Wu Zhenhua, Lagorio Cecilia, Braunstein Lidia A, Cohen Reuven, Havlin Shlomo, Stanley H Eugene
Center for Polymer Studies, Boston University, Boston, Massachusetts 02215, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2007 Jun;75(6 Pt 2):066110. doi: 10.1103/PhysRevE.75.066110. Epub 2007 Jun 27.
We propose numerical methods to evaluate the upper critical dimension d(c) of random percolation clusters in Erdös-Rényi networks and in scale-free networks with degree distribution P(k) approximately k(-lambda), where k is the degree of a node and lambda is the broadness of the degree distribution. Our results support the theoretical prediction, d(c) = 2(lambda - 1)(lambda - 3) for scale-free networks with 3 < lambda < 4 and d(c) = 6 for Erdös-Rényi networks and scale-free networks with lambda > 4 . When the removal of nodes is not random but targeted on removing the highest degree nodes we obtain d(c) = 6 for all lambda > 2 . Our method also yields a better numerical evaluation of the critical percolation threshold p(c) for scale-free networks. Our results suggest that the finite size effects increases when lambda approaches 3 from above.
我们提出了数值方法，以评估厄多斯-雷尼网络和度分布为(P(k)\approx k^{-\lambda})的无标度网络中随机渗流簇的上临界维度(d(c))，其中(k)是节点的度，(\lambda)是度分布的广度。我们的结果支持理论预测，即对于(3\lt\lambda\lt4)的无标度网络，(d(c)=2(\lambda - 1)(\lambda - 3))；对于厄多斯-雷尼网络和(\lambda\gt4)的无标度网络，(d(c)=6)。当节点的移除不是随机的，而是针对移除最高度节点时，对于所有(\lambda\gt2)，我们得到(d(c)=6)。我们的方法还能对无标度网络的临界渗流阈值(p(c))进行更好的数值评估。我们的结果表明，当(\lambda)从上方趋近于(3)时，有限尺寸效应会增加。
