On the properties of small-world network models

  • A. Barrat
  • M. Weigt

DOI: 10.1007/s100510050067

Cite this article as:
Barrat, A. & Weigt, M. Eur. Phys. J. B (2000) 13: 547. doi:10.1007/s100510050067


We study the small-world networks recently introduced by Watts and Strogatz [Nature 393, 440 (1998)], using analytical as well as numerical tools. We characterize the geometrical properties resulting from the coexistence of a local structure and random long-range connections, and we examine their evolution with size and disorder strength. We show that any finite value of the disorder is able to trigger a “small-world” behaviour as soon as the initial lattice is big enough, and study the crossover between a regular lattice and a “small-world” one. These results are corroborated by the investigation of an Ising model defined on the network, showing for every finite disorder fraction a crossover from a high-temperature region dominated by the underlying one-dimensional structure to a mean-field like low-temperature region. In particular there exists a finite-temperature ferromagnetic phase transition as soon as the disorder strength is finite. [0.5cm]

PACS. 05.50.+q Lattice theory and statistics (Ising, Potts, etc.) - 64.60.Cn Order-disorder transformations; statistical mechanics of model systems - 05.70.Fh Phase transitions: general studies

Copyright information

© EDP Sciences, Springer-Verlag, Società Italiana di Fisica 2000

Authors and Affiliations

  • A. Barrat
    • 1
  • M. Weigt
    • 2
  1. 1.Laboratoire de Physique ThéoriqueUniversité Paris-SudOrsay CedexFrance
  2. 2.CNRS-Laboratoire de Physique Théorique de l'E.N.S.Paris Cedex 05France