Journal
EUROPHYSICS LETTERS
Volume 50, Issue 5, Pages 574-579Publisher
E D P SCIENCES
DOI: 10.1209/epl/i2000-00308-1
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The small-world transition is a first-order transition at zero density p of shortcuts, whereby the normalised shortest-path distance L = (l) over bar/L undergoes a discontinuity in the thermodynamic limit. On finite systems the apparent transition is shifted by Delta p similar to L-d Equivalently a persistence size L* similar to p(-1/d) can be defined in connection with finite-size effects. Assuming L* similar to 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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