. & Barabási, A.-L. Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47–97 (2002)

    ADS  MathSciNet  Article  Google Scholar 

  2. 2

    Dorogovtsev, S. N. & Mendes, J. F. F. Evolution of Networks: From Biological Nets to the Internet and the WWW. (Oxford Univ. Press, Oxford, 2003)

    Google Scholar 

  3. 3

    Pastor-Satorras, R. & Vespignani, A. Evolution and Structure of the Internet: a Statistical Physics Approach (Cambridge Univ. Press, Cambridge, 2004)

    Google Scholar 

  4. 4

    Newman, M. E. J. The structure and function of complex networks. SIAM Rev. 45, 167–256 (2003)

    ADS  MathSciNet  Article  Google Scholar 

  5. 5

    Amaral, L. A. N. & Ottino, J. M. Complex networks—augmenting the framework for the study of complex systems. Eur. Phys. J. B 38, 147–162 (2004)

    ADS  CAS  Article  Google Scholar 

  6. 6

    Albert, R., Jeong, H. & Barabási, A.-L. Diameter of the World Wide Web. Nature 401, 130–131 (1999)

    ADS  CAS  Article  Google Scholar 

  7. 7

    Faloutsos, M., Faloutsos, P. & Faloutsos, C. On power-law relationships of the Internet topology. Comput. Commun. Rev. 29, 251–262 (1999)

    Article  Google Scholar 

  8. 8

    Erdös, P. & Rényi, A. On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci. 5, 17–61 (1960)

    MathSciNet  MATH  Google Scholar 

  9. 9

    Bollobás, B. Random Graphs (Academic, London, 1985)

    Google Scholar 

  10. 10

    Milgram, S. The small-world problem. Psychol. Today 2, 60–67 (1967)

    Google Scholar 

  11. 11

    Watts, D. J. & Strogatz, S. H. Collective dynamics of ‘small-world’ networks. Nature 393, 440–442 (1998)

    ADS  CAS  Article  Google Scholar 

  12. 12

    Bunde, A. & Havlin, S. Fractals and Disordered Systems Ch. 2 (eds Bunde, A. & Havlin, S.) 2nd edn (Springer, Heidelberg, 1996)

    Google Scholar 

  13. 13

    Vicsek, T. Fractal Growth Phenomena 2nd edn, Part IV (World Scientific, Singapore, 1992)

    Google Scholar 

  14. 14

    Feder, J. Fractals (Plenum, New York, 1988)

    Google Scholar 

  15. 15

    Barabási, A.-L. & Albert, R. Emergence of scaling in random networks. Science 286, 509–512 (1999)

    ADS  MathSciNet  Article  Google Scholar 

  16. 16

    Xenarios, I. et al. DIP: the database of interacting proteins. Nucleic Acids Res. 28, 289–291 (2000)

    CAS  Article  Google Scholar 

  17. 17

    Database of Interacting Proteins (DIP)http://dip.doe-mbi.ucla.edu (2000).

  18. 18

    Jeong, H., Tombor, B., Albert, R., Oltvai, Z. N. & Barabási, A.-L. The large-scale organization of metabolic networks. Nature 407, 651–654 (2000)

    ADS  CAS  Article  Google Scholar 

  19. 19

    Overbeek, R. et al. WIT: integrated system for high-throughput genome sequence analysis and metabolic reconstruction. Nucleic Acid Res. 28, 123–125 (2000)

    CAS  Article  Google Scholar 

Download references

Acknowledgements

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.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Hernán A. Makse.

Ethics declarations

Competing interests

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)

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Song, C., Havlin, S. & Makse, H. Self-similarity of complex networks. Nature 433, 392–395 (2005). https://doi.org/10.1038/nature03248

Download citation

Further reading

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing