Waclaw B, Bogacz L, Janke W
Institut für Theoretische Physik, Universität Leipzig, Postfach 100920, 04009 Leipzig, Germany.
Phys Rev E Stat Nonlin Soft Matter Phys. 2008 Dec;78(6 Pt 1):061125. doi: 10.1103/PhysRevE.78.061125. Epub 2008 Dec 23.
We discuss how various models of scale-free complex networks approach their limiting properties when the size N of the network grows. We focus mainly on equilibrated networks and their finite-size degree distributions. Our results show that the position of the cutoff in the degree distribution, k_{cutoff} , scales with N in a different way than predicted for N-->infinity ; that is, subleading corrections to the scaling k_{cutoff} approximately N;{alpha} are strong even for networks of order N approximately 10;{9} nodes. We observe also a logarithmic correction to the scaling for degenerated graphs with the degree distribution pi(k) approximately k;{-3} . On the other hand, the distribution of the maximal degree k_{max} may have a different scaling than the cutoff and, moreover, it approaches the thermodynamic limit much faster. We argue that k_{max} approximately N;{alpha;{'}} with an exponent alpha;{'}=min[alpha,1(gamma-1)] , where gamma is the exponent in the power law pi(k) approximately k;{-gamma} . We also present some results on the cutoff function and the distribution of the maximal degree in equilibrated networks.
我们讨论了无标度复杂网络的各种模型在网络规模(N)增长时如何趋近其极限性质。我们主要关注平衡网络及其有限规模度分布。我们的结果表明，度分布中的截止位置(k_{cutoff})与(N)的标度关系与(N\to\infty)时的预测不同；也就是说，即使对于规模约为(N\approx10^{9})个节点的网络，对标度(k_{cutoff}\approx N^{\alpha})的次主导修正也很强。我们还观察到对于度分布(\pi(k)\approx k^{-3})的退化图，标度存在对数修正。另一方面，最大度(k_{max})的分布可能具有与截止不同的标度，而且，它更快地趋近于热力学极限。我们认为(k_{max}\approx N^{\alpha'})，其中指数(\alpha'=\min[\alpha,1/(\gamma - 1)])，这里(\gamma)是幂律(\pi(k)\approx k^{-\gamma})中的指数。我们还给出了关于平衡网络中截止函数和最大度分布的一些结果。
