Eur. Phys. J. B 13, 547-560 (2000)
https://doi.org/10.1007/s100510050067

On the properties of small-world network models

¹ Laboratoire de Physique Théorique (UMR 8627) , bâtiment 210, Université Paris-Sud, 91405 Orsay Cedex, France
² CNRS-Laboratoire de Physique Théorique de l'E.N.S., 24 rue Lhomond, 75231 Paris Cedex 05, France

Received: 29 March 1999
Revised: 21 May 1999
Published online: 15 February 2000

Abstract

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.