First-order transition in small-world networks

• Published in: Europhysics letters 2000-06, Vol.50 (5), p.574-579
• Main Authors: Menezes, M. Argollo de, Moukarzel, C. F, Penna, T. J. P
• Format: Article
• Language: English
• Subjects: 05.10.-a; 05.40.-a; 64.60.-i; Computational methods in statistical physics and nonlinear dynamics; Condensed matter: structure, mechanical and thermal properties; Equations of state, phase equilibria, and phase transitions; Exact sciences and technology; Fluctuation phenomena, random processes, noise, and brownian motion; General studies of phase transitions; Physics; Statistical physics, thermodynamics, and nonlinear dynamical systems
• ISSN: 0295-5075; 1286-4854
• DOI: 10.1209/epl/i2000-00308-1

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.