Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Letter
  • Published:

Self-similarity of complex networks

Nature volume 433, pages 392–395 (2005)Cite this article

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.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

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.

Similar content being viewed by others

References

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

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Book  Google Scholar 

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

    Book  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    MATH  Google Scholar 

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

    Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    Book  Google Scholar 

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

    Book  Google Scholar 

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

    Book  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  CAS  Google Scholar 

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

  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)

    Article  ADS  CAS  Google Scholar 

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

    Article  CAS  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

Authors and Affiliations

  1. Levich Institute and Physics Department, City College of New York, New York, 10031, New York, USA

    Chaoming Song & Hernán A. Makse

  2. Minerva Center and Department of Physics, Bar-Ilan University, Ramat Gan, 52900, Israel

    Shlomo Havlin

Authors
  1. Chaoming Song
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shlomo Havlin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hernán A. Makse
    View author publications

    You can also search for this author in PubMed Google Scholar

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

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nature03248

This article is cited by

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

Advanced 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