First-order transition in small-world networks

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.