Cohen Reuven, Dawid Daryush Jonathan, Kardar Mehran, Bar-Yam Yaneer
Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel.
Phys Rev E Stat Nonlin Soft Matter Phys. 2009 Jun;79(6 Pt 2):066112. doi: 10.1103/PhysRevE.79.066112. Epub 2009 Jun 23.
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."
我们给出了在一维基础晶格上定义的广义类Watts-Strogatz图中渗流的精确解。我们发现在系统中长程键数量趋于零的极限情况下存在一个非经典临界点，渗流概率存在不连续性，平均有限簇尺寸发散。我们表明，根据占据的长程键与未占据的最近邻键的比例，临界行为分为三种情况之一，每种情况由不同的临界指数表征。这三种情况可以通过临界点附近的单个标度函数统一起来。这些结果可用于确定在通信或运输链中确保连通性所需的长程链接数量。例如，我们可以解决“传话”游戏中的通信问题。
