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Geometric phase-transition on systems with sparse long-range connections

Author

Listed:
  • Argollo de Menezes, M
  • Moukarzel, C.F
  • Penna, T.J.P

Abstract

Small-world networks are regular structures with a fraction p of regular connections per site replaced by totally random ones (“shortcuts”). This kind of structure seems to be present on networks arising in nature and technology. In this work we show that the small-world transition is a first-order transition at zero density p of shortcuts, whereby the normalized shortest-path distance L=ℓ̄/L undergoes a discontinuity in the thermodynamic limit. On finite systems the apparent transition is shifted by Δp∼L−d. Equivalently a “persistence size” L∗∼p−1/d can be defined in connection with finite-size effects. Assuming L∗∼p−τ, simple rescaling arguments imply that τ=1/d. We confirm this result by extensive numerical simulation in one to four dimensions, and argue that τ=1/d implies that this transition is first-order.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:phsmap:v:295:y:2001:i:1:p:132-139
    DOI: 10.1016/S0378-4371(01)00065-6
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    References listed on IDEAS

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    1. M. E. J. Newman & D. J. Watts, 1999. "Renormalization Group Analysis of the Small-World Network Model," Working Papers 99-04-029, Santa Fe Institute.
    2. A. Barrat & M. Weigt, 2000. "On the properties of small-world network models," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 13(3), pages 547-560, February.
    3. 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.
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    Cited by:

    1. Moukarzel, Cristian F., 2005. "Effective dimensions in networks with long-range connections," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 356(1), pages 157-161.

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