Exact analytical solution of average path length for Apollonian networks

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

Published 2007-06-24, updated 2008-01-31Version 2

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.