Dorogovtsev S N, Mendes J F, Samukhin A N
Departamento de Física and Centro de Física do Porto, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto,
Phys Rev E Stat Nonlin Soft Matter Phys. 2001 Dec;64(6 Pt 2):066110. doi: 10.1103/PhysRevE.64.066110. Epub 2001 Nov 19.
We describe the anomalous phase transition of the emergence of the giant connected component in scale-free networks growing under mechanism of preferential linking. We obtain exact results for the size of the giant connected component and the distribution of vertices among connected components. We show that all the derivatives of the giant connected component size S over the rate b of the emergence of new edges are zero at the percolation threshold b(c), and S infinity exp[-d(gamma)(b-b(c))(-1/2)], where the coefficient d is a function of the degree distribution exponent gamma. In the entire phase without the giant component, these networks are in a "critical state." The probability P(k) that a vertex belongs to a connected component of a size k is of a power-law form. At the phase transition point, P(k) approximately 1/(k ln k)(2). In the phase with the giant component, P(k) has an exponential cutoff at k(c) approximately 1/S. In the simplest particular case, we present exact results for growing exponential networks.
我们描述了在优先连接机制下增长的无标度网络中巨型连通分量出现的异常相变。我们得到了巨型连通分量的大小以及连通分量中顶点分布的精确结果。我们表明，在渗流阈值(b(c))处，巨型连通分量大小(S)关于新边出现速率(b)的所有导数均为零，且(S)趋于(\exp[ - d(\gamma)(b - b(c))^{(-1/2)}])，其中系数(d)是度分布指数(\gamma)的函数。在没有巨型分量的整个阶段，这些网络处于“临界状态”。一个顶点属于大小为(k)的连通分量的概率(P(k))具有幂律形式。在相变点，(P(k))近似为(1/(k \ln k)^2)。在有巨型分量的阶段，(P(k))在(k(c))近似为(1/S)处有指数截断。在最简单的特殊情况下，我们给出了增长指数网络的精确结果。
