Letter | Published:

Collective dynamics of ‘small-world’ networks

Nature volume 393, pages 440442 (04 June 1998) | Download Citation

Subjects

Abstract

Networks of coupled dynamical systems have been used to model biological oscillators1,2,3,4, Josephson junction arrays5,6, excitable media7, neural networks8,9,10, spatial games11, genetic control networks12 and many other self-organizing systems. Ordinarily, the connection topology is assumed to be either completely regular or completely random. But many biological, technological and social networks lie somewhere between these two extremes. Here we explore simple models of networks that can be tuned through this middle ground: regular networks ‘rewired’ to introduce increasing amounts of disorder. We find that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs. We call them ‘small-world’ networks, by analogy with the small-world phenomenon13,14 (popularly known as six degrees of separation15). The neural network of the worm Caenorhabditis elegans, the power grid of the western United States, and the collaboration graph of film actors are shown to be small-world networks. Models of dynamical systems with small-world coupling display enhanced signal-propagation speed, computational power, and synchronizability. In particular, infectious diseases spread more easily in small-world networks than in regular lattices.

Access optionsAccess options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

References

  1. 1.

    The Geometry of Biological Time(Springer, New York, 1980).

  2. 2.

    Chemical Oscillations, Waves, and Turbulence(Springer, Berlin, 1984).

  3. 3.

    & Coupled oscillators and biological synchronization. Sci. Am. 269(6), 102–109 (1993).

  4. 4.

    , & Dynamics of a ring of pulse-coupled oscillators: a group theoretic approach. Phys. Rev. Lett. 79, 2791–2794 (1997).

  5. 5.

    , & Taming spatiotemporal chaos with disorder. Nature 378, 465–467 (1995).

  6. 6.

    New results on frequency-locking dynamics of disordered Josephson arrays. Physica B 222, 315–319 (1996).

  7. 7.

    , & Acellular automaton model of excitable media including curvature and dispersion. Science 247, 1563–1566 (1990).

  8. 8.

    , & Stochastic resonance without tuning. Nature 376, 236–238 (1995).

  9. 9.

    & Rapid local synchronization of action potentials: Toward computation with coupled integrate-and-fire neurons. Proc. Natl Acad. Sci. USA 92, 6655–6662 (1995).

  10. 10.

    & Asynchronous states in neural networks of pulse-coupled oscillators. Phys. Rev. E 48(2), 1483–1490 (1993).

  11. 11.

    & Evolutionary games and spatial chaos. Nature 359, 826–829 (1992).

  12. 12.

    Metabolic stability and epigenesis in randomly constructed genetic nets. J. Theor. Biol. 22, 437–467 (1969).

  13. 13.

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

  14. 14.

    (ed.) The Small World(Ablex, Norwood, NJ, 1989).

  15. 15.

    Six Degrees of Separation: A Play(Vintage Books, New York, 1990).

  16. 16.

    Random Graphs(Academic, London, 1985).

  17. 17.

    & AY's Neuroanatomy of C. elegans for Computation(CRC Press, Boca Raton, FL, 1992).

  18. 18.

    & Social Network Analysis: Methods and Applications(Cambridge Univ. Press, 1994).

  19. 19.

    & Computer Relaying for Power Systems(Wiley, New York, 1988).

  20. 20.

    & The spread and persistence of infectious diseases in structured populations. Math. Biosci. 90, 341–366 (1988).

  21. 21.

    Amathematical model for predicting the geographic spread of new infectious agents. Math. Biosci. 90, 367–383 (1988).

  22. 22.

    Disease in metapopulation models: implications for conservation. Ecology 77, 1617–1632 (1996).

  23. 23.

    , & Toward a unified theory of sexual mixing and pair formation. Math. Biosci. 107, 379–405 (1991).

  24. 24.

    & Measures of concurrency in networks and the spread of infectious disease. Math. Biosci. 133, 165–195 (1996).

  25. 25.

    , & in Parallel Problem Solving from Nature(eds Davido, Y., Schwefel, H.-P. & Männer, R.) 344–353 (Lecture Notes in Computer Science 866, Springer, Berlin, 1994).

  26. 26.

    The Evolution of Cooperation(Basic Books, New York, 1984).

  27. 27.

    , , & Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties. Nature 338, 334–337 (1989).

Download references

Acknowledgements

We thank B. Tjaden for providing the film actor data, and J. Thorp and K. Bae for the Western States Power Grid data. This work was supported by the US National Science Foundation (Division of Mathematical Sciences).

Author information

Author notes

    • Duncan J. Watts

    Present address: Paul F. Lazarsfeld Center for the Social Sciences, Columbia University, 812 SIPA Building, 420 W118 St, New York, New York 10027, USA.

Affiliations

  1. Department of Theoretical and Applied Mechanics, KimballHall, Cornell University, Ithaca, New York 14853, USA

    • Steven H. Strogatz

Authors

  1. Search for Duncan J. Watts in:

  2. Search for Steven H. Strogatz in:

Corresponding author

Correspondence to Duncan J. Watts.

About this article

Publication history

Received

Accepted

Published

DOI

https://doi.org/10.1038/30918

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.