First-order transition in small-world networks

Details First-order transition in small-world networks First-order transition in small-world networks

1999-03-30

arxiv.org/abs/cond-mat/9903426v4

Europhys. Lett. vol.50, 5 (2000).

Physics

Condensed Matter

Disordered Systems and Neural Networks

4 pages, 3 figures, To appear in Europhysics Letters

Scientific paper

10.1209/epl/i2000-00308-1

The small-world transition is a first-order transition at zero density $p$ of shortcuts, whereby the normalized shortest-path distance 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.

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