摘要翻译：
本文提出了一种计算ERD\H{o}S-R\'{e}nyi网络和具有度分布的无标度网络中随机逾渗团簇的上临界维数$D_c$的数值方法，其中$k$是节点的度，$\lambda$是度分布的宽度。我们的结果报告了理论预测，对于$3<\lambda<4$的无标度网络，$d_c=2(\lambda-1)/(\lambda-3)$；对于ERD\h{o}s-r\'{e}nyi网络和$\lambda>4$的无标度网络，$d_c=6$。当节点的移除不是随机的，而是以移除最高级别的节点为目标时，对于所有$\lambda>2$的节点，我们获得$D_C=6$。我们的方法也给出了无标度网络的临界渗流阈值$P_c$的较好的数值计算结果。我们的结果表明，当$\lambda$从上面接近3时，有限尺寸效应增加。
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英文标题：
《Numerical evaluation of the upper critical dimension of percolation in
  scale-free networks》
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作者：
Zhenhua Wu, Cecilia Lagorio, Lidia A. Braunstein, Reuven Cohen, Shlomo
  Havlin, H. Eugene Stanley
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最新提交年份：
2007
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分类信息：

一级分类：Physics        物理学
二级分类：Disordered Systems and Neural Networks        无序系统与神经网络
分类描述：Glasses and spin glasses; properties of random, aperiodic and quasiperiodic systems; transport in disordered media; localization; phenomena mediated by defects and disorder; neural networks
眼镜和旋转眼镜；随机、非周期和准周期系统的性质；无序介质中的传输；本地化；由缺陷和无序介导的现象；神经网络
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一级分类：Physics        物理学
二级分类：Statistical Mechanics        统计力学
分类描述：Phase transitions, thermodynamics, field theory, non-equilibrium phenomena, renormalization group and scaling, integrable models, turbulence
相变，热力学，场论，非平衡现象，重整化群和标度，可积模型，湍流
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英文摘要：
  We propose a numerical method to evaluate the upper critical dimension $d_c$ of random percolation clusters in Erd\H{o}s-R\'{e}nyi networks and in scale-free networks with degree distribution ${\cal P}(k) \sim k^{-\lambda}$, where $k$ is the degree of a node and $\lambda$ is the broadness of the degree distribution. Our results report the theoretical prediction, $d_c = 2(\lambda - 1)/(\lambda - 3)$ for scale-free networks with $3 < \lambda < 4$ and $d_c = 6$ for Erd\H{o}s-R\'{e}nyi 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.
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PDF链接：
https://arxiv.org/pdf/705.1547