First-order transition in small-world networks

C. F. Moukarzel; T. J. P. Penna

doi:10.1209/epl/i2000-00308-1

All issues

Volume 50 / No 5 (June I 2000)

Europhys. Lett., 50 5 (2000) 574-579

Abstract

Issue		Europhys. Lett. Volume 50, Number 5, June I 2000


Page(s)		574 - 579
Section		General
DOI		https://doi.org/10.1209/epl/i2000-00308-1
Published online		01 September 2002

Europhys. Lett., 50 (5), pp. 574-579 (2000)

First-order transition in small-world networks

M. Argollo de Menezes, C. F. Moukarzel and T. J. P. Penna

Instituto de Física, Universidade Federal Fluminense CEP 24210-340, Niterói, RJ, Brazil

Received: 7 December 1999
Accepted: 25 March 2000

Abstract

The small-world transition is a first-order transition at zero density p of shortcuts, whereby the normalised shortest-path distance $\mathcal{L} = \overline{l}/L$ undergoes a discontinuity in the thermodynamic limit. On finite systems the apparent transition is shifted by $\Delta p \sim L^{-d}$ . Equivalently a “persistence size” $L^* \sim p^{-1/d}$ can be defined in connection with finite-size effects. Assuming $L^* \sim p^{-\tau}$ , simple rescaling arguments imply that $\tau=1/d$ . We confirm this result by extensive numerical simulation in one to four dimensions, and argue that $\tau=1/d$ implies that this transition is first order.

PACS: 05.10.-a – Computational methods in statistical physics and nonlinear dynamics / 05.40.-a – Fluctuation phenomena, random processes, noise, and Brownian motion / 64.60.-i – General studies of phase transitions

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