DOI: 10.1038/30918. Collective dynamics of ‘small-world’ networks

Collective dynamics of ‘small-world’ networks

10.1038/30918

Crossref journal-article

Springer Science and Business Media LLC

Nature (297)

Bibliography

Watts, D. J., & Strogatz, S. H. (1998). Collective dynamics of âsmall-worldâ networks. Nature, 393(6684), 440â442.

Authors 2

Duncan J. Watts (first)
Steven H. Strogatz (additional)

References 27 Referenced 32,182

Winfree, A. T. The Geometry of Biological Time(Springer, New York, 1980). (10.1007/978-3-662-22492-2) / The Geometry of Biological Time by AT Winfree (1980)
Kuramoto, Y. Chemical Oscillations, Waves, and Turbulence(Springer, Berlin, 1984). (10.1007/978-3-642-69689-3) / Chemical Oscillations, Waves, and Turbulence by Y Kuramoto (1984)
Strogatz, S. H. & Stewart, I. Coupled oscillators and biological synchronization. Sci. Am. 269(6), 102–109 (1993). (10.1038/scientificamerican1293-102) / Sci. Am. by SH Strogatz (1993)
Bressloff, P. C., Coombes, S. & De Souza, B. Dynamics of a ring of pulse-coupled oscillators: a group theoretic approach. Phys. Rev. Lett. 79, 2791–2794 (1997). (10.1103/PhysRevLett.79.2791) / Phys. Rev. Lett. by PC Bressloff (1997)
Braiman, Y., Lindner, J. F. & Ditto, W. L. Taming spatiotemporal chaos with disorder. Nature 378, 465–467 (1995). (10.1038/378465a0) / Nature by Y Braiman (1995)
Wiesenfeld, K. New results on frequency-locking dynamics of disordered Josephson arrays. Physica B 222, 315–319 (1996). (10.1016/0921-4526(96)85057-5) / Physica B by K Wiesenfeld (1996)
Gerhardt, M., Schuster, H. & Tyson, J. J. Acellular automaton model of excitable media including curvature and dispersion. Science 247, 1563–1566 (1990). (10.1126/science.2321017) / Science by M Gerhardt (1990)
Collins, J. J., Chow, C. C. & Imhoff, T. T. Stochastic resonance without tuning. Nature 376, 236–238 (1995). (10.1038/376236a0) / Nature by JJ Collins (1995)
Hopfield, J. J. & Herz, A. V. M. Rapid local synchronization of action potentials: Toward computation with coupled integrate-and-fire neurons. Proc. Natl Acad. Sci. USA 92, 6655–6662 (1995). (10.1073/pnas.92.15.6655) / Proc. Natl Acad. Sci. USA by JJ Hopfield (1995)
Abbott, L. F. & van Vreeswijk, C. Asynchronous states in neural networks of pulse-coupled oscillators. Phys. Rev. E 48(2), 1483–1490 (1993). (10.1103/PhysRevE.48.1483) / Phys. Rev. E by LF Abbott (1993)
Nowak, M. A. & May, R. M. Evolutionary games and spatial chaos. Nature 359, 826–829 (1992). (10.1038/359826a0) / Nature by MA Nowak (1992)
Kauffman, S. A. Metabolic stability and epigenesis in randomly constructed genetic nets. J. Theor. Biol. 22, 437–467 (1969). (10.1016/0022-5193(69)90015-0) / J. Theor. Biol. by SA Kauffman (1969)
Milgram, S. The small world problem. Psychol. Today 2, 60–67 (1967). / Psychol. Today by S Milgram (1967)
Kochen, M. (ed.) The Small World(Ablex, Norwood, NJ, 1989). / The Small World by M Kochen (1989)
Guare, J. Six Degrees of Separation: A Play(Vintage Books, New York, 1990). / Six Degrees of Separation: A Play by J Guare (1990)
Bollabás, B. Random Graphs(Academic, London, 1985). / Random Graphs by B Bollabás (1985)
Achacoso, T. B. & Yamamoto, W. S. AY's Neuroanatomy of C. elegans for Computation(CRC Press, Boca Raton, FL, 1992). / AY's Neuroanatomy of C. elegans for Computation by TB Achacoso (1992)
Wasserman, S. & Faust, K. Social Network Analysis: Methods and Applications(Cambridge Univ. Press, 1994). (10.1017/CBO9780511815478) / Social Network Analysis: Methods and Applications by S Wasserman (1994)
Phadke, A. G. & Thorp, J. S. Computer Relaying for Power Systems(Wiley, New York, 1988). / Computer Relaying for Power Systems by AG Phadke (1988)
Sattenspiel, L. & Simon, C. P. The spread and persistence of infectious diseases in structured populations. Math. Biosci. 90, 341–366 (1988). (10.1016/0025-5564(88)90074-0) / Math. Biosci. by L Sattenspiel (1988)
Longini, I. M. J Amathematical model for predicting the geographic spread of new infectious agents. Math. Biosci. 90, 367–383 (1988). (10.1016/0025-5564(88)90075-2) / Math. Biosci. by IMJ Longini (1988)
Hess, G. Disease in metapopulation models: implications for conservation. Ecology 77, 1617–1632 (1996). (10.2307/2265556) / Ecology by G Hess (1996)
Blythe, S. P., Castillo-Chavez, C. & Palmer, J. S. Toward a unified theory of sexual mixing and pair formation. Math. Biosci. 107, 379–405 (1991). (10.1016/0025-5564(91)90015-B) / Math. Biosci. by SP Blythe (1991)
Kretschmar, M. & Morris, M. Measures of concurrency in networks and the spread of infectious disease. Math. Biosci. 133, 165–195 (1996). (10.1016/0025-5564(95)00093-3) / Math. Biosci. by M Kretschmar (1996)
Das, R., Mitchell, M. & Crutchfield, J. P. 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). (10.1007/3-540-58484-6_278) / Parallel Problem Solving from Nature by R Das (1994)
Axelrod, R. The Evolution of Cooperation(Basic Books, New York, 1984). / The Evolution of Cooperation by R Axelrod (1984)
Gray, C. M., König, P., Engel, A. K. & Singer, W. Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties. Nature 338, 334–337 (1989). (10.1038/338334a0) / Nature by CM Gray (1989)

Dates

Type	When
Created	23 years ago (July 26, 2002, 4:37 a.m.)
Deposited	2 years, 3 months ago (May 16, 2023, 9:21 p.m.)
Indexed	6 hours, 33 minutes ago (Aug. 22, 2025, 12:45 a.m.)
Issued	27 years, 2 months ago (June 1, 1998)
Published	27 years, 2 months ago (June 1, 1998)
Published Print	27 years, 2 months ago (June 1, 1998)

Funders 0

None

BibTeX

@article{Watts_1998, title={Collective dynamics of ‘small-world’ networks}, volume={393}, ISSN={1476-4687}, url={http://dx.doi.org/10.1038/30918}, DOI={10.1038/30918}, number={6684}, journal={Nature}, publisher={Springer Science and Business Media LLC}, author={Watts, Duncan J. and Strogatz, Steven H.}, year={1998}, month=jun, pages={440–442} }

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