# On the properties of small-world network models

Abstract : We study the small-world networks recently introduced by Watts and Strogatz [Nature {\bf 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.
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Journal articles

https://hal-insu.archives-ouvertes.fr/insu-03726416
Contributor : Martin Weigt Connect in order to contact the contributor
Submitted on : Monday, July 18, 2022 - 3:00:34 PM
Last modification on : Friday, August 5, 2022 - 11:55:19 AM

### Citation

A. Barrat, M. Weigt. On the properties of small-world network models. The European Physical Journal B, 2000, 13, pp.547-560. ⟨insu-03726416⟩

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