title: Unusual percolation in simple small-world networks.
abstract: We present an exact solution of percolation in a generalized class of Watts-Strogatz graphs defined on a one-dimensional underlying lattice. We find a nonclassical critical point in the limit of the number of long-range bonds in the system going to zero, with a discontinuity in the percolation probability and a divergence in the mean finite-cluster size. We show that the critical behavior falls into one of three regimes depending on the proportion of occupied long-range to unoccupied nearest-neighbor bonds, with each regime being characterized by different critical exponents. The three regimes can be united by a single scaling function around the critical point. These results can be used to identify the number of long-range links necessary to secure connectivity in a communication or transportation chain. As an example, we can resolve the communication problem in a game of "telephone."
authors: Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel.
authors: Reuven
authors: R
authors: Cohen
authors: Daryush Jonathan
authors: DJ
authors: Dawid
authors: Mehran
authors: M
authors: Kardar
authors: Yaneer
authors: Y
authors: Bar-Yam