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Scaling and Percolation in the Small-World Network Model

Author

Listed:
  • M. E. J. Newman
  • D. J. Watts

Abstract

In this paper we study the small-world network model of Watts and Strogatz, which mimics some aspects of the structure of networks of social interactions. We argue that there is one non-trivial length-scale in the model, analogous to the correlation length in other systems, which is well-defined in the limit of infinite system size and which diverges continuously as the randomness in the network tends to zero, giving a normal critical point in this limit. This length-scale governs the cross-over from large- to small-world behavior in the model, as well as the number of vertices in a neighborhood of given radius on the network. We derive the value of the single critical exponent controlling behavior in the critical region and the finite size scaling form for the average vertex-vertex distance on the network, and, using series expansion and Pade approximants, find an approximate analytic form for the scaling function. We calculate the effective dimension of small-world graphs and show that this dimension varies as a function of the length-scale on which it is measured, in a manner reminiscent of multifractals. We also study the problem of site percolation on small-world networks as a simple model of disease propagation, and derive an approximate expression for the percolation probability at which a giant component of connected vertices first forms (in epidemiological terms, the point at which an epidemic occurs). The typical cluster radius satisfies the expected finite size scaling form with a cluster size exponent close to that for a random graph. All our analytic results are confirmed by extensive numerical simulations of the model. Appears in Phys. Rev. E 60, 7332-7342 (1999).

Suggested Citation

  • M. E. J. Newman & D. J. Watts, 1999. "Scaling and Percolation in the Small-World Network Model," Working Papers 99-05-034, Santa Fe Institute.
  • Handle: RePEc:wop:safiwp:99-05-034
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    Citations

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    Cited by:

    1. Li, Chunguang, 2009. "Memorizing morph patterns in small-world neuronal network," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(2), pages 240-246.
    2. Uwe Cantner & Andreas Meder, 2006. "Die Wirkung von Forschungskooperationen auf den Unternehmenserfolg - eine Fallstudie zum Landkreis Saalfeld Rudolstadt," Jenaer Schriften zur Wirtschaftswissenschaft (Expired!) 24/2006, Friedrich-Schiller-Universität Jena, Wirtschaftswissenschaftliche Fakultät.
    3. Doménech, Antonio, 2009. "A topological phase transition between small-worlds and fractal scaling in urban railway transportation networks?," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(21), pages 4658-4668.
    4. Yang, Jianmei & Zhuang, Dong & Xie, Weicong & Chen, Guangrong, 2013. "A study of design approach of spreading schemes for viral marketing based on human dynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(24), pages 6494-6505.
    5. Zengwang Xu & Daniel Sui, 2007. "Small-world characteristics on transportation networks: a perspective from network autocorrelation," Journal of Geographical Systems, Springer, vol. 9(2), pages 189-205, June.
    6. Lu, Zhe-Ming & Guo, Shi-Ze, 2012. "A small-world network derived from the deterministic uniform recursive tree," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(1), pages 87-92.
    7. Alfarano, Simone & Milakovic, Mishael, 2009. "Network structure and N-dependence in agent-based herding models," Journal of Economic Dynamics and Control, Elsevier, vol. 33(1), pages 78-92, January.
    8. Floortje Alkemade & Carolina Castaldi, 2005. "Strategies for the Diffusion of Innovations on Social Networks," Computational Economics, Springer;Society for Computational Economics, vol. 25(1), pages 3-23, February.
    9. De Kamps, Marc & Ladley, Daniel & Simaitis, Aistis, 2014. "Heterogeneous beliefs in over-the-counter markets," Journal of Economic Dynamics and Control, Elsevier, vol. 41(C), pages 50-68.
    10. Huang, Wei & Chen, Shengyong & Wang, Wanliang, 2014. "Navigation in spatial networks: A survey," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 393(C), pages 132-154.
    11. Delre, S.A. & Jager, W. & Bijmolt, T.H.A. & Janssen, M.A., 2007. "Targeting and timing promotional activities: An agent-based model for the takeoff of new products," Journal of Business Research, Elsevier, vol. 60(8), pages 826-835, August.
    12. Grabowski, Franciszek, 2010. "Logistic equation of arbitrary order," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(16), pages 3081-3093.
    13. Peres, Renana, 2014. "The impact of network characteristics on the diffusion of innovations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 402(C), pages 330-343.
    14. Zhou, Guangye & Li, Chengren & Li, Tingting & Yang, Yi & Wang, Chen & He, Fangjun & Sun, Jingchang, 2016. "Outer synchronization investigation between WS and NW small-world networks with different node numbers," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 457(C), pages 506-513.
    15. Huo, Liang’an & Song, Naixiang, 2016. "Dynamical interplay between the dissemination of scientific knowledge and rumor spreading in emergency," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 461(C), pages 73-84.
    16. Argollo de Menezes, M & Moukarzel, C.F & Penna, T.J.P, 2001. "Geometric phase-transition on systems with sparse long-range connections," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 295(1), pages 132-139.
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    18. repec:eee:phsmap:v:482:y:2017:i:c:p:433-443 is not listed on IDEAS
    19. Cristopher Moore & M. E. J. Newman, 2000. "Exact Solution of Site and Bond Percolation on Small-World Networks," Working Papers 00-01-007, Santa Fe Institute.
    20. Lahtinen, Jani & Kertész, János & Kaski, Kimmo, 2005. "Sandpiles on Watts–Strogatz type small-worlds," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 349(3), pages 535-547.
    21. Zheng, Yan Hong & Lu, Qi Shao, 2008. "Spatiotemporal patterns and chaotic burst synchronization in a small-world neuronal network," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(14), pages 3719-3728.
    22. repec:spr:scient:v:102:y:2015:i:2:d:10.1007_s11192-014-1475-4 is not listed on IDEAS
    23. Zhong, Li-Xin & Xu, Wen-Juan & Chen, Rong-Da & Qiu, Tian & Shi, Yong-Dong & Zhong, Chen-Yang, 2015. "Coupled effects of local movement and global interaction on contagion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 436(C), pages 482-491.

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    Keywords

    Small world; social interaction; networks; graph theory; phase transitions; percolation;

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