Renormalization group analysis of the small-world network model. (English) Zbl 0940.82029

Summary: The authors study the small-world network model, which mimics the transition between regular-lattice and random-lattice behavior in social networks of increasing size. They contend that the model displays a critical point with a divergent characteristic length as the degree of randomness tends to zero. They also propose a real-space renormalization group transformation for the model and demonstrate that the transformation is exact in the limit of large system size. They use this result to calculate the exact value of the single critical exponent for the system, and to derive the scaling form for the average number of ‘degrees of separation’ between two nodes on the network as a function of the three independent variables. They confirm their results by extensive numerical simulation.


82B31 Stochastic methods applied to problems in equilibrium statistical mechanics
91D30 Social networks; opinion dynamics
82B28 Renormalization group methods in equilibrium statistical mechanics
Full Text: DOI arXiv


[1] Watts, D.J.; Strogatz, S.H., Nature, 393, 440, (1998)
[2] Milgram, S., Psychol. today, 2, 60, (1967)
[3] L. Adamic, Xerox PARC working paper http://www.parc.xerox.com/istl/groups/iea/www/SmallWorld.html, 1999.
[4] D.J. Watts, Small Worlds: The Dynamics of Networks Between Order and Randomness, Princeton University Press, Princeton, 1999. · Zbl 1046.00006
[5] Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer, Berlin, 1984. · Zbl 0558.76051
[6] R.V. Kulkarni, E. Almaas, D. Stroud, cond-mat/9905066.
[7] R. Monasson, submitted to Eur. Phys. J. B. Also cond-mat/9903347.
[8] M. Argollo de Menezes, C.F. Moukarzel, T.J.P. Penna, First-order transition in small-world networks, submitted to Phys. Rev. Lett. Also cond-mat/9903426. · Zbl 0978.82034
[9] Barthélémy, M.; Amaral, L.A.N., Phys. rev. lett., 82, 3180, (1999)
[10] A. Barrat, submitted to Phys. Rev. Lett. Also cond-mat/9903323.
[11] M. Barthélémy, L.A.N. Amaral, Phys. Rev. Lett. 82 (1999) 5180.
[12] B. Bollobás, Random Graphs, Academic Press, New York, 1985.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.