摘要翻译：
给出了Apollonian网络平均路径长度的精确公式。借助于由自相似结构导出的递推关系，我们得到了Apollonian网络平均路径长度$\bar{d}_t$的精确解。与著名的数值结果$\bar{d}_t\propto(\ln N_t)^{3/4}$[phys.rev.lett.\textbf{94}，018702(2005)]相比，我们的严格解表明，在网络大小$N_t$的无限极限下，平均路径长度按$\bar{d}_t\propto\ln N_t$对数增长。大量的数值计算与我们的封闭解完全一致。
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英文标题：
《Exact analytical solution of average path length for Apollonian networks》
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作者：
Zhongzhi Zhang, Lichao Chen, Shuigeng Zhou, Lujun Fang, Jihong Guan,
  Tao Zou
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最新提交年份：
2008
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分类信息：

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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一级分类：Physics        物理学
二级分类：Physics and Society        物理学与社会
分类描述：Structure, dynamics and collective behavior of societies and groups (human or otherwise). Quantitative analysis of social networks and other complex networks. Physics and engineering of infrastructure and systems of broad societal impact (e.g., energy grids, transportation networks).
社会和团体（人类或其他）的结构、动态和集体行为。社会网络和其他复杂网络的定量分析。具有广泛社会影响的基础设施和系统（如能源网、运输网络）的物理和工程。
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英文摘要：
  The exact formula for the average path length of Apollonian networks is found. With the help of recursion relations derived from the self-similar structure, we obtain the exact solution of average path length, $\bar{d}_t$, for Apollonian networks. In contrast to the well-known numerical result $\bar{d}_t \propto (\ln N_t)^{3/4}$ [Phys. Rev. Lett. \textbf{94}, 018702 (2005)], our rigorous solution shows that the average path length grows logarithmically as $\bar{d}_t \propto \ln N_t$ in the infinite limit of network size $N_t$. The extensive numerical calculations completely agree with our closed-form solution.
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PDF链接：
https://arxiv.org/pdf/706.3491