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arXiv · 0706.3491

Exact analytical solution of average path length for Apollonian networks

Zhongzhi Zhang·Lichao Chen·Shuigeng Zhou·Lujun Fang·Jihong Guan·Tao Zou
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Abstract

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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Zhongzhi Zhang, Lichao Chen, Shuigeng Zhou, Lujun Fang, Jihong Guan, Tao Zou. 2008-01-31. Exact analytical solution of average path length for Apollonian networks. https://doi.org/10.1103/physreve.77.017102

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